Prescribed signal concentration on the boundary: eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation

Abstract

We consider chemotaxis-Navier–Stokes systems with logistic proliferation and signal consumption of the form

for parameter choices $\kappa \ge 0$ and $\mu >0$. Herein, we moreover impose a nonnegative and time-constant prescribed concentration $c_\star \in C^2({\overline{\Omega }})$ for the signal chemical on the boundary of the domain $\Omega \subset {\mathbb {R}}^{\mathcal {N}}$ with ${\mathcal {N}}\in \{2,3\}$. After first extending the previously known result on time-global existence of weak solutions for the Stokes variant to the full Navier–Stokes setting, we proceed with an investigation of eventual regularity properties in the slightly more restrictive setting of $c_\star$ being also constant in space. We show that sufficiently strong logistic influence, in the sense that for $\omega >0$ and $\mu _0>0$ there is some $\eta =\eta (\omega ,\mu _0,c_\star )>0$ with the property that whenever

$$\begin{aligned} \mu _0\le \mu \quad \text {and}\quad \frac{\kappa }{\min \{\mu ,\mu ^{\frac{{\mathcal {N}}+6}{6}+\omega }\}}<\eta \end{aligned}$$

are satisfied the global weak solution eventually becomes a smooth and classical solution with waiting time depending on $\omega ,\mu _0,\eta ,c_\star$ and the initial data.

1 Introduction

Self-organization and pattern formation in populations of simple bacteria, such as Bacillus subtilis, has been detected in multiple experiments [10, 14, 38] and, while the precise mathematical description of these cross-diffusive motility schemes, i.e. frameworks where a gradient in the concentration of one agent leads to a flux in another, and the corresponding dynamics has been a focal point of mathematical biology for almost half a century, some observations from the experiments can still not be recovered analytically in the solution behavior of the corresponding system of evolution equations; in particular, despite the rigorous efforts undertaken to reproduce the complex patterns ascertained by the experiments on the analytical level, much of the available results indicate that spatial homogenization should be the expected long-term behavior.

To shed more light on the mathematical specifications, let us come back to the prototypical example of populations of the aerobic Bacillus subtilis suspended in a drop of water. In [38] the formation of plume-like aggregation patterns was observed and a chemotaxis-Navier–Stokes model of the form

(1.1)

(with linear diffusion, i.e. $m=1$) was proposed to analytically describe the dynamics involved. Herein, the unknown functions n, c, u, P denote the bacterial density, the concentration of the signal chemical, i.e. oxygen, the fluid velocity field and the pressure of the fluid, respectively, while $\phi$ is a fixed potential function incorporating the influence of gravitational forces on the bacteria into the model and the smoothly bounded domain $\Omega \subset {\mathbb {R}}^{\mathcal {N}}$ represents the drop of water. For analytical studies of the corresponding initial-boundary value problem system (1.1) is usually considered under no-flux boundary conditions for n and c and no-slip boundary condition for u. In this setting the mathematically favorable structure of the boundary conditions can be used to derive a quasi-energy inequality, which provides sufficiently strong regularity information on the solution components and thereby allows for the construction of reasonably thorough solution theories. In fact, the two-dimensional setting has been covered quite exhaustively including long-term stabilization [19, 29, 45, 47, 54]. For space dimension ${\mathcal {N}}=3$ on the other hand, the existence of global weak solutions which eventually become smooth and stabilize has been established [49, 50] and the possibility for singularities to occur on small time-scales has been shown to arise only on sets of measure zero [52]. Similar results remain valid for frameworks with a more general diffusion term of porous medium type [11, 22, 48]. Even when logistic proliferation of the bacteria population (i.e. the addition $+\kappa n-\mu n^2$ on the right hand side of the first equation) is introduced to the system, the stabilization towards a spatially homogeneous steady state remains the prevalent facet of long-term behavior [27, 39], even though such precedents are known to provide rich nontrivial dynamical behavior in related chemotaxis frameworks [26, 30, 46]. See also the surveys [1, 21] for an expansive overview.

These prevailing results on convergence to trivial steady states appear to be mismatched with the experimental findings and while this could possibly be explained as a discrepancy between long-term and short-term dynamics (which for the systems in question are lacking in results) the derivation of the model in [38] was introduced with different boundary conditions than the ones usually found in current examinations. Indeed, in said introductory work it was considered to impose an inhomogeneous Dirichlet boundary condition for the chemical at the fluid-air interface, or to even augment the boundary condition to differentiate between fluid-air interface and fluid-solid-ground interface, while retaining the no-flux condition for n and the no-slip condition for u. The physical motivation behind the use of Dirichlet data was motivated by the assumption that a significant difference in the oxygen-diffusion coefficients between water and air should lead to an oxygen concentration equal to the saturation value on the boundary of the water drop.

Similar thoughts also give rise to settings where an influx of oxygen into the domain is provided by means of an explicit Robin boundary condition of the prototypical form $\frac{\partial c}{\partial \nu }=1-c$ on $\partial \Omega$, as introduced by the influential work [5]. The studies of (1.1) and related systems with more realistic boundary conditions are still quite fresh and the findings are rather sparse, mostly covering existence theory or qualitative results in adjusted, more favorable settings. In fact, classical solutions in 2D and global weak solutions in 3D to (1.1) have been established in [5] under the inclusion of logistic proliferation terms and in [7] without. Investigations including fluid interaction and nonlinear diffusion of the cells can be found in [37, 55], where global weak solutions to a variant of (1.1) with Stokes flow and nonlinear diffusion satisfying $m>\frac{7}{6}$ in 3D and global weak solutions for the full Navier–Stokes flow and any superlinear diffusion, i.e. $m>1$, in 2D were shown to exist, respectively. Additional considerations of Robin boundary conditions with relation to long-term behavior appear for a simpler fluid-free doubly elliptic version of (1.1) in [6], where for any solution of arbitrary mass $M:=\int _{\Omega }\!n>0$ a non-constant unique positive solution was obtained, in essence implying that the fluid-free version of (1.1) when considered with Robin boundary data for c features a non-trivial steady state. This is further underlined by the discoveries on the corresponding fluid-free parabolic-elliptic variant in [12], where it was verified that the solution of the doubly elliptic system indeed plays the role of the large-time limit of the solution to the parabolic-elliptic system. While these results are a fruitful and important step for further research with regard to non-trivial patterns, the mathematical difficulty arising with the Robin boundary condition present an obstacle, which currently limits progress for more detailed results on large-time behavior in the setting of (1.1).

Confined by the lack of appropriate tools the prescription of inhomogeneous Dirichlet boundary data for the signal, as already suggested in [38], has become another promising path to advance the insight on the underlying mechanics and improve the analytical instruments. One might hope that, despite the presence of inhomogeneous boundary data, the mathematical challenges are slightly less problematic than in the Robin case. Results in this direction currently, however, also mainly cover existence results and the 3D case is mostly restricted to works with Stokes fluid or more generalized solution concepts with possibly nonsmooth functions. In particular, global generalized solutions for ${\mathcal {N}}=3$ were constructed in [43] for the chemotaxis-Stokes system, while the companion work [42] features global weak solutions for ${\mathcal {N}}=2$ and $m\ge 1$ and also for ${\mathcal {N}}\ge 3$ under the assumption of $m>\frac{3{\mathcal {N}}-2}{2{\mathcal {N}}}$. Concerning the chemotaxis-Stokes system with logistic proliferation the existence of global weak solutions has been attained in the predecessor work [4]. In the Navier–Stokes case, the recent work [3] provides an existence result for global weak solutions under the assumption $m>\frac{7}{6}$ and, additionally, provides at least some long-term information in the sense that one can find a ball in a certain topology, whose radius mainly depends the boundary data and the total population size and which eventually absorbs each trajectory. Existence results with significantly better regularity straying away from the setting of no-flux/no-flux/no-slip conditions to date seem limited to the fluid-free version [23, 24, 28]. Let us also briefly mention some simulation based approaches in [8, 31].

Main results Encouraged by the recent findings on eventual regularity in a Keller–Segel–Navier–Stokes system (i.e. (1.1) with signal production instead of consumption) with proliferation in [53], where, in contrast to previous methods, the relaxation is enforced by sufficiently strong degradation instead of signal consumption, and our earlier existence result in [4], we will now consider the chemotaxis-Navier–Stokes system with logistic proliferation of the form

(1.2)

in a smoothly bounded domain $\Omega \subset {\mathbb {R}}^{\mathcal {N}}$ with ${\mathcal {N}}\in \{2,3\}$ and $\nu$ denoting the outward normal vector field on $\partial \Omega$. Aiming for global existence of weak solutions and some form of eventual regularization, we require $\kappa \ge 0$, $\mu >0$, that the time constant function $c_\star$ fulfills

$$\begin{aligned}&c_\star \in C^{2}\!\left( {\overline{\Omega }}\right) \quad \text {with}\quad c_\star \ge 0, \end{aligned}$$

(1.3)

that the gravitational potential function $\phi$ satisfies

$$\begin{aligned} \phi \in W^{2,\infty }(\Omega ) \end{aligned}$$

(1.4)

and that the initial data $n_0$, $c_0$, $u_0$ satisfy

(1.5)

with $q>{\mathcal {N}}$, $\varrho \in (\frac{{\mathcal {N}}}{4},1)$. Herein, $A:=-{\mathcal {P}} \Delta$ is the Stokes operator with its domain $D(A):=W^{2,2}\left( \Omega ; {\mathbb {R}}^{{\mathcal {N}}}\right) \cap W_{0}^{1,2}\left( \Omega ; {\mathbb {R}}^{{\mathcal {N}}}\right) \cap L_{\sigma }^{2}(\Omega )$ with $L_{\sigma }^{2}(\Omega ):=\left\{ \varphi \in L^{2\;\!}\!\left( \Omega ;{\mathbb {R}}^{\mathcal {N}}\right) \,\vert \,\nabla \cdot \varphi =0\right\}$ and ${\mathcal {P}}$ represents the Helmholtz projection of $L^{2\;\!}\!\left( \Omega ;{\mathbb {R}}^{{\mathcal {N}}}\right)$ onto $L_{\sigma }^{2}(\Omega ).$ In this setting we can formulate a quite natural general solution statement for global weak solutions.

Theorem 1.1

Assume ${\mathcal {N}}\in \{2,3\}$ and let $\Omega \subset {\mathbb {R}}^{\mathcal {N}}$ be a bounded domain with smooth boundary. Suppose that $\kappa \ge 0$, $\mu >0$ and that the functions $c_\star$ and $\phi$ comply with (1.3) and (1.4), respectively. Then, for any $n_0,c_0$ and $u_0$ agreeing with (1.5), the system (1.2) possesses at least one global weak solution (n, c, u) in the sense of Definition 4.2.

Confining ourselves to a more restrictive setting, where $c_\star$ is not only constant in time but also constant in space and additionally the ratio $\frac{\kappa }{\mu }$ of the logistic parameters satisfies the smallness condition specified in (1.6), we can furthermore ascertain that all solutions obtained by Theorem 1.1 adhere to an eventual regularization process. While this, in particular, entails that after some waiting time $T_0>0$ the weak solutions also solve the problem in the classical sense, we can, moreover, derive some qualitative large-time information. To be precise, we show the following:

Theorem 1.2

Assume ${\mathcal {N}}\in \{2,3\}$ and let $\Omega \subset {\mathbb {R}}^{\mathcal {N}}$ be a bounded domain with smooth boundary. Suppose that $c_\star \ge 0$ is constant and that $\phi$ satisfies (1.4). Then, for all $\omega >0$ and $\mu _0>0$ there exists $\eta =\eta (\omega ,\mu _0,c_\star )>0$ with the following property: Suppose that $\kappa \ge 0$ and $\mu \ge \mu _0$ are such that

$$\begin{aligned} \kappa <\eta \cdot \min \big \{\mu ,\,\mu ^{\frac{{\mathcal {N}}+6}{6}+\omega }\big \}, \end{aligned}$$

(1.6)

that $(n_0,c_0,u_0)$ complies with (1.5) and that (n, c, u) denotes the global weak solution of (1.2) provided by Theorem 1.1. Then there is $T_0=T_0(\omega ,\mu _0,\eta ,n_0,c_0,u_0,c_\star )>0$ such that

$$\begin{aligned} n\in C^{2,1}\!\left( {\overline{\Omega }}\times [T_0,\infty )\right) ,\quad c\in C^{2,1}\!\left( {\overline{\Omega }}\times [T_0,\infty )\right) ,\quad u\in C^{2,1}\!\left( {\overline{\Omega }}\times [T_0,\infty );{\mathbb {R}}^{\mathcal {N}}\right) , \end{aligned}$$

and such that with some $P\in C^{1,0}\!\left( {\overline{\Omega }}\times [T_0,\infty )\right)$, the quadruple (n, c, u, P) forms a classical solution of (1.2) in ${\overline{\Omega }}\times [T_0,\infty )$.

Outline. The main parts of the work can be summarized in the following way. In Sects. 2, 3 and 4 we focus our interest on the global existence of weak solution to (1.2). These solutions are constructed as limit objects from solutions to a family of suitably regularized problems. The derivations of the necessary a priori information therein are extensions of the methods employed in the predecessor work [4] to cover the nonlinear convection term in the fluid equation.

Sections 5 and 6 are then devoted to the proof of Theorem 1.2. In Sect. 5 we will make use of the strong degradation in the first equation to first obtain eventual smallness information on certain time-averaged quantities. Together with a functional of the form

$$\begin{aligned} \int _{\Omega }\!\psi \big (n_\varepsilon -\tfrac{\kappa }{\mu }\big )(\cdot ,t)+\int _{\Omega }\!|\nabla c_\varepsilon |^{2p} (\cdot ,t) \end{aligned}$$

with a suitably chosen function $\psi$, in detail introduced in (5.55), we can then transfer these relaxation properties also to

$$\begin{aligned} \int _{\Omega }\!n_\varepsilon ^p(\cdot ,t)+\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}(\cdot ,t) \end{aligned}$$

for some p slightly larger than $\frac{{\mathcal {N}}}{2}$, which is known to act, to some extent, as a critical exponent with respect to classical regularity. In Sect. 6 we then draw on standard regularity theory to finalize the proof of Theorem 1.2. The arguments employed in both of the Sects. 5 and 6 are based on insights provided by [53], where a Keller–Segel–Navier–Stokes system with Neumann boundary conditions for the chemical signal and signal production by the bacteria was considered. The presence of inhomogeneous boundary conditions, however, makes the setting more intricate and requires further technical advancements for multiple parts of the strategy.

2 Time-local existence in approximate problems and basic a priori estimates

The weak solutions of (1.2) which will be provided by Theorem 1.1 will be constructed as a limit object from solutions to suitably regularized systems. To this end, we fix a family $(\rho _\varepsilon )_{\varepsilon \in (0,1)}\subset C_0^\infty (\Omega )$ of smooth cut-off functions satisfying

$$\begin{aligned} 0\le \rho _\varepsilon (x)\le 1\quad \text {for all }x\in \Omega \quad \text {such that }\rho _\varepsilon \nearrow 1\text { as }\varepsilon \searrow 0, \end{aligned}$$

and introduce the corresponding family of approximating problems to (1.2) given by

with $f_\varepsilon (s):=\frac{1}{(1+\varepsilon s)^3}$ and $g_\varepsilon (s):=\frac{s}{1+\varepsilon s}$ for $s\ge 0$ and $Y_{\varepsilon }:=(1+\varepsilon A)^{-1}$ denoting the Yosida approximation of the identity with respect to the Stokes operator (see e.g. [34, Sec. II.3.4]). Taking into account the adaptations necessary to treat the convective term of (2.1c) with arguments presented in [45, Lemma 2.2], we can essentially follow the steps of [4, Lemma 3.1] to obtain the following time-local existence result for (1.2).

Lemma 2.1

Let ${\mathcal {N}}\in \{2,3\}$, $\Omega \subset {\mathbb {R}}^{\mathcal {N}}$ be a bounded domain with smooth boundary, $q>{\mathcal {N}}$, $\varrho \in (\frac{{\mathcal {N}}}{4},1)$, $\kappa \ge 0$ and $\mu >0$. Assume that $c_\star$ and $\phi$ satisfy (1.3) and (1.4), respectively, and that $n_0$, $c_0$ and $u_0$ comply with (1.5). Then for any $\varepsilon \in (0,1)$, there exist $T_{max,\;\!\varepsilon }\in (0,\infty ]$ and a uniquely determined triple $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of functions

$$\begin{aligned} n_\varepsilon&\in C^{0}\!\left( {\overline{\Omega }}\times [0,T_{max,\;\!\varepsilon })\right) \cap C^{2,1}\!\left( {\overline{\Omega }}\times (0,T_{max,\;\!\varepsilon })\right) ,\\ c_\varepsilon&\in C^{0}\!\left( {\overline{\Omega }}\times [0,T_{max,\;\!\varepsilon })\right) \cap C^{2,1}\!\left( {\overline{\Omega }}\times (0,T_{max,\;\!\varepsilon })\right) \cap L_{loc}^{\infty }\!\left( [0,T_{max,\;\!\varepsilon });W^{1,q}(\Omega )\right) ,\\ u_\varepsilon&\in C^{0}\!\left( {\overline{\Omega }}\times [0,T_{max,\;\!\varepsilon });{\mathbb {R}}^{\mathcal {N}}\right) \cap C^{2,1}\!\left( {\overline{\Omega }}\times (0,T_{max,\;\!\varepsilon });{\mathbb {R}}^{\mathcal {N}}\right) , \end{aligned}$$

which, together with some $P_\varepsilon \in C^{1,0}\!\left( {\overline{\Omega }}\times (0,T_{max,\;\!\varepsilon })\right)$, solve (2.1) in the classical sense and satisfy $n_\varepsilon >0$ and $c_\varepsilon \ge 0$ in ${\overline{\Omega }}\times [0,T_{max,\;\!\varepsilon })$. Moreover, either $T_{max,\;\!\varepsilon }=\infty$ or

$$\begin{aligned} \limsup _{t\nearrow T_{max,\;\!\varepsilon }}\left( \Vert n_\varepsilon (\cdot ,t)\Vert _{L^{\infty }(\Omega )}+\Vert c_\varepsilon (\cdot ,t)\Vert _{W^{1,q}(\Omega )}+\Vert A^\varrho u_\varepsilon (\cdot ,t)\Vert _{L^{2}(\Omega )}\right) =\infty . \end{aligned}$$

(2.2)

Henceforth, if not stated differently, we will always assume ${\mathcal {N}}\in \{2,3\}$, that $\Omega \subset {\mathbb {R}}^{\mathcal {N}}$ is a bounded domain with smooth boundary and that $\phi$ and $(n_0,c_0,u_0)$ are fixed and satisfy (1.4) and (1.5) for some $q>{\mathcal {N}}$, respectively. Accordingly, the choices for the generic constants appearing below may also, in an expected natural manner, rely on ${\mathcal {N}},\Omega$, $\phi$ and the initial data in addition to the dependencies specified in the statements. Moreover, we will always denote by $T_{max,\;\!\varepsilon }$ the maximal existence time provided by Lemma 2.1. Let us additionally introduce the divergence-free spaces $W_{0,\sigma }^{1,1}\big (\Omega ;{\mathbb {R}}^{\mathcal {N}}\big ):=W_0^{1,1}\big (\Omega ;{\mathbb {R}}^{\mathcal {N}}\big )\cap L_\sigma ^2(\Omega )$ and $C_{0,\sigma }^\infty \big (\Omega \times [0,\infty );{\mathbb {R}}^{\mathcal {N}}\big ):= \big \{\varphi \in C_0^\infty \big (\Omega \times [0,\infty );{\mathbb {R}}^{\mathcal {N}}\big )\,\vert \,\nabla \cdot \varphi =0\big \}$, which appear a few times during our investigations.

Before we strive after our first main objective, i.e. the confirmation of time-global solvability for each fixed $\varepsilon \in (0,1)$, let us briefly discuss the following basic set of a priori estimates for $n_\varepsilon$ and $c_\varepsilon$, which will lay the cornerstone for many estimations yet to come.

Lemma 2.2

Let $\kappa \ge 0$ and $\mu >0$. Assume $c_\star$ fulfills (1.3) and set

$$\begin{aligned} M_0:=\max \left\{ \int _{\Omega }\!n_0,\,\frac{\kappa |\Omega |}{\mu }\right\} \qquad \text {and}\qquad K_0:=\max \big \{\Vert c_\star \Vert _{L^{\infty \;\!}\!\left( \partial \Omega \right) },\,\Vert c_0\Vert _{L^{\infty }(\Omega )}\big \}. \end{aligned}$$

Then for any $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) fulfills

$$\begin{aligned} \int _{\Omega }\!n_\varepsilon (\cdot ,t)\le M_0,\qquad \text {and}\qquad \Vert c_\varepsilon (\cdot ,t)\Vert _{L^{\infty }(\Omega )}\le K_0\quad \text {for all }t\in (0,T_{max,\;\!\varepsilon }) \end{aligned}$$

and

$$\begin{aligned} \int _t^{t+\tau }\!\!\int _{\Omega }\!n_\varepsilon ^2\le \frac{(\kappa +1)M_0}{\mu }\quad \text {for all }t\in (0,T_{max,\;\!\varepsilon }-\tau ), \end{aligned}$$

where $\tau :=\min \{1,\frac{1}{2}T_{max,\;\!\varepsilon }\}$.

Proof

First, we integrate (2.1a) over $\Omega$ and derive from the divergence-free property of $u_\varepsilon$ and the boundary conditions that an integration by parts implies

$$\begin{aligned}&\frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!n_\varepsilon =\kappa \int _{\Omega }\!n_\varepsilon - \mu \int _{\Omega }\!n_\varepsilon ^2\quad \text {on }(0,T_{max,\;\!\varepsilon })\text { for all }\varepsilon \in (0,1), \end{aligned}$$

(2.3)

which when combined with the Cauchy–Schwarz inequality shows that

$$\begin{aligned}&\frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!n_\varepsilon \le \kappa \int _{\Omega }\!n_\varepsilon -\frac{\mu }{|\Omega |}\left( \int _{\Omega }\!n_\varepsilon \right) ^2\quad \text {on }(0,T_{max,\;\!\varepsilon })\text { for all }\varepsilon \in (0,1). \end{aligned}$$

Therefore, for all $\varepsilon \in (0,1)$ the function $y_\varepsilon (t):=\int _{\Omega }\!n_\varepsilon (\cdot ,t)$, $t\ge 0$ satisfies

$$\begin{aligned} y_\varepsilon '(t)\le \kappa y_\varepsilon (t)-\frac{\mu }{|\Omega |}y_\varepsilon ^2(t)\quad \text {on }(0,T_{max,\;\!\varepsilon }) \end{aligned}$$

and a comparison argument with $M_0:=\max \big \{\int _{\Omega }\!n_0,\frac{\kappa |\Omega |}{\mu }\big \}$ as upper solution entails the desired bound for $\int _{\Omega }\!n_\varepsilon (\cdot ,t)$ on $(0,T_{max,\;\!\varepsilon })$ for all $\varepsilon \in (0,1)$. Afterwards, we integrate (2.3) with respect to time to conclude that by nonnegativity of $n_\varepsilon$ throughout $\Omega \times (0,T_{max,\;\!\varepsilon })$ and $\tau \le 1$

$$\begin{aligned} \mu \int _t^{t+\tau }\!\!\int _{\Omega }\!n_\varepsilon ^2\le \int _{\Omega }\!n_\varepsilon (\cdot ,t)+\kappa \int _t^{t+\tau }\!\!\int _{\Omega }\!n_\varepsilon \le M_0+\kappa M_0\tau \le (\kappa +1)M_0 \end{aligned}$$

holds for all $t\in (0,T_{max,\;\!\varepsilon }-\tau )$ and all $\varepsilon \in (0,1)$, which obviously provides us with the asserted time-space bound for $n_\varepsilon ^2$. Finally, thanks to $g_\varepsilon (s)\ge 0$ for all $s\ge 0$ and once more the nonnegativity of $n_\varepsilon$, we immediately have

$$\begin{aligned} \Vert c_\varepsilon (\cdot ,t)\Vert _{L^{\infty }(\Omega )}\le \max \big \{\Vert c_\star \Vert _{L^{\infty \;\!}\!\left( \partial \Omega \right) },\Vert c_0\Vert _{L^{\infty }(\Omega )}\big \}=:K_0 \quad \text {for all } t\in (0,T_{max,\;\!\varepsilon })\text { and }\varepsilon \in (0,1) \end{aligned}$$

by the maximum principle, completing the proof. $\square$

3 The pursuit of appropriate compactness features

This section is dedicated to the derivation of sufficiently strong compactness properties, which, beyond the result on time-global existence for fixed $\varepsilon \in (0,1)$ in Lemma 3.4, will also be the base for the limit process undertaken in Sect. 4. Moreover, comparison result from Lemma 3.1 and the differential inequality in Lemma 3.2 will later also play an important role in our analysis on the eventual smallness of certain quantities for constant $c_\star$ (refer to Sect. 5).

3.1 Exploiting the quadratic degradation of $n_\varepsilon$

We prepare a well-known ODE comparison lemma allowing time-dependent right-hand sides, which are bounded when averaged over small intervals. A formulation close to the result presented here can be found in [51, Lemma 3.4] and only minor adjustments are necessary to adapt the proof given in the reference. (See also [53, Lemma 3.1].)

Lemma 3.1

Let $t_0\in {\mathbb {R}}$, $T\in (t_0,\infty ]$ and $a>0$. Suppose that $y\in C^{1}\!\left( (t_0,T)\right) \cap C^{0}\!\left( [t_0,T)\right)$, $h\in C^{0}\!\left( [0,T)\right)$, $h\ge 0$, $\tau \in (t_0,T-t_0)$, $C>0$ satisfy

$$\begin{aligned} y'(t)+ay(t)\le h(t)\quad \text {for all }\ t\in (t_0,T) \end{aligned}$$

and

$$\begin{aligned} \int _{t}^{t+\tau } h(s){{\,\mathrm{d\!}\,}}s\le C\quad \text {for all }\ t\in (t_0,T-\tau ). \end{aligned}$$

Then

$$\begin{aligned} y(t)\le e^{-a(t-t_0)}y(t_0)+\frac{C}{1-e^{-a}}\quad \text {for all }\ t\in [t_0,T) \end{aligned}$$

and in particular $y(t)\le y(t_0)+\frac{C}{1-e^{-a}}$ throughout $[t_0,T)$.

The previous lemma at hand, we can make use of the control over $\int _t^{t+\tau }\!\!\int _{\Omega }\!n_\varepsilon ^2$ provided by Lemma 2.2 to derive some $L^2$ information on $u_\varepsilon$, which additionally also entails a first time-space estimate for $\nabla u_\varepsilon$.

Lemma 3.2

There exist $R_1>0$ and $R_2>0$ such that for each $\kappa \ge 0$, $\mu >0$, $c_\star$ satisfying (1.3) and all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) fulfills

$$\begin{aligned} \frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!|u_\varepsilon (\cdot ,t)|^2+\frac{R_1}{2}\int _{\Omega }\!|u_\varepsilon (\cdot ,t)|^2+\frac{1}{2}\int _{\Omega }\!|\nabla u_\varepsilon (\cdot ,t)|^2\le R_2\Vert n_\varepsilon (\cdot ,t)\Vert _{L^{\frac{6}{5}}(\Omega )}^2\quad \text {on }(0,T_{max,\;\!\varepsilon }). \end{aligned}$$

(3.1)

Moreover, assuming $\kappa \ge 0$, $\mu >0$ and that $c_\star$ complies with (1.3) and introducing the constants

$$\begin{aligned} M_0:=\max \left\{ \int _{\Omega }\!n_0,\frac{\kappa |\Omega |}{\mu }\right\} ,\quad R_3:=R_2|\Omega |^\frac{2}{3}\frac{(\kappa +1)M_0}{\mu },\quad R_4:=\int _{\Omega }\!|u_0|^2+\frac{R_3}{1-e^{-\frac{R_1}{2}}}, \end{aligned}$$

for all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ satisfies

$$\begin{aligned} \int _{\Omega }\!|u_\varepsilon (\cdot ,t)|^2\le R_4 \quad \text {for all }t\in (0,T_{max,\;\!\varepsilon }) \end{aligned}$$

(3.2)

and

$$\begin{aligned} \int _{t}^{t+\tau }\!\!\int _{\Omega }\!|\nabla u_\varepsilon |^2\le 2(R_4+R_3)\quad \text {for all }t\in (0,T_{max,\;\!\varepsilon }-\tau ), \end{aligned}$$

(3.3)

where $\tau :=\min \big \{1,\frac{1}{2}T_{max,\;\!\varepsilon }\big \}$.

Proof

First, we note that by a Poincaré inequality there is $C_1>0$ such that

$$\begin{aligned} C_1\int _{\Omega }\!|\varphi |^2\le \int _{\Omega }\!|\nabla \varphi |^2\quad \text {for all }\varphi \in W_0^{1,2}\big (\Omega ;{\mathbb {R}}^{\mathcal {N}}\big ) \end{aligned}$$

(3.4)

and that in light of Sobolev embeddings one can find $C_2>0$ satisfying

$$\begin{aligned} \Vert \varphi \Vert _{L^{6}(\Omega )}\le C_2\Vert \nabla \varphi \Vert _{L^{2}(\Omega )}\quad \text {for all }\varphi \in W_0^{1,2}\big (\Omega ;{\mathbb {R}}^{\mathcal {N}}\big ) \end{aligned}$$

(3.5)

and, additionally, according to (1.4) we can also fix $C_3>0$ such that $\Vert \nabla \phi \Vert _{L^{\infty }(\Omega )}\le C_3$. With these constants prepared we test (2.1c) against $u_\varepsilon$ and integrate by parts to see that

$$\begin{aligned} \frac{1}{2}\frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!|u_\varepsilon |^2+\int _{\Omega }\!|\nabla u_\varepsilon |^2=\int _{\Omega }\!n_\varepsilon (\nabla \phi \cdot u_\varepsilon ) \end{aligned}$$

(3.6)

for all $t\in (0,T_{max,\;\!\varepsilon })$ and $\varepsilon \in (0,1)$, where we made use of the homogeneous Dirichlet boundary condition for $u_\varepsilon$ and that $\nabla \cdot u_\varepsilon =0$ on $\Omega \times (0,T_{max,\;\!\varepsilon })$, which together not only show that $\int _{\Omega }\!\nabla P_\varepsilon \cdot u_\varepsilon =0$ for all $t\in (0,T_{max,\;\!\varepsilon })$ and $\varepsilon \in (0,1)$, but also that for all $\varepsilon \in (0,1)$

$$\begin{aligned} \int _{\Omega }\!(Y_\varepsilon u_\varepsilon \cdot \nabla )u_\varepsilon \cdot u_\varepsilon&=-\int _{\Omega }\!(\nabla \cdot Y_\varepsilon u_\varepsilon )|u_\varepsilon |^2-\frac{1}{2}\int _{\Omega }\!Y_\varepsilon u_\varepsilon \cdot \nabla |u_\varepsilon |^2= \frac{1}{2}\int _{\Omega }\!(\nabla \cdot Y_\varepsilon u_\varepsilon )|u_\varepsilon |^2=0 \end{aligned}$$

on $(0,T_{max,\;\!\varepsilon })$. We then apply Hölder’s inequality to (3.6) and make use of (3.5) and our choice for $C_3>0$ to obtain

$$\begin{aligned} \frac{1}{2}\frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!|u_\varepsilon |^2+\int _{\Omega }\!|\nabla u_\varepsilon |^2\le \Vert \nabla \phi \Vert _{L^{\infty }(\Omega )}\Vert n_\varepsilon \Vert _{L^{\frac{6}{5}}(\Omega )}\Vert u_\varepsilon \Vert _{L^{6}(\Omega )}\le C_2C_3\Vert n_\varepsilon \Vert _{L^{\frac{6}{5}}(\Omega )}\Vert \nabla u_\varepsilon \Vert _{L^{2}(\Omega )} \end{aligned}$$

on $(0,T_{max,\;\!\varepsilon })$ for all $\varepsilon \in (0,1)$, which, by means of Young’s inequality, can further be turned into

$$\begin{aligned} \frac{1}{2}\frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!|u_\varepsilon |^2+\frac{1}{2}\int _{\Omega }\!|\nabla u_\varepsilon |^2\le \frac{C_2^2C_3^2}{2}\Vert n_\varepsilon \Vert _{L^{\frac{6}{5}}(\Omega )}^2\quad \text {on }(0,T_{max,\;\!\varepsilon })\text { for all }\varepsilon \in (0,1). \end{aligned}$$

Then, estimating half of the term with the spatial gradient from below as in (3.4) and multiplying the inequality by 2, we conclude that for all $\varepsilon \in (0,1)$ we have

$$\begin{aligned} \frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!|u_\varepsilon |^2+\frac{C_1}{2}\int _{\Omega }\!|u_\varepsilon |^2+\frac{1}{2}\int _{\Omega }\!|\nabla u_\varepsilon |^2\le C_2^2C_3^2\Vert n_\varepsilon \Vert _{L^{\frac{6}{5}}(\Omega )}^2\quad \text {on }(0,T_{max,\;\!\varepsilon }), \end{aligned}$$

which coincides with the asserted differential inequality in (3.1) upon setting $R_1:=C_1$ and $R_2:=C_2^2C_3^2$. To see that also the desired boundedness properties hold with the constants provided by the lemma, we introduce for $\varepsilon \in (0,1)$ the functions $y_\varepsilon (t)=\int _{\Omega }\!|u_\varepsilon (\cdot ,t)|^2$, $t\ge 0$ and $h_\varepsilon (t):=R_2\Vert n_\varepsilon (\cdot ,t)\Vert _{L^{\frac{6}{5}}(\Omega )}^2$, $t\ge 0$ and conclude from the differential inequality above that

$$\begin{aligned} y_\varepsilon '(t)+\frac{R_1}{2}y_\varepsilon (t)\le h_\varepsilon (t)\quad \text {for all }t\in (0,T_{max,\;\!\varepsilon })\text { and }\varepsilon \in (0,1). \end{aligned}$$

In light of Hölder’s inequality and Lemma 2.2 we then easily verify that

$$\begin{aligned} \int _{t}^{t+\tau }\!h_\varepsilon (s){{\,\mathrm{d\!}\,}}s\le R_2|\Omega |^\frac{2}{3}\int _t^{t+\tau }\!\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{2}(\Omega )}^2{{\,\mathrm{d\!}\,}}s\le R_3 \end{aligned}$$

(3.7)

for all $t\in (0,T_{max,\;\!\varepsilon }-\tau )$ and $\varepsilon \in (0,1)$, so that an application of Lemma 3.1 implies that

$$\begin{aligned} y_\varepsilon (t)\le y_\varepsilon (0)+\frac{R_3}{1-e^{-\frac{R_1}{2}}}\le R_4\quad \text {for all }t\in (0,T_{max,\;\!\varepsilon })\text { and }\varepsilon \in (0,1), \end{aligned}$$

proving (3.2). To verify (3.3), we finally integrate (3.1) with respect to time and draw on (3.2) and (3.7) to find that

$$\begin{aligned} \frac{1}{2}\int _t^{t+\tau }\!\!\int _{\Omega }\!|\nabla u_\varepsilon |^2\le \int _{\Omega }\!|u_\varepsilon (\cdot ,t)|^2+R_2\int _t^{t+\tau }\!\Vert n_\varepsilon \Vert _{L^{\frac{6}{5}}(\Omega )}^2\le R_4+R_3 \end{aligned}$$

for all $t\in (T_{max,\;\!\varepsilon }-\tau )$ and $\varepsilon \in (0,1)$, which after rearrangement completes the proof. $\square$

The next Lemma will show that the quadratic degradation of $n_\varepsilon$ when combined with the bound on the gradient of $u_\varepsilon$ from the previous Lemma entails remarkably strong boundedness information for the signal chemical $c_\varepsilon$ and thereby provides a key point in the estimation process towards both time-global existence for fixed $\varepsilon \in (0,1)$ and suitably strong precompactness properties for the limit procedure undertaken in Sect. 4.

Lemma 3.3

Let $\kappa \ge 0$, $\mu >0$ and assume that $c_\star$ fulfills (1.3). Then there is $C=C(\kappa ,\mu ,c_\star )>0$ such that for all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) fulfills

$$\begin{aligned} \int _{\Omega }\!|\nabla c_\varepsilon (\cdot ,t)|^2 \le C\quad \text {for all }t\in (0,T_{max,\;\!\varepsilon }) \end{aligned}$$

and

$$\begin{aligned} \int _t^{t+\tau }\!\!\int _{\Omega }\!|\Delta c_\varepsilon |^2\le C\qquad \text {and}\qquad \int _t^{t+\tau }\!\!\int _{\Omega }\!|\nabla c_\varepsilon |^4\le C\quad \text {for all }t\in (0,T_{max,\;\!\varepsilon }-\tau ), \end{aligned}$$

where $\tau :=\min \{1,\frac{1}{2}T_{max,\;\!\varepsilon }\}$.

Proof

Testing (2.1b) against $-\Delta c_\varepsilon$ and integrating by parts we obtain from the imposed boundary conditions that

$$\begin{aligned} \frac{1}{2}\frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!|\nabla c_\varepsilon |^2&=\int _{\partial \Omega }\!c_{\varepsilon t}\frac{\partial c_\varepsilon }{\partial \nu }-\int _{\Omega }\!|\Delta c_\varepsilon |^2-\int _{\Omega }\!(D^2c_\varepsilon \cdot u_\varepsilon )\cdot \nabla c_\varepsilon -\int _{\Omega }\!\nabla c_\varepsilon \cdot (\nabla u_\varepsilon \cdot \nabla c_\varepsilon )\nonumber \\&\qquad +\int _{\Omega }\!g_\varepsilon (n_\varepsilon )c_\varepsilon \Delta c_\varepsilon \end{aligned}$$

(3.8)

on $(0,T_{max,\;\!\varepsilon })$ for all $\varepsilon \in (0,1)$. Here, Young’s inequality, the fact that $g_\varepsilon (s)\le s$ for $s\ge 0$ and Lemma 2.2 entail that for all $\varepsilon \in (0,1)$ there is $C_1=C_1(c_\star )>0$ such that

$$\begin{aligned} \int _{\Omega }\!g_\varepsilon (n_\varepsilon )c_\varepsilon \Delta c_\varepsilon \le \frac{1}{4}\int _{\Omega }\!|\Delta c_\varepsilon |^2+\Vert c_\varepsilon \Vert _{L^{\infty }(\Omega )}^2\int _{\Omega }\!n_\varepsilon ^2\le \frac{1}{4}\int _{\Omega }\!|\Delta c_\varepsilon |^2+C_1\int _{\Omega }\!n_\varepsilon ^2 \end{aligned}$$

(3.9)

on $(0,T_{max,\;\!\varepsilon })$. Moreover, elliptic regularity theory [17] along with Young’s inequality provides $C_2>0$ and $C_3>0$ satisfying

$$\begin{aligned} \int _{\Omega }\!|\nabla c_\varepsilon |^2\le C_2\big (\Vert \Delta c_\varepsilon \Vert _{L^{2}(\Omega )}\Vert c_\varepsilon \Vert _{L^{2}(\Omega )}+1\big )\le \frac{1}{8}\Vert \Delta c_\varepsilon \Vert _{L^{2}(\Omega )}^2+C_3 \end{aligned}$$

(3.10)

on $(0,T_{max,\;\!\varepsilon })$ for all $\varepsilon \in (0,1)$. Similarly, noting that $\nabla \cdot u_\varepsilon =0$ on $\Omega \times (0,T_{max,\;\!\varepsilon })$ and $u_\varepsilon =0$ on $\partial \Omega \times (0,T_{max,\;\!\varepsilon })$ imply $\int _{\Omega }\!(D^2c_\varepsilon \cdot u_\varepsilon )\cdot \nabla c_\varepsilon =\frac{1}{2}\int _{\Omega }\!\nabla \cdot (|\nabla c_\varepsilon |^2u_\varepsilon )=0$, we obtain from Hölder’s inequality, the Gagliardo–Nirenberg inequality and Young’s inequality that there are $C_4>0$ and $C_5>0$ such that

$$\begin{aligned} -\int _{\Omega }\!(D^2c_\varepsilon \cdot u_\varepsilon )\cdot \nabla c_\varepsilon -\int _{\Omega }\!\nabla c_\varepsilon \cdot (\nabla u_\varepsilon \cdot \nabla c_\varepsilon )&\le \Vert \nabla u_\varepsilon \Vert _{L^{2}(\Omega )}\Vert \nabla c_\varepsilon \Vert _{L^{4}(\Omega )}^2\nonumber \\&\le C_4\Vert \nabla u_\varepsilon \Vert _{L^{2}(\Omega )}\big (\Vert \Delta c_\varepsilon \Vert _{L^{2}(\Omega )}\Vert c_\varepsilon \Vert _{L^{\infty }(\Omega )}\nonumber \\&\quad +\Vert c_\varepsilon \Vert _{L^{\infty }(\Omega )}^{2}\big )\nonumber \\&\le \frac{1}{8}\Vert \Delta c_\varepsilon \Vert ^2_{L^{2}(\Omega )}+C_2\Vert \nabla u_\varepsilon \Vert _{L^{2}(\Omega )}^2+C_5 \end{aligned}$$

(3.11)

is valid on $(0,T_{max,\;\!\varepsilon })$ for all $\varepsilon \in (0,1)$. Accordingly, combining the fact that due to $c_\star (x)$ not depending on the time variable we have $\frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}c_\varepsilon \vert _{\partial \Omega }=0$ on $(0,T_{max,\;\!\varepsilon })$ with (3.8)–(3.11) for all $\varepsilon \in (0,1)$ we have

$$\begin{aligned} \frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!|\nabla c_\varepsilon |^2+2\int _{\Omega }\!|\nabla c_\varepsilon |^2+\int _{\Omega }\!|\Delta c_\varepsilon |^2\le 2C_1\int _{\Omega }\!n_\varepsilon ^2+2C_2\int _{\Omega }\!|\nabla u_\varepsilon |^2+2(C_3+C_5) \end{aligned}$$

(3.12)

on $(0,T_{max,\;\!\varepsilon })$. Hence, the spatio-temporal bounds of Lemmas 2.2 and 3.2 make Lemma 3.1 applicable and we thereby obtain $C_6=C_6(\kappa ,\mu ,c_\star )>0$ such that for all $\varepsilon \in (0,1)$

$$\begin{aligned} \int _{\Omega }\!|\nabla c_\varepsilon (\cdot ,t)|^2\le C_6\quad \text {holds for all }t\in (0,T_{max,\;\!\varepsilon }). \end{aligned}$$

Afterwards we return to (3.12) and integrate with respect to time to conclude the remaining asserted bounds of the lemma, since by means of the Gagliardo–Nirenberg inequality there is $C_7>0$ such that $\Vert \nabla c_\varepsilon \Vert _{L^{4}(\Omega )}^4\le C_7\Vert \Delta c_\varepsilon \Vert _{L^{2}(\Omega )}^2\Vert c_\varepsilon \Vert _{L^{\infty }(\Omega )}^2+C_7\Vert c_\varepsilon \Vert _{L^{\infty }(\Omega )}^4$ holds on $(0,T_{max,\;\!\varepsilon })$ for all $\varepsilon \in (0,1)$. $\square$

3.2 Time global existence of approximate solutions

With a small change in variables we can get rid of the inhomogeneous boundary condition of (2.1) and instead consider a closely related system, which is accessible to the powerful machinery of semigroup estimates for the Neumann and Dirichlet heat semigroups and the Stokes semigroup. The estimates prepared in previous Lemmas are then already sufficiently strong to conclude that for each fixed $\varepsilon \in (0,1)$ the case $T_{max,\;\!\varepsilon }<\infty$ cannot occur.

Lemma 3.4

Let $\kappa \ge 0$ and $\mu >0$ and assume that $c_\star$ complies with (1.3). Then for all $\varepsilon \in (0,1)$ the solution of (2.1) is global in time, i.e. $T_{max,\;\!\varepsilon }=\infty$.

Proof

Introducing the substitution ${\widehat{c}}_{\varepsilon }:=c_\star -c_\varepsilon$ we rewrite system (2.1) in the equivalent form

(3.13)

where, by means of the assumed regularity of $c_\star$, all important properties can be easily transferred back to (2.1). Now, assuming that for fixed $\varepsilon \in (0,1)$ the maximal time of existence $T_{max,\;\!\varepsilon }$ is a finite number we only have to slightly adjust and augment the proof of global existence in the Stokes setting (as presented in [4, Lemma 3.7]) with arguments accounting for the nonlinear convection term in the fluid equation. In our current setting the additional estimations necessary for the convection term can easily be established by the methods demonstrated in e.g. [45, Section 4.2] and [49, Lemma 3.9]. $\square$

3.3 Refined compactness features of $\{n_\varepsilon \}_{\varepsilon \in (0,1)}$

While we are well-prepared for the treatment of the signal chemical and fluid components in the limit procedure of Sect. 4, we still lack any information on the spatial gradient of $n_\varepsilon$. Inspecting an inequality for the functional $\int _{\Omega }\!\,(n_\varepsilon \ln n_\varepsilon )(\cdot ,t)$ we can draw on the previously established space-time bounds to extract additional space-time information on $\nabla \sqrt{n_\varepsilon }$, which in a second interpolation step can be further refined to a bound on $\nabla n_\varepsilon$ in $L_{loc}^{\frac{4}{3}}\!\left( {\overline{\Omega }}\times [0,\infty )\right)$.

Lemma 3.5

Let $\kappa \ge 0$ and $\mu >0$ and assume that $c_\star$ fulfills (1.3). For each $T>0$ there is $C_T=C_T(\kappa ,\mu ,c_\star ,T)>0$ such that for all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) satisfies

$$\begin{aligned} \int _0^{T}\!\!\int _{\Omega }\!\frac{|\nabla n_\varepsilon |^2}{n_\varepsilon }+\int _0^{T}\!\!\int _{\Omega }\!n_\varepsilon ^2\ln \big (n_\varepsilon ^2+1\big )\le C_T\quad \text {and}\quad \int _0^{T}\!\!\int _{\Omega }\!|\nabla n_\varepsilon |^\frac{4}{3}\le C_T. \end{aligned}$$

Proof

Testing (2.1a) against $(\ln n_\varepsilon +1)$ and integrating by parts we find that for all $\varepsilon \in (0,1)$

$$\begin{aligned} \frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!n_\varepsilon \ln n_\varepsilon&=-\int _{\Omega }\!\frac{|\nabla n_\varepsilon |^2}{n_\varepsilon }+\int _{\Omega }\!\rho _\varepsilon f_\varepsilon (n_\varepsilon )(\nabla n_\varepsilon \cdot \nabla c_\varepsilon )+\kappa \int _{\Omega }\!n_\varepsilon \ln n_\varepsilon +\kappa \int _{\Omega }\!n_\varepsilon \\&\quad -\mu \int _{\Omega }\!n_\varepsilon ^2\ln n_\varepsilon -\mu \int _{\Omega }\!n_\varepsilon ^2\\&\le - \frac{3}{4}\int _{\Omega }\!\frac{|\nabla n_\varepsilon |^2}{n_\varepsilon }+\frac{1}{4\mu } \int _{\Omega }\!|\nabla c_\varepsilon |^4+\kappa \int _{\Omega }\!n_\varepsilon \ln n_\varepsilon +\kappa \int _{\Omega }\!n_\varepsilon \\&\quad -\mu \int _{\Omega }\!n_\varepsilon ^2\ln n_\varepsilon \quad \text {on }(0,\infty ), \end{aligned}$$

where we also used the divergence-free property of $u_\varepsilon$, that $\rho _\varepsilon = 0$ on $\partial \Omega$ for all $\varepsilon \in (0,1)$, the fact that $|\rho _\varepsilon (x)f_\varepsilon (s)|=\frac{\rho _\varepsilon (x)}{(1+\varepsilon s)^3}\le 1$ for all $s\ge 0$, $x\in \Omega$ and $\varepsilon \in (0,1)$ and Young’s inequality. Moreover, since clearly there is $C_1=C_1(\kappa ,\mu )>0$ such that

$$\begin{aligned} \kappa s\ln s+\kappa s-\frac{\mu }{2}s^2 \ln s\le C_1 \quad \text {for all } s>0, \end{aligned}$$

we further obtain

$$\begin{aligned} \frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!n_\varepsilon \ln n_\varepsilon +\frac{3}{4}\int _{\Omega }\!\frac{|\nabla n_\varepsilon |^2}{n_\varepsilon }+\frac{\mu }{2}\int _{\Omega }\!n_\varepsilon ^2\ln n_\varepsilon \le \frac{1}{4\mu } \int _{\Omega }\!|\nabla c_\varepsilon |^4+C_1|\Omega | \end{aligned}$$

for all $t\in (0,\infty )$ and all $\varepsilon \in (0,1)$. Integrating this differential inequality over (0, T) yields

$$\begin{aligned}&\frac{3}{4}\int _0^{T}\!\!\int _{\Omega }\!\frac{|\nabla n_\varepsilon |^2}{n_\varepsilon }+\frac{\mu }{2}\int _0^{T}\!\!\int _{\Omega }\!n_\varepsilon ^2\ln n_\varepsilon \\&\quad \le \ \frac{1}{4\mu } \int _0^{T}\!\!\int _{\Omega }\!|\nabla c_\varepsilon |^4-\int _{\Omega }\!n_\varepsilon (\cdot ,T)\ln n_\varepsilon (\cdot ,T)+\int _{\Omega }\!n_\varepsilon (\cdot ,0)\ln n_\varepsilon (\cdot ,0)\\&\qquad +C_1|\Omega |T\quad \text {for all }\varepsilon \in (0,1), \end{aligned}$$

which, by employing the bound provided by Lemma 3.3 and the estimates $-\frac{1}{e}\le s\ln s$ for all $s\ge 0$ and $2s^2\ln s\ge s^2\ln (s^2+1)-1$ for all $s\ge 0$, proves the first assertion. To verify the second, we rewrite the intergral under consideration and draw on Young’s inequality to estimate

$$\begin{aligned} \int _0^{T}\!\!\int _{\Omega }\!|\nabla n_\varepsilon |^\frac{4}{3}&=\int _0^{T}\!\!\int _{\Omega }\!\frac{|\nabla n_\varepsilon ||\nabla n_\varepsilon |^{\frac{1}{3}}\sqrt{n_\varepsilon }}{\sqrt{n_\varepsilon }} \le \int _0^{T}\!\!\int _{\Omega }\!\frac{|\nabla n_\varepsilon |^2}{n_\varepsilon }+\frac{1}{4} \int _0^{T}\!\!\int _{\Omega }\!|\nabla n_\varepsilon |^{\frac{4}{3}}+\frac{1}{16}\int _0^{T}\!\!\int _{\Omega }\!n_\varepsilon ^2 \end{aligned}$$

for all $\varepsilon \in (0,1)$. Reordering and making use of Lemma 2.2 as well as the first part of this Lemma we conclude the desired bound. $\square$

3.4 Regularity estimates for the time derivatives

As final ingredient for an Aubin–Lions type argument, we now prepare uniform bounds for the time derivatives in suitable spaces.

Lemma 3.6

Let $\kappa \ge 0$, $\mu >0$ and suppose that $c_\star$ satifies (1.3). For every $T>0$ there exists a constant $C_T=C_T(\kappa ,\mu ,c_\star ,T)>0$ such that for any $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) fulfills

$$\begin{aligned}&\int _0^T\Vert \partial _tn_\varepsilon \Vert _{ (W_0^{1,4}(\Omega ))^*}\le C_T, \end{aligned}$$

(3.14)

$$\begin{aligned}&\int _0^T\Vert \partial _tc_\varepsilon \Vert _{(W_0^{1,2}(\Omega ))^*}^2\le C_T, \end{aligned}$$

(3.15)

and

$$\begin{aligned} \int _0^T\Vert \partial _t u_\varepsilon \Vert _{(W_{0,\sigma }^{1,2}(\Omega ))^*}^\frac{4}{3}\le C_T. \end{aligned}$$

(3.16)

Proof

Given $T>0$ we first note that by the Gagliardo–Nirenberg inequality, Young’s inequality, the Poincaré inequality and Lemma 3.2 there are $C_1>0$, $C_2(\kappa ,\mu )>0$ and $C_3(\kappa ,\mu ,T)>0$ such that

$$\begin{aligned} \int _0^T\!\!\Vert u_\varepsilon \Vert _{L^{4}(\Omega )}^2\le C_1\!\int _0^T\!\!\Vert \nabla u_\varepsilon \Vert _{L^{2}(\Omega )}^{\frac{{\mathcal {N}}}{2}}\Vert u_\varepsilon \Vert _{L^{2}(\Omega )}^{\frac{4-{\mathcal {N}}}{2}}\le C_2(\kappa ,\mu )\int _0^T\!\!\Vert \nabla u_\varepsilon \Vert _{L^{2}(\Omega )}^2\le C_3(\kappa ,\mu ,T) \end{aligned}$$

(3.17)

for all $\varepsilon \in (0,1)$. Now, we fix $\varphi \in L^{\infty \;\!}\!\big ((0,T);W_0^{1,4}(\Omega )\big )$ satisfying $\Vert \varphi \Vert _{L^{\infty \;\!}\!((0,T);W_0^{1,4}(\Omega ))}\le 1$ and in light of (2.1a), $u_\varepsilon$ being divergence-free, the boundary conditions for $n_\varepsilon$ and $u_\varepsilon$ as well as the fact that $\rho _\varepsilon =0$ holds on $\partial \Omega$, obtain that

$$\begin{aligned} \Big |\int _{\Omega }\!\partial _tn_\varepsilon \varphi \Big |&=\Big |\int _{\Omega }\!\Big (-u_\varepsilon \cdot \!\nabla n_\varepsilon +\Delta n_\varepsilon -\nabla \!\cdot \big (\rho _\varepsilon f_\varepsilon (n_\varepsilon )n_\varepsilon \nabla c_\varepsilon \big )+\kappa n_\varepsilon -\mu n_\varepsilon ^2\Big )\varphi \Big |\\&=\Big |\int _{\Omega }\!n_\varepsilon (u_\varepsilon \cdot \nabla \varphi )-\int _{\Omega }\!\nabla n_\varepsilon \cdot \nabla \varphi +\int _{\Omega }\!\rho _\varepsilon f_\varepsilon (n_\varepsilon )n_\varepsilon (\nabla c_\varepsilon \cdot \nabla \varphi )\\&\quad +\kappa \int _{\Omega }\!n_\varepsilon \varphi -\mu \int _{\Omega }\!n_\varepsilon ^2\varphi \Big | \end{aligned}$$

for all $t>0$ and $\varepsilon \in (0,1)$. Making use of Hölder’s inequality and the fact that for all $\varepsilon \in (0,1)$ we have $|\rho _\varepsilon f_\varepsilon (n_\varepsilon )|\le 1$ on $\Omega \times (0,\infty )$, we find upon integration over (0, T) that

$$\begin{aligned} \int _0^T\!\Big |\int _{\Omega }\!\partial _tn_\varepsilon \varphi \Big |&\le \int _0^T\!\! \Big (\Vert n_\varepsilon \Vert _{L^{2}(\Omega )}\Vert u_\varepsilon \Vert _{L^{4}(\Omega )}+\Vert \nabla n_\varepsilon \Vert _{L^{\frac{4}{3}}(\Omega )}+\Vert n_\varepsilon \Vert _{L^{2}(\Omega )}\Vert \nabla c_\varepsilon \Vert _{L^{4}(\Omega )}\Big )\Vert \nabla \varphi \Vert _{L^{4}(\Omega )}\\&\qquad +\int _0^T\!\!\Big (\kappa \Vert n_\varepsilon \Vert _{L^{1}(\Omega )}+ \mu \Vert n_\varepsilon \Vert _{L^{2}(\Omega )}^2\Big )\Vert \varphi \Vert _{L^{\infty }(\Omega )}\quad \text {for all }\varepsilon \in (0,1), \end{aligned}$$

whence a combination of Young’s inequality with the embedding $W_0^{1,4}(\Omega )\hookrightarrow L^{\infty }(\Omega )$, (3.17) and the bounds from Lemmas 2.2, 3.3 and 3.5 entails (3.14).

Similarly, fixing ${\tilde{\varphi }}\in L^{\infty \;\!}\!\big ((0,T);W_0^{1,2}(\Omega )\big )$ with $\Vert {\tilde{\varphi }}\Vert _{L^{\infty \;\!}\!((0,T);W_0^{1,2}(\Omega ))}\le 1$ and testing (2.1b) against ${\tilde{\varphi }}$ we find $C_4(\kappa ,\mu ,c_\star )>0$ fulfilling

$$\begin{aligned} \int _0^{T}\!\Big |\int _{\Omega }\!\partial _tc_\varepsilon {\tilde{\varphi }}\Big |^2&= \int _0^{T}\!\Big |\int _{\Omega }\!\Big (-u_\varepsilon \cdot \nabla c_\varepsilon +\Delta c_\varepsilon -g_\varepsilon (n_\varepsilon ) c_\varepsilon \Big ){\tilde{\varphi }}\Big |^2\\&\le C_4(\kappa ,\mu ,c_\star )\int _0^{T}\!\!\Big (\Vert \nabla c_\varepsilon \Vert _{L^{4}(\Omega )}^4+\Vert \Delta c_\varepsilon \Vert _{L^{2}(\Omega )}^2 +\Vert n_\varepsilon \Vert _{L^{2}(\Omega )}^2+1\Big ) \end{aligned}$$

for all $\varepsilon \in (0,1)$, where we relied on bounds from Lemmas 2.2 and 3.2. Hence, (3.15) is again a direct consequence of the spatio-temporal bounds provided by Lemmas 2.2 and 3.3.

Finally, we fix $\psi \in C_{0}^{\infty }(\Omega ;{\mathbb {R}}^{\mathcal {N}})$ with $\nabla \cdot \psi \equiv 0$ throughout $\Omega$ and employ Hölder’s inequality similarly to before (see also [40, Lemma 5.5]) to obtain

$$\begin{aligned} \Big \vert \int _{\Omega }\!\partial _tu_\varepsilon \psi \Big \vert&=\Big |\int _{\Omega }\!\nabla u_\varepsilon \cdot \nabla \psi +\int _{\Omega }\!(Y_{\varepsilon } u_{\varepsilon } \otimes u_{\varepsilon })\cdot \nabla \psi +\int _{\Omega }\!n_\varepsilon (\nabla \phi \cdot \psi )\Big |\nonumber \\&\le \Vert \nabla u_\varepsilon \Vert _{L^{2}(\Omega )}\Vert \nabla \psi \Vert _{L^{2}(\Omega )}+\Vert Y_\varepsilon u_\varepsilon \Vert _{L^{6}(\Omega )}\Vert u_\varepsilon \Vert _{L^{3}(\Omega )}\Vert \nabla \psi \Vert _{L^{2}(\Omega )}\nonumber \\&\quad +\Vert \nabla \phi \Vert _{L^{\infty }(\Omega )}\Vert n_\varepsilon \Vert _{L^{2}(\Omega )}\Vert \psi \Vert _{L^{2}(\Omega )} \end{aligned}$$

(3.18)

on $(0,\infty )$ for all $\varepsilon \in (0,1)$. Here, the embedding $W_{0,\sigma }^{1,2}(\Omega )\hookrightarrow L^{6}(\Omega )$ and the nonexpansiveness of $Y_\varepsilon$ on $L_\sigma ^2(\Omega )$ on the one hand (cf. [27, Lemma 2.15]), and the Gagliardo–Nirenberg inequality and the Poincaré inequality on the other entail that there is $C_5>0$ such that

$$\begin{aligned}&\Vert Y_\varepsilon u_\varepsilon (\cdot ,t)\Vert _{L^{6}(\Omega )}\le C_5\Vert \nabla u_\varepsilon (\cdot ,t)\Vert _{L^{2}(\Omega )}\quad \text {and}\quad \\&\quad \Vert u_\varepsilon (\cdot ,t)\Vert _{L^{3}(\Omega )}^\frac{4}{3}\le C_5\Vert \nabla u_\varepsilon (\cdot ,t)\Vert _{L^{2}(\Omega )}^\frac{2{\mathcal {N}}}{9}\Vert u_\varepsilon (\cdot ,t) \Vert _{L^{2}(\Omega )}^\frac{2(6-{\mathcal {N}})}{9} \end{aligned}$$

for all $t>0$ and $\varepsilon \in (0,1)$, whence we derive from a combination of these inequalities that

$$\begin{aligned}&\Vert Y_\varepsilon u_\varepsilon (\cdot ,t)\Vert _{L^{6}(\Omega )}^\frac{4}{3}\Vert u_\varepsilon (\cdot ,t)\Vert _{L^{3}(\Omega )}^\frac{4}{3}\\&\quad \le C_5\Vert \nabla u_\varepsilon (\cdot ,t)\Vert _{L^{2}(\Omega )}^{\frac{4}{3}+\frac{2{\mathcal {N}}}{9}}\Vert u_\varepsilon (\cdot ,t)\Vert _{L^{2}(\Omega )}^\frac{2(6-{\mathcal {N}})}{9}\quad \text {for all }t>0\text { and }\varepsilon \in (0,1). \end{aligned}$$

Thus, we conclude from (3.18), the fact that $\frac{4}{3}+\frac{2{\mathcal {N}}}{9}\le 2$, Young’s inequality, Lemmas 3.2 and  2.2 that there are $C_6(\kappa ,\mu )>0$ and $C_7(\kappa ,\mu ,T)>0$ satisfying

$$\begin{aligned}&\int _0^T\!\!\Vert \partial _tu_\varepsilon \Vert _{(W_{0,\sigma }^{1,2}(\Omega ))^*}^\frac{4}{3}\\&\quad \le C_6(\kappa ,\mu )\int _0^T\!\Vert \nabla u_\varepsilon \Vert _{L^{2}(\Omega )}^2+C_6(\kappa ,\mu )\int _0^T\!\Vert n_\varepsilon \Vert _{L^{2}(\Omega )}^2+C_6(\kappa ,\mu ) T\le C_7(\kappa ,\mu ,T) \end{aligned}$$

for all $\varepsilon \in (0,1)$, completing the proof. $\square$

4 Existence of a global weak solution

The preparations of the previous sections at hand, we can now focus on the proof of Theorem 1.1.

4.1 Convergence: properties of the limit object (n, c, u)

Making use of the precompactness properties entailed by the uniform estimates of Sects. 2 and 3, we can extract a sequence along which $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ converges to some limit object in topologies suitable for our purposes.

Proposition 4.1

Let $\kappa \ge 0$, $\mu >0$ and assume that $c_\star$ complies with (1.3). There exist a sequence $(\varepsilon _j)_{j\in {\mathbb {N}}}\subset (0,1)$ with $\varepsilon _j\searrow 0$ as $j\rightarrow \infty$ and functions (n, c, u) satisfying

$$\begin{aligned} n&\in L_{loc}^{2}\!\left( {\overline{\Omega }}\times [0,\infty )\right) \quad \text {with}\quad \nabla n\in L_{loc}^{\frac{4}{3}}\!\left( {\overline{\Omega }}\times [0,\infty );{\mathbb {R}}^{\mathcal {N}}\right) \\ c&\in L_{loc}^{\infty }\!\left( \Omega \times (0,\infty )\right) \quad \text {with}\quad c-c_\star \in L_{loc}^{4}\big ([0,\infty );W_0^{1,4}(\Omega )\big )\\ u&\in L_{loc}^{2}\big ([0,\infty );W_{0,\sigma }^{1,2}\big (\Omega ;{\mathbb {R}}^{\mathcal {N}}\big )\big ) \end{aligned}$$

such that the solutions $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) fulfill

$$\begin{aligned} n_\varepsilon \rightarrow \,&n&\text {in }\ L_{loc}^{p}\!\left( {\overline{\Omega }}\times [0,\infty )\right) \text { for any }p\in [1,2]\text { and a.e. in}\ \Omega \times (0,\infty ),\end{aligned}$$

(4.1)

$$\begin{aligned} \nabla n_\varepsilon \rightharpoonup \&\nabla n&\ \;&\text {in }\ L_{loc}^{\frac{4}{3}}\!\left( {\overline{\Omega }}\times [0,\infty );{\mathbb {R}}^{\mathcal {N}}\right) ,\end{aligned}$$

(4.2)

$$\begin{aligned} \rho _\varepsilon f_\varepsilon (n_\varepsilon )n_\varepsilon \rightarrow \,&n&\,&\text {in }\ L_{loc}^{p}\!\left( {\overline{\Omega }}\times [0,\infty )\right) \text { for any }p\in [1,2],\end{aligned}$$

(4.3)

$$\begin{aligned} g_\varepsilon (n_\varepsilon )\rightarrow \,&n&\,&\text {in }\ L_{loc}^{p}\!\left( {\overline{\Omega }}\times [0,\infty )\right) \text { for any }p\in [1,2],\end{aligned}$$

(4.4)

$$\begin{aligned} c_\varepsilon \rightarrow \,&c&\text {in }\ L_{loc}^{q}\!\left( {\overline{\Omega }}\times [0,\infty )\right) \text { for any }q\in [1,\infty )\text { and a.e. in}\ \Omega \times (0,\infty ),\end{aligned}$$

(4.5)

$$\begin{aligned} c_\varepsilon {\mathop {\rightharpoonup }\limits ^{\star }}\,&c&\text {in }\ L^{\infty \;\!}\!\left( \Omega \times (0,\infty )\right) ,\end{aligned}$$

(4.6)

$$\begin{aligned} \nabla c_\varepsilon \rightharpoonup \,&\nabla c&\text {in }\ L_{loc}^{4}\!\left( {\overline{\Omega }}\times [0,\infty );{\mathbb {R}}^{\mathcal {N}}\right) ,\end{aligned}$$

(4.7)

$$\begin{aligned} u_\varepsilon \rightarrow \,&u&\text {in }\ L_{loc}^{2}\!\left( {\overline{\Omega }}\times [0,\infty );{\mathbb {R}}^{\mathcal {N}}\right) \text { and a.e. in}\ \Omega \times (0,\infty ),\end{aligned}$$

(4.8)

$$\begin{aligned} \nabla u_\varepsilon \rightharpoonup \,&\nabla u&\text {in }\ L_{loc}^{2}\!\left( {\overline{\Omega }}\times [0,\infty );{\mathbb {R}}^{{\mathcal {N}}\times {\mathcal {N}}}\right) ,\end{aligned}$$

(4.9)

$$\begin{aligned} Y_{\varepsilon } u_\varepsilon \otimes u_\varepsilon \rightarrow \,&u\otimes u&\text {in } \ L_{loc}^{1}\!\left( {\overline{\Omega }}\times [0,\infty );{\mathbb {R}}^{{\mathcal {N}}\times {\mathcal {N}}}\right) , \end{aligned}$$

(4.10)

as $\varepsilon =\varepsilon _j\searrow 0$.

Proof

A combination of Lemmas 2.2, 3.5 and 3.6 with an Aubin–Lions type lemma (e.g. [33, Corollary 8.4]) entails that

$$\begin{aligned} \{n_\varepsilon \}_{\varepsilon \in (0,1)}\quad \text {is relatively compact in }L_{loc}^{\frac{4}{3}}\!\left( {\overline{\Omega }}\times [0,\infty )\right) \end{aligned}$$

and that hence there is some sequence $(\varepsilon _j)_{j\in {\mathbb {N}}}$ with $\varepsilon _j\searrow 0$ as $j\rightarrow \infty$ and $n\in L_{loc}^{\frac{4}{3}}\!\left( {\overline{\Omega }}\times [0,\infty )\right)$ such that $n_\varepsilon \rightarrow n$ in $L_{loc}^{\frac{4}{3}}\!\left( {\overline{\Omega }}\times [0,\infty )\right)$ and a.e. in $\Omega \times (0,\infty )$. Thereafter, we conclude also from Lemma 3.5 that (4.2) holds. Next, in light of a result from de la Vallée–Poussin and Vitali’s theorem we can refine the logarithmic bound present in Lemma 3.5 to obtain (4.1) along a further (non-relabeled) subsequence (compare [4, Proposition 6.1]) and, in turn, also (4.3) and (4.4) due to $|\rho _\varepsilon f_\varepsilon (n_\varepsilon )n_\varepsilon |\le n_\varepsilon$ and $|g_\varepsilon (n_\varepsilon )|\le n_\varepsilon$ in $\Omega \times (0,\infty )$. Employing similar arguments for $c_\varepsilon$, while relying on Lemmas 2.2, 3.3 and 3.6 show (4.5), (4.6) and (4.7) to be true and, additionally, drawing on Lemmas 3.2 and 3.6 for $u_\varepsilon$ we conclude (4.8) and (4.9) in the same manner. For the remaining convergence property concerning $u_\varepsilon$, we consider the properties of the Yosida approximation (see e.g. [34, Section II.3.4]) together with Lemma 3.2 and (4.8) to obtain that $\left\| Y_{\varepsilon } u_{\varepsilon }-u\right\| _{L^{2}(\Omega )} \le \left\| Y_{\varepsilon }(u_{\varepsilon }-u)\right\| _{L^{2}(\Omega )}+\left\| \left( Y_{\varepsilon }-Y\right) u\right\| _{L^{2}(\Omega )}\le \left\| u_{\varepsilon }-u\right\| _{L^{2}(\Omega )}+\left\| \left( Y_{\varepsilon }-Y\right) u\right\| _{L^{2}(\Omega )} \rightarrow 0$, from which, upon another combination with (4.8), we infer (4.10). The asserted regularity properties of the limit objects and $c-c_\star$ are evidently entailed by (4.1), (4.2), (4.6), (4.7), (4.8), (4.9) and (1.3) and the fact that $c_\varepsilon -c_\star =0$ on $\partial \Omega \times [0,\infty )$. $\square$

4.2 Specifications of global weak solutions: the proof of Theorem 1.1

With Proposition 4.1 we have already completed the cornerstone of the proof of Theorem 1.1 and, in essence, only have to collect the provided information appropriately. To contemplate this in more detail, let us briefly recap the structures and properties required of a global weak solution in the following definition.

Definition 4.2

Let $\kappa \ge 0$ and $\mu >0$. Assume that $\phi$ and $c_\star$ comply with (1.4) and (1.3), respectively, and assume that $(n_0,c_0,u_0)$ fulfills (1.5). Let

$$\begin{aligned} \left\{ \begin{array}{r@{\ }l} n&{}\in L_{loc}^{2}\!\left( {\overline{\Omega }}\times [0,\infty )\right) \cap L_{loc}^{1}\!\left( [0,\infty );W^{1,1}(\Omega )\right) ,\\ c&{}\in L_{loc}^{1}\!\left( [0,\infty );W^{1,1}(\Omega )\right) \quad \text {with}\quad c-c_\star \in L_{loc}^{1}\big ([0,\infty );W_0^{1,1}(\Omega )\big ),\\ u&{}\in L_{loc}^{1}\big ([0,\infty );W_{0,\sigma }^{1,1}\big (\Omega ;{\mathbb {R}}^{\mathcal {N}}\big )\big ) \end{array}\right. \end{aligned}$$

be such that $n\ge 0$ and $c\ge 0$ a.e. in $\Omega \times (0,\infty )$. Then (n, c, u) will be called a global weak solution of (1.2) provided that

$$\begin{aligned}&nc\in L_{loc}^{1}\!\left( {\overline{\Omega }}\times [0,\infty )\right) ,\\&\quad \big \{n\nabla c,\; nu,\; cu\big \}\subset L_{loc}^{1}\!\left( {\overline{\Omega }}\times [0,\infty );{\mathbb {R}}^{\mathcal {N}}\right) ,\quad u \otimes u\in L_{loc}^{1}\!\left( {\overline{\Omega }}\times [0,\infty );{\mathbb {R}}^{{\mathcal {N}}\times {\mathcal {N}}}\right) , \end{aligned}$$

that

$$\begin{aligned} -\int _0^\infty \!\!\int _{\Omega }\!n\varphi _t-\int _{\Omega }\!n_0\varphi (\cdot ,0)&=\,-\int _0^\infty \!\!\int _{\Omega }\!\nabla n\cdot \nabla \varphi +\int _0^\infty \!\!\int _{\Omega }\!n (\nabla c\cdot \nabla \varphi )+\kappa \int _0^\infty \!\!\int _{\Omega }\!n\varphi \nonumber \\&\quad -\mu \int _0^\infty \!\!\int _{\Omega }\!n^2\varphi +\int _0^\infty \!\!\int _{\Omega }\!n (u\cdot \nabla \varphi ) \end{aligned}$$

(4.11)

holds for all $\varphi \in C_0^\infty \big ({\overline{\Omega }}\times [0,\infty )\big )$, that

$$\begin{aligned} -\int _0^\infty \!\!\int _{\Omega }\!c\varphi _t-\int _{\Omega }\!c_0\varphi (\cdot ,0)=-\int _0^\infty \!\!\int _{\Omega }\!\nabla c\cdot \nabla \varphi -\int _0^\infty \!\!\int _{\Omega }\!nc\varphi +\int _0^\infty \!\!\int _{\Omega }\!c (u\cdot \nabla \varphi ) \end{aligned}$$

(4.12)

is valid for all $\varphi \in C_0^\infty \big (\Omega \times [0,\infty )\big )$, and that

$$\begin{aligned} -\int _0^\infty \!\!\int _{\Omega }\!u\cdot \psi _t-\int _{\Omega }\!u_0\cdot \psi (\cdot ,0)=&-\int _0^\infty \!\!\int _{\Omega }\!\nabla u\cdot \nabla \psi +\int _0^\infty \!\!\int _{\Omega }\!(u \otimes u)\cdot \nabla \psi \nonumber \\&+\int _0^\infty \!\!\int _{\Omega }\!n(\nabla \phi \cdot \psi ) \end{aligned}$$

(4.13)

for all $\psi \in C_{0}^\infty \big (\Omega \times [0,\infty );{\mathbb {R}}^{\mathcal {N}}\big )$ satisfying $\nabla \cdot \psi =0$.

The precise specifications of a global weak solution at hand, we can now draw on the regularity and convergence information provided by Proposition 4.1 to verify that the obtained limit object indeed fulfills the desired integral equalities.

Proof of Theorem 1.1

First, we choose test-functions $\varphi \in C_0^\infty \big ({\overline{\Omega }}\times [0,\infty )\big )$, ${\tilde{\varphi }}\in C_0^\infty \big (\Omega \times [0,\infty )\big )$ and $\psi \in C_{0,\sigma }^\infty \big ({\overline{\Omega }}\times [0,\infty );{\mathbb {R}}^{\mathcal {N}}\big )$ and then fix $T>0$ such that ${{\varphi }}\equiv 0$, $\tilde{\varphi }\equiv 0$ and $\psi \equiv 0$ in $\Omega \times (T,\infty )$. Testing the equations of system (2.1) against $\varphi ,{\tilde{\varphi }},\psi$, respectively, we can then rely on the convergence properties provided by Proposition 4.1 to pass to the limit in the integral identities corresponding to (4.11)–(4.13) (compare e.g. [49, Lemma 4.1] and the proof of [4, Theorem 1.1].) and verify that indeed the limit (n, c, u) is a weak solution to system (1.2). $\square$

5 Eventual smallness of certain quantities

For the remainder of the work we will now consider a slightly more restrictive setting, in which we assume $c_\star \ge 0$ to not only be constant in time but also constant in space. Moreover, the second important assumption, necessary to get most of our analytical machinery to work, will consist of assuming that the degradation in the term $+\kappa n_\varepsilon -\mu n_\varepsilon ^2$ of the first equation of (1.2) is strong enough in the sense that $\frac{\kappa }{\mu }$ is sufficiently small. (Refer back to (1.6) for the precise statement.)

5.1 Waiting times and smallness properties for the bacteria and fluid components

We start with a refinement of the global bounds of Lemma 2.2 for large times, removing the dependence on initial data from the bounds. (See also [53, Lemma 2.1 ii)].)

Lemma 5.1

Let $\kappa \ge 0$, $\mu >0$ and $c_\star \ge 0$. If ${\hat{\kappa }}>0$ satisfies ${\hat{\kappa }}\ge \kappa$, then for each $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) satisfies

$$\begin{aligned} \int _{\Omega }\!n_\varepsilon (\cdot ,t)\le 2|\Omega |\frac{{\hat{\kappa }}}{\mu }\qquad \text {and}\qquad \int _{t}^{t+1}\!\!\int _{\Omega }\!n_\varepsilon ^2\le 2|\Omega |\frac{{\hat{\kappa }}({\hat{\kappa }}+1)}{\mu ^2}\qquad \text {for all }t\ge \frac{\ln (2)}{{\hat{\kappa }}}. \end{aligned}$$

(5.1)

Proof

Starting similarly to Lemma 2.2, given ${\hat{\kappa }}>0$ satisfying ${\hat{\kappa }}\ge \kappa$ we integrate (2.1a) over $\Omega$ to derive

$$\begin{aligned} \frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!n_\varepsilon =\kappa \int _{\Omega }\!n_\varepsilon -\mu \int _{\Omega }\!n_\varepsilon ^2\le {\hat{\kappa }}\int _{\Omega }\!n_\varepsilon -\mu \int _{\Omega }\!n_\varepsilon ^2\quad \text {for all }t>0\text { and }\varepsilon \in (0,1). \end{aligned}$$

(5.2)

In light of the Cauchy–Schwarz inequality we have $(\int _{\Omega }\!n_\varepsilon )^2\le |\Omega |\int _{\Omega }\!n_\varepsilon ^2$ and hence $y_\varepsilon (t):=\int _{\Omega }\!n_\varepsilon (\cdot ,t)$, $t\ge 0$, $\varepsilon \in (0,1)$ satisfies the Bernoulli-type ODE

$$\begin{aligned} y_\varepsilon '(t)\le {\hat{\kappa }} y_\varepsilon (t)-\frac{\mu }{|\Omega |}y_\varepsilon ^2(t)\quad \text {for all }t>0\text { and }\varepsilon \in (0,1), \end{aligned}$$

which, due to the nonnegativity of $n_\varepsilon$, entails that

$$\begin{aligned} y_\varepsilon (t)&\le \Big (\frac{\mu }{{\hat{\kappa }}|\Omega |}\big (1-e^{-{\hat{\kappa }} t}\big )+\frac{1}{y_\varepsilon (0)}e^{-{\hat{\kappa }}t}\Big )^{-1}\\&\le \Big (\frac{\mu }{{\hat{\kappa }}|\Omega |}\big (1-e^{-{\hat{\kappa }} t}\big )\Big )^{-1}\quad \text {for all }t>0\text { and }\varepsilon \in (0,1). \end{aligned}$$

Hence, noticing that $(1-e^{-{\hat{\kappa }} t})\ge \frac{1}{2}$ for $t\ge \frac{\ln (2)}{{\hat{\kappa }}}$ we may easily confirm that the first assertion in (5.1) is true. To verify the second assertion we integrate (5.2) in time and use the previously obtained bound on $\int _{\Omega }\!n_\varepsilon$ for large times, to obtain that

$$\begin{aligned} \int _{\Omega }\!n_\varepsilon (\cdot ,t+1)+\mu \!\int _{t}^{t+1}\!\!\int _{\Omega }\!n_\varepsilon ^2\le&\int _{\Omega }\!n_\varepsilon (\cdot ,t)+{\hat{\kappa }}\!\int _{t}^{t+1}\!\!\int _{\Omega }\!n_\varepsilon \le 2|\Omega |\frac{{\hat{\kappa }}}{\mu }\\&+{\hat{\kappa }}\Big (2|\Omega | \frac{{\hat{\kappa }}}{\mu }\Big )=2|\Omega |\frac{{\hat{\kappa }}({\hat{\kappa }}+1)}{\mu } \end{aligned}$$

for all $t\ge \frac{\ln (2)}{{\hat{\kappa }}}$ and $\varepsilon \in (0,1)$. Drawing once more on the nonnegativity of $n_\varepsilon$ and dividing by $\mu >0$, the conclusion of the second part of (5.1) is also immediate. $\square$

The following result on eventual smallness of temporally averaged $L^p$ norms of $n_\varepsilon$ for p slightly beyond $\frac{{\mathcal {N}}+6}{6}$ is a direct consequence from the above and can be attained by suitable use of Hölder interpolations. It is also the main reason for the particular shape of the condition in (1.6). The proof is quite similar to [53, Lemma 2.2], but since it is so indispensible for the assumptions imposed in Theorem 1.2 we provide a detailed proof either way.

Lemma 5.2

Let $\omega >0$ and $\mu _0>0$. There is $\theta _{\diamond }^{(1)}=\theta _{\diamond }^{(1)}(\omega )\in (0,\min \{1-\frac{{\mathcal {N}}}{6},\omega \})$ with the property: For all $\delta >0$ there is $\eta _{\diamond }^{(1)}=\eta _{\diamond }^{(1)}(\omega ,\delta )>0$ such that whenever $\kappa \ge 0$ and $\mu \ge \mu _0$ satisfy (1.6) for some $\eta <\eta _{\diamond }^{(1)}(\omega ,\delta )$, there exists $t_{\diamond }^{(1)}=t_{\diamond }^{(1)}(\omega ,\mu _0,\delta ,\eta )>0$ such that for each $p\in \big [\frac{{\mathcal {N}}+6}{6},\frac{{\mathcal {N}}+6}{6}+\theta _{\diamond }^{(1)}\big ]$, $c_\star \ge 0$ and all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) fulfills

$$\begin{aligned} \int _{t}^{t+1}\!\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s< \delta \quad \text {for all }t\ge t_{\diamond }^{(1)}. \end{aligned}$$

Proof

For fixed $\omega >0$ we pick $\theta _1=\theta _{\diamond }^{(1)}(\omega )\in (0,\min \{1-\frac{{\mathcal {N}}}{6},\omega \})$ and set

$$\begin{aligned} a=a(\omega ):=\frac{2({\mathcal {N}}+6\theta _1)}{{\mathcal {N}}+6+6\theta _1}\in (0,1)\quad \text {and}\quad C_1=C_1(\omega ):=\max \big \{1,|\Omega |^{2\theta _1}\big \} \end{aligned}$$

(5.3)

and for given $\delta >0$ we choose $\eta _1=\eta _{\diamond }^{(1)}(\omega ,\delta )>0$ satisfying

$$\begin{aligned} \eta _1\le 1\quad \text {and}\quad 4|\Omega |^{2-a}\eta _1^{2-a}<\frac{\delta }{C_1}. \end{aligned}$$

(5.4)

Then, assuming that $\kappa \ge 0$ and $\mu >0$ satisfy (1.6) for some $\eta <\eta _1$ we define

$$\begin{aligned} {\hat{\kappa }}={\hat{\kappa }}(\omega ,\mu ,\delta ,\eta ):=\eta \min \left\{ \mu ,\mu ^{\frac{{\mathcal {N}}+6}{6}+\omega }\right\} \quad \text {and}\quad t_0=t_0(\omega ,\mu ,\delta ,\eta ):=\frac{\ln (2)}{{\hat{\kappa }}}, \end{aligned}$$

(5.5)

so that ${\hat{\kappa }}$ satisfies ${\hat{\kappa }}>\kappa$ and ${\hat{\kappa }}>0$ and thus an application of Lemma 5.1 shows

$$\begin{aligned} \int _{\Omega }\!n_\varepsilon (\cdot ,t)\le 2|\Omega |\frac{{\hat{\kappa }}}{\mu }\quad \text {for all }t\ge t_0\text { and }\varepsilon \in (0,1) \end{aligned}$$

(5.6)

as well as

$$\begin{aligned} \int _{t}^{t+1}\!\!\int _{\Omega }\!n_\varepsilon ^2\le 2|\Omega |\frac{{\hat{\kappa }}({\hat{\kappa }}+1)}{\mu ^2}\quad \text {for all }t\ge t_0\text { and }\varepsilon \in (0,1). \end{aligned}$$

(5.7)

Therefore, since $\frac{{\mathcal {N}}}{6}+1+\theta _1<2$ an interpolation between Lebesgue spaces combined with (5.6) entails that

$$\begin{aligned} \int _t^{t+1}\!\!\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{\frac{{\mathcal {N}}}{6}+1+\theta _1}(\Omega )}^2{{\,\mathrm{d\!}\,}}s&\le \!\int _t^{t+1}\!\!\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{2}(\Omega )}^{2a}\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{1}(\Omega )}^{2(1-a)}{{\,\mathrm{d\!}\,}}s\\&\le \left( 2|\Omega |\frac{{\hat{\kappa }}}{\mu }\right) ^{2(1-a)}\!\!\int _t^{t+1}\!\!\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{2}(\Omega )}^{2a}{{\,\mathrm{d\!}\,}}s \end{aligned}$$

for all $t\ge t_0$ and $\varepsilon \in (0,1)$. Moreover, as $\theta _1<1-\frac{{\mathcal {N}}}{6}$ implies $a<1$, again drawing on Hölder’s inequality and (5.7) we find that

$$\begin{aligned} \int _t^{t+1}\!\!\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{\frac{{\mathcal {N}}}{6}+1+\theta _1}(\Omega )}^2{{\,\mathrm{d\!}\,}}s&\le \left( 2|\Omega |\frac{{\hat{\kappa }}}{\mu }\right) ^{2(1-a)}\left( \int _t^{t+1}\!\!\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{2}(\Omega )}^2{{\,\mathrm{d\!}\,}}s\right) ^a\\&\le \left( 2|\Omega |\frac{{\hat{\kappa }}}{\mu }\right) ^{2(1-a)}\left( 2|\Omega |\frac{{\hat{\kappa }}({\hat{\kappa }}+1)}{\mu ^2}\right) ^a\\&=(2|\Omega |)^{2-a}\frac{{\hat{\kappa }}^{2-a}({\hat{\kappa }}+1)^a}{\mu ^2} \end{aligned}$$

for all $t\ge t_0$ and $\varepsilon \in (0,1)$. Now consider the following three cases. If $\mu \le 1$, then, according to (5.5), ${\hat{\kappa }}=\eta \mu ^{\frac{{\mathcal {N}}}{6}+1+\omega }$ and hence ${\hat{\kappa }}+1=\eta \mu ^{\frac{{\mathcal {N}}}{6}+1+\omega }+1\le \eta +1\le 2$ due to the restriction $\eta _1\le 1$ in (5.4). Thus, noticing that by choice of $\theta _1<\omega$ we have

$$\begin{aligned} \big (\tfrac{{\mathcal {N}}}{6}+1+\omega \big )(2-a)=2\cdot \frac{{\mathcal {N}}+6+6\omega }{{\mathcal {N}}+6+6\theta _1}>2, \end{aligned}$$

we obtain from $\mu \le 1$ and the second inequality in (5.4) that

$$\begin{aligned}&(2|\Omega |)^{2-a}\frac{{\hat{\kappa }}^{2-a}({\hat{\kappa }}+1)^a}{\mu ^2} \le (2|\Omega |)^{2-a}\frac{(\eta \mu ^{\frac{{\mathcal {N}}}{6}+1+\omega })^{2-a}2^a}{\mu ^2}\\&\quad =4|\Omega |^{2-a}\eta ^{2-a}\mu ^{(\frac{{\mathcal {N}}}{6}+1+\omega )(2-a)-2}<\frac{\delta }{C_1}. \end{aligned}$$

For the second case consider ${\hat{\kappa }}\le 1$ and $\mu >1$, i.e. we have ${\hat{\kappa }}=\eta \mu$ in (5.5). Here, also ${\hat{\kappa }}+1\le 2$, so that again (5.4) entails

$$\begin{aligned} (2|\Omega |)^{2-a}\frac{{\hat{\kappa }}^{2-a}({\hat{\kappa }}+1)^a}{\mu ^2}\le 4|\Omega |^{2-a}\frac{(\eta \mu )^{2-a}}{\mu ^2}=4|\Omega |^{2-a}\eta ^{2-a}\mu ^{-a}<\frac{\delta }{C_1} \end{aligned}$$

in light of $\mu ^{-a}\le 1$.

In the third case assume ${\hat{\kappa }}>1$ and $\mu >1$. Then, ${\hat{\kappa }}+1\le 2{\hat{\kappa }}$ and hence

$$\begin{aligned} (2|\Omega |)^{2-a}\frac{{\hat{\kappa }}^{2-a}({\hat{\kappa }}+1)^a}{\mu ^2}\le 4|\Omega |^{2-a}\frac{{\hat{\kappa }}^{2-a+a}}{\mu ^2}=4|\Omega |^{2-a}\eta ^2\le 4|\Omega |^{2-a}\eta ^{2-a}<\frac{\delta }{C_1}, \end{aligned}$$

where we once more relied on both of the restrictions imposed on $\eta$ in (5.4). In conclusion, all of the cases lead to

$$\begin{aligned} \int _t^{t+1}\!\!\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{\frac{{\mathcal {N}}+6}{6}+\theta _1}(\Omega )}^2{{\,\mathrm{d\!}\,}}s \le \frac{\delta }{C_1}\quad \text {for all }t\ge t_0\text { and }\varepsilon \in (0,1). \end{aligned}$$

Since $C_1\ge 1$ and $t_0$ is increases with decreasing values of $\mu$ we can make use of the assumption $\mu \ge \mu _0$ and set $t_{\diamond }^{(1)}(\omega ,\mu _0,\delta ,\eta ):=t_0(\omega ,\mu _0,\delta ,\eta )$ to obtain the asserted inequality for $p=p_0:=\frac{{\mathcal {N}}}{6}+1+\theta _1$. If $p\in \big [\frac{{\mathcal {N}}}{6}+1,p_0\big )$ is arbitrary, an application of Hölder’s inequality leads to the desired result since then

$$\begin{aligned} \int _t^{t+1}\!\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s\le |\Omega |^{\frac{2(p_0-p)}{p_0 p}}\int _t^{t+1}\!\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{p_0}(\Omega )}^2{{\,\mathrm{d\!}\,}}s \le |\Omega |^{\frac{2(p_0-p)}{p_0 p}}\frac{\delta }{C_1}\quad \end{aligned}$$

for all $t\ge t_{\diamond }^{(1)}$ and $\varepsilon \in (0,1)$, where

$$\begin{aligned} \frac{2(p_0-p)}{p_0 p}=\frac{2\theta _1}{p_0p}\le \frac{2\theta _1}{(\frac{{\mathcal {N}}}{6}+1)^2}< 2\theta _1 \end{aligned}$$

allows for the estimation of $|\Omega |^\frac{2(p_0-p)}{p_0 p}\le \max \big \{1,|\Omega |^{2\theta _1}\big \}=C_1$. $\square$

Having a first authentic smallness property for $n_\varepsilon$ at hand, we can return to the differential inequality for $u_\varepsilon$ provided by Lemma 3.2 and the comparison Lemma 3.1 to obtain a large-time estimate of similar character for $\nabla u_\varepsilon$. (Compare [53, Lemma 3.2].)

Lemma 5.3

Let $\omega >0$ and $\mu _0>0$. For $\delta >0$ there is $\eta _{\diamond }^{(2)}=\eta _{\diamond }^{(2)}(\omega ,\delta )>0$ with the following property: Whenever $\kappa \ge 0$ and $\mu \ge \mu _0$ satisfy (1.6) for some $\eta <\eta _{\diamond }^{(2)}(\omega ,\delta )$, there exsits $t_{\diamond }^{(2)}=t_{\diamond }^{(2)}(\omega ,\mu _0,\delta ,\eta ,n_0,u_0)>0$ such that for each $c_\star \ge 0$ and all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) fulfills

$$\begin{aligned} \int _t^{t+1}\Vert \nabla u_\varepsilon (\cdot ,s)\Vert _{L^{2}(\Omega )}^2{{\,\mathrm{d\!}\,}}s< \delta \quad \text {for all }t\ge t_{\diamond }^{(2)}. \end{aligned}$$

Proof

In order to prepare later estimates let us draw on Lemma 3.2 to introduce constants $R_1>0$ and $R_2>0$ such that

$$\begin{aligned} \frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!|u_\varepsilon (\cdot ,t)|^2+\frac{R_1}{2}\int _{\Omega }\!|u_\varepsilon (\cdot ,t)|^2+\frac{1}{2}\int _{\Omega }\!|\nabla u_\varepsilon (\cdot ,t)|^2\le R_2\Vert n_\varepsilon (\cdot ,t)\Vert _{L^{\frac{6}{5}}(\Omega )}^2 \end{aligned}$$

(5.8)

is valid for all $t>0$, $\kappa \ge 0$, $\mu >0$ and $\varepsilon \in (0,1)$. Now, provided $\omega >0$ we take $\theta _1=\theta _{\diamond }^{(1)}(\omega )\in (0,1-\frac{{\mathcal {N}}}{6})$ from Lemma 5.2 and accordingly denote by $p=p(\omega ):=\frac{{\mathcal {N}}+6}{6}+\theta _1$ the largest admissible parameter, for which said Lemma 5.2 establishes an eventual smallness property. Then, $p>\frac{6}{5}$ and given any $\delta >0$ we pick $\delta _1(\omega ,\delta )>0$ such that both

$$\begin{aligned} \frac{R_2|\Omega |^\frac{5p-6}{3p}\delta _1}{1-e^{\frac{-R_1}{2}}}<\frac{\delta }{8}\quad \text {and}\quad R_2|\Omega |^\frac{5p-6}{3p}\delta _1<\frac{\delta }{4} \end{aligned}$$

(5.9)

are valid and let $\eta _1=\eta _1(\omega ,\delta ):=\eta _{\diamond }^{(1)}(\omega ,\delta _1)$ be the corresponding number provided by Lemma 5.2. Supposing that $\kappa \ge 0$ and $\mu \ge \mu _0>0$ satisfy (1.6) for some $\eta <\eta _1$ we then set $t_1=t_1(\omega ,\mu _0,\delta ,\eta ):=t_{\diamond }^{(1)}(\omega ,\mu _0,\delta _1,\eta )>0$, as provided by Lemma 5.2, such that

$$\begin{aligned} \int _t^{t+1}\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s<\delta _1\quad \text {for all }t\ge t_1\text { and }\varepsilon \in (0,1). \end{aligned}$$

(5.10)

Hence, letting $y_\varepsilon (t):=\int _{\Omega }\!|u_\varepsilon (\cdot ,t)|^2{{\,\mathrm{d\!}\,}}x$ and $h_\varepsilon (t):=R_2\Vert n_\varepsilon (\cdot ,t)\Vert _{L^{\frac{6}{5}}(\Omega )}^2$ for $t\ge 0$ and $\varepsilon \in (0,1)$, we find from (5.8) that

$$\begin{aligned} y_{\varepsilon }'(t)+\frac{R_1}{2}y_{\varepsilon }(t)+\frac{1}{2}\int _{\Omega }\!|\nabla u_\varepsilon |^2\le h_\varepsilon (t)\quad \text {for all }t>0\text { and }\varepsilon \in (0,1). \end{aligned}$$

(5.11)

Moreover, since in view of (1.6) we have $\frac{\kappa }{\mu }\le \eta$, we find that letting

$$\begin{aligned} M_0:=\max \Big \{\int _{\Omega }\!n_0,\frac{\kappa |\Omega |}{\mu }\Big \},\quad R_3:=R_2|\Omega |^\frac{2}{3}\frac{(\kappa +1)M_0}{\mu },\quad R_4:=\int _{\Omega }\!|u_0|^2+\frac{R_3}{1-e^{-\frac{R_1}{2}}}, \end{aligned}$$

we also have $M_0\le \int _{\Omega }\!n_0+\eta |\Omega |=:M_1(\eta ,n_0)$ and, additionaly using $\mu \ge \mu _0$, $R_3\le R_2|\Omega |^\frac{2}{3}(\eta +\frac{1}{\mu _0})M_1$. Drawing on Lemma 3.2 once more, thereby entails that

$$\begin{aligned} y_\varepsilon (t)=\int _{\Omega }\!|u_\varepsilon (\cdot ,t)|^2\le R_4\le \int _{\Omega }\!|u_0|^2+\frac{R_2|\Omega |^\frac{2}{3}(\eta +\frac{1}{\mu _0})M_1}{\big (1-e^{-\frac{R_1}{2}}\big )}=: R_5(\mu _0,\eta ,n_0,u_0) \end{aligned}$$

(5.12)

for all $t>0$ and $\varepsilon \in (0,1)$. We can then draw on Lemma 3.1 and the eventual smallness of $\int _t^{t+1}h_\varepsilon (s){{\,\mathrm{d\!}\,}}s$ provided by (5.10) to refine the estimate on $y_\varepsilon$ for large times. In fact, due to Hölder’s inequality and (5.10) we have

$$\begin{aligned}&\int _t^{t+1}h_\varepsilon (s){{\,\mathrm{d\!}\,}}s\\&\quad \le R_2|\Omega |^{\frac{5p-6}{3p}}\int _t^{t+1}\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s\le R_2|\Omega |^{\frac{5p-6}{3p}}\delta _1\quad \text {for all }t\ge t_1\text { and }\varepsilon \in (0,1). \end{aligned}$$

Therefore, employing Lemma 3.1 to (5.11) for $t_0=t_1$ we obtain

$$\begin{aligned} y_\varepsilon (t)\le e^{-\frac{R_1}{2}(t-t_1)}y_\varepsilon (t_1)+\frac{R_2|\Omega |^\frac{5p-6}{3p}\delta _1}{1-e^{-\frac{R_1}{2}}}\quad \text {for all }t\ge t_1\text { and }\varepsilon \in (0,1), \end{aligned}$$

(5.13)

featuring terms which are either controlled exponentially in time or by our choice of $\delta _1$. Hence, choosing $t_2=t_2(\omega ,\mu _0,\delta ,\eta ,n_0,u_0)\ge t_1$ so large that

$$\begin{aligned} e^{-\frac{R_1}{2}(t_2-t_1)}R_5<\frac{\delta }{8}, \end{aligned}$$

we find that in light of (5.13), (5.12) and (5.9)

$$\begin{aligned} y_\varepsilon (t)\le e^{-\frac{R_1}{2}(t_2-t_1)}R_5+\frac{R_2|\Omega |^\frac{5p-6}{3p}\delta _1}{1-e^{-\frac{R_1}{2}}}<\frac{\delta }{8}+\frac{\delta }{8}=\frac{\delta }{4}\quad \text {for all }t\ge t_2\text { and }\varepsilon \in (0,1). \end{aligned}$$

Therefore, integrating the differential inequality (5.11) in time yields

$$\begin{aligned} \frac{1}{2}\int _t^{t+1}\int _{\Omega }\!|\nabla u_\varepsilon |^2&\le y_\varepsilon (t)+\int _t^{t+1}h_\varepsilon (s){{\,\mathrm{d\!}\,}}s\le \frac{\delta }{4}+R_2|\Omega |^{\frac{5p-6}{3p}}\delta _1<\frac{\delta }{4}+\frac{\delta }{4}=\frac{\delta }{2} \end{aligned}$$

for all $t\ge t_2$ and $\varepsilon \in (0,1)$, which evidently concludes the proof upon letting $t_{\diamond }^{(2)}=t_2$. $\square$

Making use of well-known embedding properties of the domain $D(A_2^\varrho )$ of the Stokes operator and corresponding smoothing properties of the Stokes semigroup, we can choose $\delta >0$ appropriately in the previous two lemmas to extend the eventual smallness results to the two important quantities below. The proof adjusts the arguments of [53, Lemma 3.3] to cover both ${\mathcal {N}}\in \{2,3\}$. We only provide the main steps and direct the reader to the referenced work for additional details.

Lemma 5.4

Let $\omega >0$ and $\mu _0>0$. There is $\theta _{\diamond }^{(3)}=\theta _{\diamond }^{(3)}(\omega )>0$ such that for any $\delta >0$ one can find $\eta _{\diamond }^{(3)}=\eta _{\diamond }^{(3)}(\omega ,\delta )>0$ such that whenever $\kappa \ge 0$ and $\mu \ge \mu _0$ satisfy (1.6) for some $\eta <\eta _{\diamond }^{(3)}(\omega ,\delta )$, there exsits $t_{\diamond }^{(3)}=t_{\diamond }^{(3)}(\omega ,\mu _0,\delta ,\eta ,n_0,u_0)>0$ such that for any $\varrho \in \big [0,\frac{{\mathcal {N}}-2}{4}+\theta _{\diamond }^{(3)}\big ]$, $r\in [{\mathcal {N}},{\mathcal {N}}+\theta _{\diamond }^{(3)}]$, each $c_\star \ge 0$ and all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) fulfills

$$\begin{aligned} \Vert A^\varrho u_\varepsilon (\cdot ,t)\Vert _{L^{2}(\Omega )}<\delta \qquad \text {and}\qquad \Vert u_\varepsilon (\cdot ,t)\Vert _{L^{r}(\Omega )}< \delta \quad \text {for all }t\ge t_{\diamond }^{(3)}. \end{aligned}$$

Proof

Given $\omega >0$ we denote by $\theta _1=\theta _1(\omega ):=\theta _{\diamond }^{(1)}(\omega )\in \left( 0,1-\frac{{\mathcal {N}}}{6}\right)$ the number from Lemma 5.2 and correspondingly define $p=p(\omega ):=\frac{{\mathcal {N}}+6}{6}+\min \{\theta _1,\omega \}$, so that p especially satisfies $\frac{{\mathcal {N}}}{2}<p<2$. Additionally, this choice for p entails that

$$\begin{aligned} \frac{{\mathcal {N}}}{2}-\frac{{\mathcal {N}}}{2p}<1+\frac{{\mathcal {N}}}{2}-\frac{{\mathcal {N}}}{p},\qquad \frac{{\mathcal {N}}}{2}-\frac{{\mathcal {N}}}{2p}<1\qquad \text {and}\qquad \frac{{\mathcal {N}}-2}{2}<1+\frac{{\mathcal {N}}}{2}-\frac{{\mathcal {N}}}{p}, \end{aligned}$$

so that we can fix $\varrho _\circ =\varrho _\circ (\omega )\in \big (\frac{{\mathcal {N}}-2}{4},\frac{1}{2}\big )$ such that

$$\begin{aligned} \frac{{\mathcal {N}}}{2}-\frac{{\mathcal {N}}}{2p}<2\varrho _\circ <1+\frac{{\mathcal {N}}}{2}-\frac{{\mathcal {N}}}{p} \end{aligned}$$

and afterwards $\theta _3=\theta _{\diamond }^{(3)}(\omega )>0$ such that still

$$\begin{aligned} \frac{{\mathcal {N}}-2}{4}+\theta _3\le \varrho _\circ \qquad \text {and}\qquad {\mathcal {N}}+\theta _3\le 2p. \end{aligned}$$

With these parameters fixed, we can draw on well-established embedding properties for fractional powers of the Stokes operator A [15, 20, 34] to conclude the existence of $C_i=C_i(\omega )>0$, $i\in \{1,\dots ,4\}$ such that

$$\begin{aligned} \big \Vert A^{-\frac{1-2\varrho _\circ }{2}}\varphi \Vert _{L^{2}(\Omega )}&\le C_1\Vert \varphi \Vert _{L^{p}(\Omega )}&\text {for all }\varphi \in L_\sigma ^{p}(\Omega ),\\ \Vert \varphi \Vert _{L^{2p}(\Omega )}&\le C_2\Vert A^{\varrho _\circ }\varphi \Vert _{L^{2}(\Omega )}&\text {for all }\varphi \in D(A^{\varrho _\circ }),\\ \Vert \nabla \varphi \Vert _{L^{2p}(\Omega )}&\le C_3\big \Vert A^{\frac{1+2\varrho _\circ }{2}}\varphi \big \Vert _{L^{2}(\Omega )}\qquad&\text {for all }\varphi \in D\big (A^{\frac{1+2\varrho _\circ }{2}}\big ) \end{aligned}$$

and, recalling ${\mathcal {N}}+\theta _3\le 2p$, such that

$$\begin{aligned} \Vert \varphi \Vert _{L^{{\mathcal {N}}+\theta _3}(\Omega )}&\le C_4\Vert A^{\varrho _\circ }\varphi \Vert _{L^{2}(\Omega )}\qquad&\text {for all }\varphi \in D(A^{\varrho _\circ }). \end{aligned}$$

(5.14)

Furthermore, using $\varrho \le \varrho _\circ <\frac{1}{2}$ one can also find $C_i=C_i(\omega )>0$, $i\in \{5,\dots ,8\}$ satisfying

$$\begin{aligned} C_5 \Vert A^{\varrho _\circ }\varphi \Vert _{L^{2}(\Omega )}&\le \big \Vert A^{\frac{1+2\varrho _\circ }{2}}\varphi \big \Vert _{L^{2}(\Omega )}\qquad&\text {for all }\varphi \in D\big (A^{\frac{1+2\varrho _\circ }{2}}\big ), \end{aligned}$$

(5.15)

$$\begin{aligned} \Vert A^{\varrho _\circ }\varphi \Vert _{L^{2}(\Omega )}&\le C_6\Vert \nabla \varphi \Vert _{L^{2}(\Omega )}\qquad&\text {for all }\varphi \in D\big (A^{\frac{1}{2}}\big ), \end{aligned}$$

(5.16)

$$\begin{aligned} \Vert A^{\varrho }\varphi \Vert _{L^{2}(\Omega )}&\le C_7\Vert A^{\varrho _\circ }\varphi \Vert _{L^{2}(\Omega )}\qquad&\text {for all }\varphi \in D\big (A^{\varrho _\circ }\big ), \end{aligned}$$

(5.17)

and, since the Helmholtz projection ${\mathcal {P}}$ is bounded on $L^{p\;\!}\!\left( \Omega ;{\mathbb {R}}^{\mathcal {N}}\right)$, such that

$$\begin{aligned} \Vert {\mathcal {P}}\varphi \Vert _{L^{p}(\Omega )}&\le C_8\Vert \varphi \Vert _{L^{p}(\Omega )}\qquad&\text {for all }\varphi \in L^{p\;\!}\!\left( \Omega ;{\mathbb {R}}^{\mathcal {N}}\right) . \end{aligned}$$

Additionally, we set

$$\begin{aligned} C_9=C_9(\omega )&:=2C_1^2C_2^2C_3^2C_8^2,\qquad&C_{10}:=\Vert \nabla \phi \Vert _{L^{\infty }(\Omega )},\nonumber \\ C_{11}=C_{11}(\omega )&:=2C_1^2C_8^2C_{10}^2\text {and}\quad&C_{12}:=\max \Big \{1,|\Omega |^{\frac{\theta _3}{{\mathcal {N}}^2}}\Big \}. \end{aligned}$$

(5.18)

In agreement with these constants and given some $\delta >0$ we pick $\delta _1(\delta )>0$ such that

$$\begin{aligned} C_4\sqrt{\delta _1}<\frac{\delta }{C_{12}}\ \quad \text {and}\quad \ \delta _1\le \frac{1}{8C_9}\ \quad \ \text {as well as }\ \quad C_7\sqrt{\delta _1}\le \delta \end{aligned}$$

(5.19)

hold, whereafter we fix $\delta _i(\delta )$, $i\in \{2,3,4\}$ satisfying

$$\begin{aligned} \frac{\delta _2}{1-e^{-\frac{C_5^2}{4}}}<\frac{\delta _1}{2}\quad \text { and }\quad C_6^2\delta _3<\frac{\delta _1}{2}\quad \text { as well as }\quad C_{11}\delta _4<\delta _2. \end{aligned}$$

(5.20)

We then let $\eta _1=\eta _1(\omega ,\delta ):=\eta _{\diamond }^{(1)}(\omega ,\delta _4)$ and $\eta _2=\eta _2(\omega ,\delta ):=\eta _{\diamond }^{(2)}(\omega ,\delta _3)$ be provided by Lemmas 5.2 and 5.3, respectively, and introduce $\eta _3=\eta _{\diamond }^{(3)}(\omega ,\delta ):=\min \{\eta _1,\eta _2\}$.

Now, assuming that $\kappa \ge 0$ and $\mu \ge \mu _0>0$ satisfy (1.6) for some $\eta <\eta _3$ we infer from Lemma 5.2 and the fact that $p\le \frac{{\mathcal {N}}+6}{6}+\theta _1$ the existence of $t_1=t_1(\omega ,\mu _0,\delta ,\eta ):=t_{\diamond }^{(1)}(\omega ,\mu _0,\delta _4,\eta )>0$ such that

$$\begin{aligned} \int _t^{t+1}\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s<\delta _4\quad \text {for all }t\ge t_1\text { and }\varepsilon \in (0,1) \end{aligned}$$

(5.21)

and with $t_2=t_2(\omega ,\mu _0,\delta ,\eta ,n_0,u_0):=t_{\diamond }^{(2)}(\omega ,\mu _0,\delta _3,\eta ,n_0,u_0)>0$ taken from Lemma 5.3, we moreover conclude that

$$\begin{aligned} \int _{t}^{t+1}\!\Vert \nabla u_\varepsilon (\cdot ,s)\Vert _{L^{2}(\Omega )}^2{{\,\mathrm{d\!}\,}}s < \delta _3\quad \text {for all }\varepsilon \in (0,1)\text { and }t\ge t_2. \end{aligned}$$

Hence, introducing $t_3=t_{\diamond }^{(3)}(\omega ,\mu _0,\delta ,\eta ,n_0,u_0):=\max \{t_1,t_2\}+1$, for each $\varepsilon \in (0,1)$ we can find $t_\varepsilon =t_\varepsilon (\omega ,\mu _0,\delta ,\eta ,n_0,u_0)\in (t_3-1,t_3)$ such that

$$\begin{aligned} \int _{\Omega }\!\big |A^{\varrho _\circ }u_\varepsilon (\cdot ,t_\varepsilon )\big |^2\le C_6^2\Vert \nabla u_\varepsilon (\cdot ,t_\varepsilon )\Vert _{L^{2}(\Omega )}^2<C_6^2\delta _3<\frac{\delta _1}{2}\le \frac{1}{4C_9}, \end{aligned}$$

(5.22)

where we made use of (5.16), (5.20) and (5.19). Thus, letting $y_\varepsilon (t):=\int _{\Omega }\!|A^{\varrho _\circ }u_\varepsilon (\cdot ,t)|^2$, $t\ge t_\varepsilon$ we find that for

$$\begin{aligned} S_{\varepsilon }:=\left\{ T_0>t_\varepsilon \ \Big |\ y_\varepsilon (t)\le \frac{1}{4C_9}\quad \text {for all }t\in [t_\varepsilon ,T_0)\right\} , \end{aligned}$$

the number $T_\varepsilon :=\sup S_{\varepsilon }$ is a well-defined with $T_\varepsilon \in (t_\varepsilon ,\infty ]$. To verify that in fact $T_\varepsilon =\infty$ for each $\varepsilon \in (0,1)$, we test (2.1c) against $A^{2\varrho _\circ }u_\varepsilon$ and proceed with estimation techniques often exercised in the context of chemotaxis-fluid system, presented with details in the proof of [53, Lemma 3.3] and in particular illustrating where the choices for the constants in (5.18) stem from, to obtain

$$\begin{aligned} \frac{1}{2}\frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!\big |A^{\varrho _\circ }u_\varepsilon (\cdot ,t)\big |^2+\frac{1}{8}\int _{\Omega }\!\big |A^{\frac{1+2\varrho _\circ }{2}}u_\varepsilon (\cdot ,t)\big |^2\le C_{11}\Vert n_\varepsilon (\cdot ,t)\Vert _{L^{p}(\Omega )}^2\quad \text {for all }t\in (t_\varepsilon ,T_\varepsilon ), \end{aligned}$$

so that with $h_\varepsilon (t):=2C_{11}\Vert n_\varepsilon (\cdot ,t)\Vert _{L^{p}(\Omega )}^2$ for $t>0$ and (5.15) we have

$$\begin{aligned} y_\varepsilon '(t)+\frac{C_5^2}{4}y_\varepsilon (t)\le h_\varepsilon (t)\quad \text {for all }t\in (t_\varepsilon ,T_\varepsilon ). \end{aligned}$$

Herein, in light of (5.21) and (5.20), $h_\varepsilon (t)$ satisfies

$$\begin{aligned} \int _t^{t+1}h_\varepsilon (s){{\,\mathrm{d\!}\,}}s\le C_{11}\delta _4<\delta _2\quad \text {for all }t>t_\varepsilon , \end{aligned}$$

and we can infer from Lemma 3.1 that

$$\begin{aligned} y_\varepsilon (t)\le e^{-\frac{C_5^2}{4}(t-t_\varepsilon )}y(t_\varepsilon )+\frac{\delta _2}{1-e^{-\frac{C_5^2}{4}}}\le \int _{\Omega }\!\big |A^{\varrho _\circ }u_\varepsilon (\cdot ,t_\varepsilon )\big |^2+\frac{\delta _2}{1-e^{-\frac{C_5^2}{4}}}<\frac{\delta _1}{2}+\frac{\delta _1}{2}=\delta _1 \end{aligned}$$

(5.23)

for all $t\in [t_\varepsilon ,T_\varepsilon )$, due to (5.22) and (5.20). By choice of $\delta _1$ in (5.19) we thereby obtain $y_\varepsilon (t)\le \frac{1}{8C_9}$ for all $t\in (t_\varepsilon ,T_\varepsilon )$, which by continuity of $y_\varepsilon$ and definition of $S_\varepsilon$ entails $T_\varepsilon =\infty$ for each $\varepsilon \in (0,1)$. Moreover, drawing once more on (5.23), (5.17) and (5.19), we obtain

$$\begin{aligned} \big \Vert A^\varrho u_\varepsilon (\cdot ,t)\big \Vert _{L^{2}(\Omega )}\le C_7\big \Vert A^{\varrho _\circ }u_\varepsilon (\cdot ,t)\Vert _{L^{2}(\Omega )}\le C_7\sqrt{\delta _1}\le \delta \quad \text {for all }t\ge t_\varepsilon , \end{aligned}$$

and, in view of (5.14) and (5.19), also

$$\begin{aligned} \Vert u_\varepsilon (\cdot ,t)\Vert _{L^{{\mathcal {N}}+\theta _3}(\Omega )}\le C_4\big \Vert A^{\varrho _\circ }u_\varepsilon (\cdot ,t)\Vert _{L^{2}(\Omega )}<C_4\sqrt{\delta _1}\le \frac{\delta }{C_{12}}\quad \text {for all }t\ge t_\varepsilon . \end{aligned}$$

(5.24)

In order to finally refine this to an estimate for arbitrary $r\in [{\mathcal {N}},{\mathcal {N}}+\theta _3]$, we employ Hölder’s inequality and rely on the fact that $0\le \frac{{\mathcal {N}}+\theta _3-r}{({\mathcal {N}}+\theta _3)r}\le \frac{\theta _3}{{\mathcal {N}}^2}$, (5.18) and (5.24) to estimate

$$\begin{aligned} \Vert u_\varepsilon (\cdot ,t)\Vert _{L^{r}(\Omega )}\le |\Omega |^{\frac{{\mathcal {N}}+\theta _3-r}{({\mathcal {N}}+\theta _3)r}}\Vert u_\varepsilon (\cdot ,t)\Vert _{L^{{\mathcal {N}}+\theta _3}(\Omega )}<C_{12}\frac{\delta }{C_{12}}=\delta \quad \text {for all }t\ge t_\varepsilon , \end{aligned}$$

so that, in accordance with $t_\varepsilon <t_3$ for each $\varepsilon \in (0,1)$, the asserted eventual smallness properties hold for all $t\ge t_3$. $\square$

5.2 Eventual stabilization properties of the signal chemical

Eventual smallness estimates for the signal chemical $c_\varepsilon$ can obviously not be expected. If, however, we focus our investigation to the difference-function ${\widehat{c}}_{\varepsilon }$ again (compare the proof of Lemma 3.4) some large time stabilization may not be too far out of reach. In fact, let us consider substituting ${\widehat{c}}_{\varepsilon }=c_\star -c_\varepsilon$ into (2.1). The corresponding family of transformed systems is then given by

where we note that the setting here is easier than in Lemma 3.4 due to $c_\star \ge 0$ being constant in space.

The first of the following two results provides eventual smallness for ${\widehat{c}}_{\varepsilon }$ in $L^p$ for arbitrary $p\ge 2$. This Lemma is the main reason as to why $\eta$ and $T_0$ in Theorem 1.2 depend on $c_\star$. While this dependence on $c_\star$ appears to be reasonably logical and without alternative, one has to undertake significant care with the estimation process to not also introduce a dependence on $c_0$ into $\eta$. The dependence of the waiting time on $c_0$, however, is quite natural and to be expected. Our proof is based on methods presented in [53, Lemma 4.1] and [3, Lemma 3.1].

Lemma 5.5

Let $\omega >0$, $\mu _0>0$, $c_\star \ge 0$ and $q\ge 1$. Then for any $\delta >0$ there is $\eta _{\diamond }^{(4)}=\eta _{\diamond }^{(4)}(\omega ,q,\delta ,c_\star )>0$ such that whenever $\kappa \ge 0$ and $\mu \ge \mu _0$ satisfy (1.6) for some $\eta <\eta _{\diamond }^{(4)}(\omega ,q,\delta ,c_\star )$, one can find $t_{\diamond }^{(4)}=t_{\diamond }^{(4)}(\omega ,\mu _0,q,\delta ,\eta ,c_0,c_\star )>0$ such that for all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (5.25) fulfills

$$\begin{aligned} \big \Vert \big (c_\star -c_\varepsilon \big )(\cdot ,t)\big \Vert _{L^{2q}(\Omega )}^2<\delta \quad \text {for all }t\ge t_{\diamond }^{(4)}. \end{aligned}$$

Proof

We first recall, that according to the Poincaré inequality there is some $C_1>0$ such that

$$\begin{aligned} C_1\int _{\Omega }\!|\varphi |^2\le \int _{\Omega }\!|\nabla \varphi |^2\quad \text {for all }\varphi \in W_0^{1,2}(\Omega ). \end{aligned}$$

(5.26)

Then, given $q\ge 1$ we pick $k=k(q)\in {\mathbb {N}}$ such that $q\in (k-1,k]$ and find from testing (5.25b) against ${\widehat{c}}_{\varepsilon }^{2k-1}$ and integrating by parts that for all $\varepsilon \in (0,1)$ we have

$$\begin{aligned} \frac{1}{2k}\frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!{{\widehat{c}}_{\varepsilon }}^{\,2k}+(2k-1)\int _{\Omega }\!{\widehat{c}}_{\varepsilon }^{\,2k-2}\big |\nabla {\widehat{c}}_{\varepsilon }\big |^2 =\int _{\Omega }\!g_\varepsilon (n_\varepsilon )c_\varepsilon {\widehat{c}}_{\varepsilon }^{\,2k-1}\quad \text {on }(0,\infty ), \end{aligned}$$

(5.27)

where the boundary integrals and the integral containing the fluid component disappear due to the fact that $u_\varepsilon$ is divergence free and that both $u_\varepsilon =0$ and ${\widehat{c}}_{\varepsilon }=0$ hold on $\partial \Omega$. Noticing that $2k-1$ is odd we divide the domain in areas essentially determined by the sign of ${\widehat{c}}_{\varepsilon }$ and infer from the nonnegativity of $c_\varepsilon$, $n_\varepsilon$ and $g_\varepsilon (s)$ for $s\ge 0$ and the expression ${\widehat{c}}_{\varepsilon }=c_\star -c_\varepsilon$ that

$$\begin{aligned} \int _{\Omega }\!g_\varepsilon (n_\varepsilon )c_\varepsilon {\widehat{c}}_{\varepsilon }^{\,2k-1}&=\int _{\{c_\varepsilon \le c_\star \}}\!g_\varepsilon (n_\varepsilon )c_\varepsilon (c_\star -c_\varepsilon )^{2k-1}\\&\quad +\int _{\{c_\varepsilon >c_\star \}}\!g_\varepsilon (n_\varepsilon ) c_\varepsilon (c_\star -c_\varepsilon )^{2k-1}\le c_\star ^{2k}\int _{\{c_\varepsilon \le c_\star \}}g_\varepsilon (n_\varepsilon ) \end{aligned}$$

on $(0,\infty )$ for all $\varepsilon \in (0,1)$, since $(c_\star -c_\varepsilon )^{2k-1}$ is always negative when $c_\varepsilon >c_\star$ due to $2k-1$ being odd. Then, we use the fact that $g_\varepsilon (s)\le s$ for all $s\ge 0$ and employ Hölder’s inequality, to obtain that with $p:=\frac{{\mathcal {N}}+6}{6}> 1$ for all $\varepsilon \in (0,1)$ we have

$$\begin{aligned} \int _{\Omega }\!g_\varepsilon (n_\varepsilon )c_\varepsilon {\widehat{c}}_{\varepsilon }^{\,2k-1}\le c_\star ^{2k}\int _{\Omega }\!g_\varepsilon (n_\varepsilon )\le c_\star ^{2k}|\Omega |^\frac{p-1}{p}\Vert n_\varepsilon \Vert _{L^{p}(\Omega )}\quad \text {on }(0,\infty ). \end{aligned}$$

(5.28)

Rewriting $(2k-1){\widehat{c}}_{\varepsilon }^{\,2k-2}|\nabla {\widehat{c}}_{\varepsilon }|^2=\frac{2k-1}{k^2}|\nabla {\widehat{c}}_{\varepsilon }^{\,k}|^2$ and estimating $1\le \frac{2k-1}{k}$ we find upon plugging (5.28) into (5.27) that

$$\begin{aligned} \frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!{\widehat{c}}_{\varepsilon }^{\,2k}(\cdot ,t)+2\int _{\Omega }\!\big |\nabla {\widehat{c}}_{\varepsilon }^{\,k}(\cdot ,t)\big |^2\le 2kc_\star ^{2k}|\Omega |^\frac{p-1}{p}\Vert n_\varepsilon (\cdot ,t)\Vert _{L^{p}(\Omega )}\quad \text {for all }\varepsilon \in (0,1)\text { and }t>0, \end{aligned}$$

which, due $k\in {\mathbb {N}}$ and (5.26), entails that letting $y_\varepsilon (t):=\int _{\Omega }\!{\widehat{c}}_{\varepsilon }^{\,2k}(\cdot ,t)=\int _{\Omega }\!|c_\star -c_\varepsilon |^{2k}(\cdot ,t)$ and $h_\varepsilon (t):=\Vert n_\varepsilon (\cdot ,t)\Vert _{L^{p}(\Omega )}$ for $t\ge 0$, $\varepsilon \in (0,1)$ and introducing $C_2:=2C_1>0$ it holds that

$$\begin{aligned} y_\varepsilon '(t)+C_{2}y_\varepsilon (t)\le 2kc_\star ^{2k} |\Omega |^\frac{p-1}{p} h_\varepsilon (t)\quad \text {for all }\varepsilon \in (0,1)\text { and }t>0. \end{aligned}$$

(5.29)

Now, we set $C_{3}:=\max \big \{1,|\Omega |\big \}$ and given $\omega >0$ and $\delta >0$ we pick a small $\delta _1=\delta _1(\delta )>0$ satisfying

$$\begin{aligned} C_{3}\delta _1< \delta \end{aligned}$$

(5.30)

and then $\delta _2(q,\delta ,c_\star )>0$ such that

$$\begin{aligned} \frac{2k c_\star ^{2k} |\Omega |^\frac{p-1}{p}\delta _2^\frac{1}{2}}{1-e^{-C_2}}<\frac{\delta _1^{k}}{2}. \end{aligned}$$

(5.31)

We then let $\eta _1=\eta _1(\omega ,q,\delta ,c_\star ):=\eta _{\diamond }^{(1)}(\omega ,\delta _2)>0$ be provided by Lemma 5.2 and assume that $\kappa \ge 0$ and $\mu >0$ are such that (1.6) is valid for some $\eta <\eta _1$. Hence, according to Lemma 5.2 and the Cauchy–Schwarz inequality, there is $t_1=t_1(\omega ,\mu _0,\delta ,\eta ):=t_{\diamond }^{(1)}(\omega ,\mu _0,\delta _2,\eta )>0$ such that

$$\begin{aligned} \int _{t}^{t+1}\!\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{p}(\Omega )}{{\,\mathrm{d\!}\,}}s\le \left( \int _t^{t+1}\!\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{p}(\Omega )}^2\right) ^\frac{1}{2}<\delta _2^\frac{1}{2}\quad \text {for all }\varepsilon \in (0,1)\text { and }t\ge t_1. \end{aligned}$$

With this eventual smallness of the short-term temporal averages of $h_\varepsilon$ we conclude from an application of Lemma 3.1 to (5.29) that

$$\begin{aligned} y_\varepsilon (t)\le e^{-C_{2}(t-t_1)}y_\varepsilon (t_1)+\frac{2k c_\star ^{2k} |\Omega |^\frac{p-1}{p}\delta _2^\frac{1}{2}}{1-e^{-C_{2}}}\quad \text {for all }\varepsilon \in (0,1)\text { and }t\ge t_1. \end{aligned}$$

(5.32)

Moreover, according to Lemma 2.2 there is $K_0=K_0(c_0,c_\star )>0$ such that

$$\begin{aligned} \int _{\Omega }\!{\widehat{c}}_{\varepsilon }^{\,2k}(\cdot ,t)= \int _{\Omega }\!\vert c_\star -c_\varepsilon \vert ^{2k}(\cdot ,t) \le \big (4^{k}K_0^{2k}+4^{k}c_\star ^{2k}\big )|\Omega |\quad \text {for all }\varepsilon \in (0,1)\text { and }t>0. \end{aligned}$$

(5.33)

Hence, taking $t_{2}=t_{2}(\omega ,\mu _0,q,\delta ,\eta ,c_0,c_\star )\ge t_1$ sufficiently large so that

$$\begin{aligned} e^{-C_{2}(t_{2}-t_1)}\big (4^{k}K_0^{2k}+4^{k}c_\star ^{2k}\big )|\Omega |<\frac{\delta _1^{k}}{2}, \end{aligned}$$

(5.34)

we find from collecting (5.31)–(5.34) that

$$\begin{aligned} \int _{\Omega }\!\vert c_\star -c_\varepsilon \vert ^{2k}(\cdot ,t)&\le e^{-C_{2}(t-t_1)}y_\varepsilon (t_1)+\frac{2k c_\star ^{2k} |\Omega |^\frac{p-1}{p}\delta _2^\frac{1}{2}}{1-e^{-C_{2}}}\nonumber \\ {}&\le e^{-C_{2}(t-t_1)}\big (4^{k}K_0^{2k}+4^{k}c_\star ^{2k}\big )|\Omega |+\frac{2k c_\star ^{2k} |\Omega |^\frac{p-1}{p}\delta _2^\frac{1}{2}}{1-e^{-C_{2}}} <\frac{\delta _1^{k}}{2}+\frac{\delta _1^{k}}{2}=\delta _1^{k} \end{aligned}$$

(5.35)

for all $\varepsilon \in (0,1)$ and $t\ge t_{2}$. Noticing that $\max \{k-1,1\}\le q\le k$ implies that $|\Omega |^\frac{k-q}{kq}\le \max \big \{1,|\Omega |\big \}=C_{3}$, we employ Hölder’s inequality together with (5.35) and (5.30) to conclude that

$$\begin{aligned} \Vert c_\star -c_\varepsilon \Vert _{L^{2q}(\Omega )}^2\le |\Omega |^{\frac{k-q}{kq}}\Vert c_\star -c_\varepsilon \Vert _{L^{2k}(\Omega )}^2\le C_{3}\delta _1<\delta \end{aligned}$$

for all $\varepsilon \in (0,1)$ and $t\ge t_{2}$, which completes the proof upon setting $\eta _{\diamond }^{(4)}(\omega ,q,\delta ,c_\star ):=\eta _1$ and $t_{\diamond }^{(4)}(\omega ,\mu _0,q,\delta ,\eta ,c_0,c_\star ):=t_2$. $\square$

In a second step we find that a combination of the eventual smallness properties above with maximal Sobolev regularity estimates for the Dirichlet heat semigroup [16] allows us to obtain asymptotic smallness information on time-averaged $L^p$ norms for the spatial gradient of ${\widehat{c}}_{\varepsilon }$ with some $p>{\mathcal {N}}$. Since the gradients of ${\widehat{c}}_{\varepsilon }$ and $c_\varepsilon$ only differ in sign, due to $c_\star$ being constant, this also readily entails time-averaged smallness information on $\nabla c_\varepsilon$. (See also [53, Lemma 4.2].)

Lemma 5.6

Let $\omega >0$, $\mu _0>0$ and $c_\star \ge 0$. Then one can find $\theta _{\diamond }^{(5)}=\theta _{\diamond }^{(5)}(\omega )>2-\frac{2}{3}{\mathcal {N}}\ge 0$ with the property that for all $\delta >0$ there is $\eta _{\diamond }^{(5)}=\eta _{\diamond }^{(5)}(\omega ,\delta ,c_\star )>0$ such that whenever $\kappa \ge 0$ and $\mu \ge \mu _0$ satisfy (1.6) for some $\eta <\eta _{\diamond }^{(5)}(\omega ,\delta ,c_\star )$, there is $t_{\diamond }^{(5)}=t_{\diamond }^{(5)}(\omega ,\mu _0,\delta ,\eta ,n_0,c_0,u_0,c_\star )>0$ such that for any $q\in [{\mathcal {N}},{\mathcal {N}}+\theta _5]$ and all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) satisfies

$$\begin{aligned} \int _t^{t+1}\big \Vert \nabla c_\varepsilon (\cdot ,s)\big \Vert _{L^{q}(\Omega )}^2{{\,\mathrm{d\!}\,}}s<\delta \quad \text {for all }t\ge t_{\diamond }^{(5)}. \end{aligned}$$

Proof

With $\theta _1=\theta _1(\omega ):=\theta _{\diamond }^{(1)}(\omega )\in (0,1-\frac{{\mathcal {N}}}{6})$ provided by Lemma 5.2 we set $\theta :=\min \{\frac{\theta _1}{2},\omega \}$, so that with $p=p(\omega ):=\frac{{\mathcal {N}}+6}{6}+\theta$ and $p_0=p_0(\omega ):=\frac{{\mathcal {N}}+6}{6}+\theta _1$ we have on the one hand $p<p_0<{\mathcal {N}}$ and on the other hand also $W^{2,p}(\Omega )\hookrightarrow W^{1,\frac{{\mathcal {N}}p}{{\mathcal {N}}-p}}(\Omega )$. Hence, letting $\theta _5=\theta _{\diamond }^{(5)}(\omega ):=\frac{{\mathcal {N}}p}{{\mathcal {N}}-p}-{\mathcal {N}}$ we find from $p>\frac{{\mathcal {N}}+6}{6}$ that

$$\begin{aligned} \theta _5>\frac{{\mathcal {N}}({\mathcal {N}}+6)}{5{\mathcal {N}}-6}-{\mathcal {N}}\ge 2-\frac{2}{3}{\mathcal {N}}\ge 0, \end{aligned}$$

in both of the cases ${\mathcal {N}}=2$ and ${\mathcal {N}}=3$ and in light of the Hölder inequality we obtain $C_1=C_1(\omega )>0$ such that for any $q\in [{\mathcal {N}},{\mathcal {N}}+\theta _5]$

$$\begin{aligned} \Vert \nabla \varphi \Vert _{L^{q}(\Omega )}\le C_1\Vert \varphi \Vert _{W^{2,p}(\Omega )}\quad \text {for all }\varphi \in W^{2,p}(\Omega ). \end{aligned}$$

(5.36)

Additionally, since $p_0$ and p only depend on $\omega$ and satisfy $p_0>p$, we can pick $k=k(\omega )\ge 1$ such that $2k\ge \frac{p_0p}{p_0-p}$, whence letting

$$\begin{aligned} C_2=C_2(\omega ):=\max \left\{ |\Omega |^{\frac{2k(p_0-p)-p_0p}{kp_0p}},|\Omega |^\frac{2k-p}{kp}\right\} \end{aligned}$$

we also obtain from an application of Hölder’s inequality that

$$\begin{aligned} \Vert \varphi \Vert _{L^{\frac{p_0p}{p_0-p}}(\Omega )}^2\le C_2\Vert \varphi \Vert _{L^{2k}(\Omega )}^2\quad \text {and}\quad \Vert \varphi \Vert _{L^{p}(\Omega )}^2\le C_2\Vert \varphi \Vert _{L^{2k}(\Omega )}^2\quad \text {for all }\varphi \in L^{2k}(\Omega ). \end{aligned}$$

(5.37)

Moreover, drawing on the maximal regularity estimates for the Dirichlet heat semigroup [16], we find $C_3(\omega )>0$ such that if $\varphi \in C^{2,1}\!\left( {\overline{\Omega }}\times [0,2]\right)$ and $h\in C^{0}\!\left( {\overline{\Omega }}\times [0,2]\right)$ fulfill

$$\begin{aligned} \left\{ \begin{array}{ll} \varphi _t=\Delta \varphi +h(x,t),&{}x\in \Omega ,\ t\in (0,2),\\ \varphi =0,&{}x\in \partial \Omega ,\ t\in (0,2),\\ \varphi (x,0)=0,&{}x\in \Omega , \end{array}\right. \end{aligned}$$

(5.38)

then

$$\begin{aligned} \int _0^2\big \Vert \varphi (\cdot ,s)\big \Vert _{W^{2,p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s\le C_3\int _0^2\big \Vert h(\cdot ,s)\big \Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s. \end{aligned}$$

(5.39)

These constants and parameters assembled we can now appropriately employ Lemmas 5.2, 5.4 and 5.5 to prepare later estimations. In fact, given $\delta >0$ we pick $\delta _1=\delta _1(\omega ,\delta ,c_\star )>0$ satisfying

$$\begin{aligned} 8C_3(1+c_\star ^2)\delta _1<\frac{\delta }{4C_1^2}, \end{aligned}$$

(5.40)

$\delta _2=\delta _2(\omega ,\delta )>0$ such that

$$\begin{aligned} 4C_1^2C_3\delta _2^2\le \frac{1}{2} \end{aligned}$$

(5.41)

and $\delta _3=\delta _3(\omega ,\delta )>0$ fulfilling

$$\begin{aligned} 32C_2C_3\delta _3<\frac{\delta }{4C_1^2}\quad \text {and}\quad C_2\delta _3< 1 \end{aligned}$$

(5.42)

and, accordingly, we denote by $\eta _1=\eta _1(\omega ,\delta ,c_\star ):=\eta _{\diamond }^{(1)}(\omega ,\delta _1)$, $\eta _3=\eta _3(\omega ,\delta ):=\eta _{\diamond }^{(3)}(\omega ,\delta _2)$ and $\eta _4=\eta _4(\omega ,\delta ,c_\star ):=\eta _{\diamond }^{(4)}(\omega ,k,\delta _3,c_\star )$ the numbers provided by Lemmas 5.2, 5.3 and 5.5, respectively, stressing once more that the choice of k only depended on $\omega$. Then, with $\eta _5=\eta _{\diamond }^{(5)}(\omega ,\delta ,c_\star ):=\min \{\eta _1,\eta _3,\eta _4\}$ we hereafter assume that $\kappa \ge 0$ and $\mu >0$ satisfy (1.6) for some $\eta <\eta _5$. Since $\eta <\eta _1$ we may invoke Lemma 5.2 to obtain $t_1=t_1(\omega ,\mu _0,\delta ,\eta ):=t_{\diamond }^{(1)}(\omega ,\mu _0,\delta _1,\eta )>0$ such that for both $p=\frac{{\mathcal {N}}+6}{6}+\theta$ and $p_0=\frac{{\mathcal {N}}+6}{6}+\theta _1$ and each $\varepsilon \in (0,1)$ we have

$$\begin{aligned} \int _t^{t+1}\!\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s<\delta _1 \quad \text {and}\quad \int _t^{t+1}\!\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{p_0}(\Omega )}^2{{\,\mathrm{d\!}\,}}s<\delta _1\quad \text {for all }t\ge t_1. \end{aligned}$$

(5.43)

Similarly, the restriction to $\eta <\eta _3$ and Lemma 5.4 entail that with $t_3=t_3(\omega ,\mu _0,\delta ,\eta ,n_0,u_0) := t_{\diamond }^{(3)}(\omega ,\mu _0,\delta _2,\eta ,n_0,u_0)>0$ we have

$$\begin{aligned} \big \Vert u_\varepsilon (\cdot ,t)\big \Vert _{L^{{\mathcal {N}}}(\Omega )}<\delta _2\quad \text {for all }\varepsilon \in (0,1)\text { and }t\ge t_3. \end{aligned}$$

(5.44)

Moreover, in light of the condition $\eta <\eta _4$, we find from an application of Lemma 5.5 that there is $t_4=t_4(\omega ,\mu _0,\delta ,\eta ,n_0,c_0,u_0,c_\star ):= t_{\diamond }^{(4)}(\omega ,\mu _0,\delta _3,\eta ,n_0,c_0,u_0,c_\star )>0$ fulfilling

$$\begin{aligned} \big \Vert \big (c_\star -c_\varepsilon \big )(\cdot ,t)\big \Vert _{L^{2k}(\Omega )}^2\le \delta _3\quad \text {for all }\varepsilon \in (0,1)\text { and }t\ge t_4. \end{aligned}$$

(5.45)

Then, for $t\ge t_5=t_{\diamond }^{(5)}(\omega ,\mu _0,\delta ,\eta ,n_0,c_0,u_0,c_\star ):=\max \{t_1,t_3,t_4\}+1$, we fix a cut-off function $\zeta \in C^{\infty }\!\left( [t-1,t+1]\right)$ satisfying ${{\,\mathrm{supp}\,}}\zeta \subset (t-1,t+1]$, $\zeta \equiv 1$ in $[t,t+1]$ and $0\le \zeta '\le 2$ in $[t-1,t+1]$ and for fixed $\varepsilon \in (0,1)$ define

$$\begin{aligned} z(x,t):=\zeta (t)\cdot {\widehat{c}}_{\varepsilon }(x,t),\quad x\in {\overline{\Omega }}, t\in [t-1,t+1]. \end{aligned}$$

Straightforward calculations drawing on (5.25b) show that z is a solution of

$$\begin{aligned} \left\{ \begin{array}{ll} z_t=\Delta z-u_\varepsilon \cdot \nabla z-g_\varepsilon (n_\varepsilon )z+\zeta g_\varepsilon (n_\varepsilon )c_\star +\zeta '{\widehat{c}}_{\varepsilon }&{}\text {in }\Omega \times (t-1,t+1),\\ z=0&{}\text {on }\partial \Omega \times (t-1,t+1),\\ z(\cdot ,t-1)=0&{} \text {in }\Omega , \end{array}\right. \end{aligned}$$

so that upon taking $\varphi (x,s)=z(x,t-1+s)$, $(x,s)\in {\overline{\Omega }}\times [0,2]$, in (5.38) we find from (5.39) that

$$\begin{aligned} \int _{t-1}^{t+1}\big \Vert z(\cdot ,s)\big \Vert _{W^{2,p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s&\le C_3\int _{t-1}^{t+1}\!\big \Vert \big (-u_\varepsilon \cdot \nabla z-g_\varepsilon (n_\varepsilon )z+\zeta g_\varepsilon (n_\varepsilon )c_\star +\zeta '{\widehat{c}}_{\varepsilon }\big )(\cdot ,s)\big \Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s\nonumber \\&\le 4C_3\bigg [\int _{t-1}^{t+1}\!\big \Vert (u_\varepsilon \cdot \nabla z)(\cdot ,s)\big \Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s+\int _{t-1}^{t+1}\!\big \Vert \big (n_\varepsilon {\widehat{c}}_{\varepsilon }\big )(\cdot ,s)\big \Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s\nonumber \\&\qquad +\int _{t-1}^{t+1}\!\big \Vert c_\star n_\varepsilon (\cdot ,s)\big \Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s+4\int _{t-1}^{t+1}\!\big \Vert {\widehat{c}}_{\varepsilon }(\cdot ,s)\big \Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s\bigg ], \end{aligned}$$

(5.46)

due to $0\le \zeta \le 1$, $\zeta '\le 2$ and $g_\varepsilon (s)\le s$ for $s\ge 0$. To treat the third integral on the right side of (5.46) we draw on the first estimate in (5.43) to find that

$$\begin{aligned} \int _{t-1}^{t+1}\!\big \Vert c_\star n_\varepsilon (\cdot ,s)\big \Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s\le c_\star ^2\int _{t-1}^{t+1}\!\big \Vert n_\varepsilon (\cdot ,s)\big \Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s<2c_\star ^2\delta _1. \end{aligned}$$

For the second integral on the right hand side of (5.46) we make use of the fact that $p<p_0$ and Hölder’s inequality to infer that

$$\begin{aligned} \int _{t-1}^{t+1}\!\big \Vert (n_\varepsilon {\widehat{c}}_{\varepsilon })(\cdot ,s)\big \Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s\le \int _{t-1}^{t+1}\!\big \Vert n_\varepsilon (\cdot ,s)\big \Vert _{L^{p_0}(\Omega )}^2\big \Vert {\widehat{c}}_{\varepsilon }(\cdot ,s)\big \Vert _{L^{\frac{p_0p}{p_0-p}}(\Omega )}^2{{\,\mathrm{d\!}\,}}s, \end{aligned}$$

whence we conclude from the first part of (5.37), (5.45) and the second parts of (5.43) and (5.42) that

$$\begin{aligned} \int _{t-1}^{t+1}\!\big \Vert (n_\varepsilon {\widehat{c}}_{\varepsilon })(\cdot ,s)\big \Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s\le C_2\delta _3\int _{t-1}^{t+1}\big \Vert n_\varepsilon (\cdot ,s)\big \Vert _{L^{p_0}(\Omega )}^2{{\,\mathrm{d\!}\,}}s<2C_2\delta _3\delta _1<2\delta _1. \end{aligned}$$

Therefore, a combination of the estimates above with (5.40) entails that

$$\begin{aligned} 4C_3\int _{t-1}^{t+1}\!\big \Vert \big (n_\varepsilon {\widehat{c}}_{\varepsilon })(\cdot ,s)&\big \Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s+4C_3\int _{t-1}^{t+1}\!\big \Vert c_\star n_\varepsilon (\cdot ,s)\big \Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s<8C_3(1+c_\star ^2)\delta _1<\frac{\delta }{4C_1^2}. \end{aligned}$$

(5.47)

In a similar fashion, relying on (5.37), (5.45) and the first estimate in (5.42) we obtain

$$\begin{aligned} 16C_3\int _{t-1}^{t+1}\!\big \Vert {\widehat{c}}_{\varepsilon }(\cdot ,s)\big \Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s\le 16C_2C_3\int _{t-1}^{t+1}\!\big \Vert {\widehat{c}}_{\varepsilon }(\cdot ,s)\big \Vert _{L^{2k}(\Omega )}^2{{\,\mathrm{d\!}\,}}s\le 32C_2C_3\delta _3<\frac{\delta }{4C_1^2}. \end{aligned}$$

(5.48)

For the estimation of the remaining term on the right hand side of (5.46) we invoke the Hölder inequality in combination with (5.44) to find that

$$\begin{aligned}&\int _{t-1}^{t+1}\!\!\big \Vert (u_\varepsilon \cdot \nabla z)(\cdot ,s)\big \Vert _{L^{p}(\Omega )}^2\!{{\,\mathrm{d\!}\,}}s\\&\quad \le \!\!\int _{t-1}^{t+1}\!\!\big \Vert u_\varepsilon (\cdot ,s)\big \Vert _{L^{{\mathcal {N}}}(\Omega )}^2\big \Vert \nabla z(\cdot ,s)\big \Vert _{L^{\frac{{\mathcal {N}}p}{{\mathcal {N}}-p}}(\Omega )}^2\!{{\,\mathrm{d\!}\,}}s\!\le \delta _2^2\!\int _{t-1}^{t+1}\!\!\big \Vert \nabla z(\cdot ,s)\big \Vert _{L^{\frac{{\mathcal {N}}p}{{\mathcal {N}}-p}}(\Omega )}^2\!{{\,\mathrm{d\!}\,}}s. \end{aligned}$$

Accordingly, recalling that $\frac{{\mathcal {N}}p}{{\mathcal {N}}-p}={\mathcal {N}}+\theta _5$, (5.36) and (5.41) entail that

$$\begin{aligned}&4C_3\int _{t-1}^{t+1}\!\big \Vert (u_\varepsilon \cdot \nabla z)(\cdot ,s)\big \Vert _{L^{p}(\Omega )}^2\nonumber \\&\quad \le 4C_1^2C_3\delta _2^2\int _{t-1}^{t+1}\big \Vert z(\cdot ,s)\big \Vert _{W^{2,p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s< \frac{1}{2}\int _{t-1}^{t+1}\big \Vert z(\cdot ,s)\big \Vert _{W^{2,p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s, \end{aligned}$$

(5.49)

which upon collecting (5.46)–(5.49) and drawing once more on (5.36) thereby shows

$$\begin{aligned} \frac{1}{2C_1^2}\int _{t-1}^{t+1}\big \Vert \nabla z(\cdot ,s)\big \Vert _{L^{q}(\Omega )}^2{{\,\mathrm{d\!}\,}}s\le \frac{1}{2}\int _{t-1}^{t+1}\big \Vert z(\cdot ,s)\big \Vert _{W^{2,p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s<\frac{\delta }{4C_1^2}+\frac{\delta }{4C_1^2}=\frac{\delta }{2C_1^2}. \end{aligned}$$

Since $z\equiv {\widehat{c}}_{\varepsilon }$ in $\Omega \times (t,t+1)$ and $\nabla {\widehat{c}}_{\varepsilon }=-\nabla c_\varepsilon$ in $\Omega \times (0,\infty )$ the assertion of the Lemma is an evident consequence of the above. $\square$

5.3 A functional inequality for $\int _{\Omega }\!\psi \big (n_\varepsilon -\tfrac{\kappa }{\mu }\big )+\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}$

In order to aim at better non time-averaged asymptotic smallness results we will rely on a entropy-like coupling of $n_\varepsilon$ and $\nabla c_\varepsilon$ in the sense that for a certain well-chosen function $\psi =\psi (s)$, basically behaving like $s^p$ for $s\ge 0$ and $p\in (1,2)$ [cf. (5.55)], the investigation of

$$\begin{aligned} y_\varepsilon (t):=\int _{\Omega }\!\psi \big (n_\varepsilon -\tfrac{\kappa }{\mu }\big )(\cdot ,t)+\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}(\cdot ,t) \end{aligned}$$

for some suitable $p\in (\frac{{\mathcal {N}}+6}{6},2)$, reveals an ODE structure with superlinear forcing terms. Combining this structure with the smallness of $y_\varepsilon$ at some point in time, the existence of which is implied by the smallness properties in the previous sections, we will then establish control on $y_\varepsilon$ for all later times.

We start with a differential inequality for $\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}$ for $p\in (1,2)$ by employing a standard testing procedure to (2.1b), where we leave the precise treatment of the boundary terms and terms with mixed variables for later lemmas.

Lemma 5.7

Let $p\in (1,2)$. Then for each $\kappa \ge 0$, $\mu >0$, $c_\star \ge 0$ and all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) satisfies

$$\begin{aligned}&\frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}+\frac{2(p-1)}{p}\int _{\Omega }\!\big |\nabla |\nabla c_\varepsilon |^p\big |^2+p\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}|D^2c_\varepsilon |^2\\&\quad \le 2p\big (2(p-1)+{\mathcal {N}}\big )\bigg (\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}|u_\varepsilon |^2+\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}n_\varepsilon ^2c_\varepsilon ^2\bigg )\\&\qquad +p\int _{\partial \Omega }\!|\nabla c_\varepsilon |^{2p-2}\frac{\partial |\nabla c_\varepsilon |^2}{\partial \nu }-2p\int _{\partial \Omega }\!|\nabla c_\varepsilon |^{2p-2}g_\varepsilon (n_\varepsilon )c_\star \frac{\partial c_\varepsilon }{\partial \nu }\quad \text {on }(0,\infty ). \end{aligned}$$

Proof

Making use of the identity $\nabla c_\varepsilon \cdot \nabla \Delta c_\varepsilon =\frac{1}{2}\Delta |\nabla c_\varepsilon |^2-|D^2c_\varepsilon |^2$ we find from (2.1b) that

$$\begin{aligned} \frac{1}{2p}\frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}&=\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}\nabla c_\varepsilon \cdot \nabla \Big (\Delta c_\varepsilon -u_\varepsilon \cdot \nabla c_\varepsilon -g_\varepsilon (n_\varepsilon )c_\varepsilon \Big )\\&=\frac{1}{2}\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}\Delta |\nabla c_\varepsilon |^2-\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}|D^2c_\varepsilon |^2\\ {}&\qquad -\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}\nabla c_\varepsilon \cdot \nabla (u_\varepsilon \cdot \nabla c_\varepsilon )-\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}\nabla c_\varepsilon \cdot \nabla \big (g_\varepsilon (n_\varepsilon )c_\varepsilon \big ) \end{aligned}$$

for all $t>0$ and $\varepsilon \in (0,1)$. Herein, integration by parts in the first, third and fourth integrals on the right hand side show that

$$\begin{aligned}&\frac{1}{2p}\frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}\nonumber \\&\quad =-\frac{p-1}{2}\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-4}\big |\nabla |\nabla c_\varepsilon |^2\big |^2+\frac{1}{2}\int _{\partial \Omega }\!|\nabla c_\varepsilon |^{2p-2}\frac{\partial |\nabla c_\varepsilon |^2}{\partial \nu }-\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}|D^2c_\varepsilon |^2\nonumber \\&\qquad +\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}\Delta c_\varepsilon (u_\varepsilon \cdot \nabla c_\varepsilon )+(p-1)\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-4}\big (\nabla |\nabla c_\varepsilon |^2\cdot \nabla c_\varepsilon \big )(u_\varepsilon \cdot \nabla c_\varepsilon )\nonumber \\&\qquad +\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}\Delta c_\varepsilon g_\varepsilon (n_\varepsilon )c_\varepsilon +(p-1)\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-4}\big (\nabla |\nabla c_\varepsilon |^2\cdot \nabla c_\varepsilon \big )g_\varepsilon (n_\varepsilon )c_\varepsilon \nonumber \\&\qquad -\int _{\partial \Omega }\!|\nabla c_\varepsilon |^{2p-2}g_\varepsilon (n_\varepsilon )c_\star \frac{\partial c_\varepsilon }{\partial \nu }\qquad \text {for all }t>0\text { and }\varepsilon \in (0,1), \end{aligned}$$

(5.50)

where we used that $c_\varepsilon =c_\star$ on $\partial \Omega \times (0,\infty )$ and that the boundary integral containing $u_\varepsilon$ disappears due to $u_\varepsilon =0$ on $\partial \Omega \times (0,\infty )$. Then, using $|\Delta c_\varepsilon |\le \sqrt{{\mathcal {N}}}|D^2c_\varepsilon |$ as well as $g_\varepsilon (s)\le s$ for $s\ge 0$, we obtain from an application of Young’s inequality that

$$\begin{aligned}&\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}\Delta c_\varepsilon (u_\varepsilon \cdot \nabla c_\varepsilon )+\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}\Delta c_\varepsilon g_\varepsilon (n_\varepsilon )c_\varepsilon \nonumber \\&\quad \le \frac{1}{2}\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}|D^2c_\varepsilon |^2+{\mathcal {N}}\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}|u_\varepsilon |^2+{\mathcal {N}}\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}n_\varepsilon ^2c_\varepsilon ^2 \end{aligned}$$

(5.51)

for all $t>0$ and $\varepsilon \in (0,1)$. Similarly, Young’s inequality also entails

$$\begin{aligned}&(p-1)\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-4}\big (\nabla |\nabla c_\varepsilon |^2\cdot \nabla c_\varepsilon \big )(u_\varepsilon \cdot \nabla c_\varepsilon )+(p-1)\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-4} \big (\nabla |\nabla c_\varepsilon |^2\cdot \nabla c_\varepsilon \big )g_\varepsilon (n_\varepsilon )c_\varepsilon \nonumber \\&\quad \le \frac{p-1}{4}\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-4}\big |\nabla |\nabla c_\varepsilon |^2 \big |^2+2(p-1)\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}|u_\varepsilon |^2+2(p-1)\int _{\Omega }\!| \nabla c_\varepsilon |^{2p-2}n_\varepsilon ^2c_\varepsilon ^2 \end{aligned}$$

(5.52)

for all $t>0$ and $\varepsilon \in (0,1)$. Accumulating the estimates (5.50)–(5.52), rearranging appropriately and finally rewriting

$$\begin{aligned} \frac{2p(p-1)}{4}\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-4}\big |\nabla |\nabla c_\varepsilon |^2\big |^2 =\frac{2(p-1)}{p}\int _{\Omega }\!\big |\nabla |\nabla c_\varepsilon |^p\big |^2 \end{aligned}$$

leads to the desired differential inequality. $\square$

Due to the inhomogeneous boundary condition for $c_\varepsilon$ the boundary terms are of a particularly troublesome nature. For well-behaved domains, however, we at least have a pointwise inequality for the normal derivative $\frac{\partial |\nabla c_\varepsilon |^2}{\partial \nu }$, which (as it turns out in Lemma 5.10 below) will be quite beneficial. The statment of this lemma is taken from [3, Lemma 3.4], whereto we refer the reader for proof.

Lemma 5.8

Let ${\mathcal {N}}\ge 2$ and $G\subset {\mathbb {R}}^{\mathcal {N}}$ be a bounded domain with $C^2$-boundary and let ${K_{\partial }}\in {\mathbb {R}}$ denote the maximum of the curvatures on $\partial G$. Then, whenever $\varphi \in C^{2}\!\left( {\overline{\Omega }}\right)$ and $\varphi _\star \in {\mathbb {R}}$ are such that $\varphi =\varphi _\star$ on $\partial G$,

$$\begin{aligned} \frac{\partial |\nabla \varphi |^2}{\partial \nu }\le 2\frac{\partial \varphi }{\partial \nu }\Delta \varphi +2{K_{\partial }} \Big |\frac{\partial \varphi }{\partial \nu }\Big |^2\quad \text {on }\partial G. \end{aligned}$$

Let us also briefly prepare the following Lemma, which is basically a direct consequence of the Hölder and Young inequalities. Exploiting once more that $\nabla (c_\star -c_\varepsilon )=-\nabla c_\varepsilon$ and $D^2(c_\star -c_\varepsilon )=-D^2c_\varepsilon$ we can thereby estimate terms of the form $\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}$ against the advantageous influence of the diffusion and a term for which we can draw on asymptotic smallness.

Lemma 5.9

Let $p\in (1,2)$. There is $C=C(p)>0$ such that for all $\xi >0$, $\lambda >0$ and each $\varphi \in C^{2}\!\left( {\overline{\Omega }}\right)$ with $\varphi =0$ on $\partial \Omega$ the inequality

$$\begin{aligned} \lambda \int _{\Omega }\!|\nabla \varphi |^{2p}\le \xi \int _{\Omega }\!|\nabla \varphi |^{2p-2}|D^2\varphi |^2+\frac{C\lambda ^{p+1}}{\xi ^p}\int _{\Omega }\!|\varphi |^{2p} \end{aligned}$$

holds.

Proof

Since $\varphi =0$ on $\partial \Omega$, an integration by parts yields

$$\begin{aligned} \lambda \int _{\Omega }\!|\nabla \varphi |^{2p}&=\lambda \int _{\Omega }\!\nabla \varphi \cdot (|\nabla \varphi |^{2p-2}\nabla \varphi )\\&=-\lambda \int _{\Omega }\!\varphi |\nabla \varphi |^{2p-2}\Delta \varphi -2 \lambda (p-1)\int _{\Omega }\!\varphi |\nabla \varphi |^{2p-4}\big ((D^2\varphi \cdot \nabla \varphi )\cdot \nabla \varphi \big ). \end{aligned}$$

Letting $C_1=C_1(p):=2(p-1)+\sqrt{{\mathcal {N}}}$, we obtain due to $|\Delta \varphi |\le \sqrt{{\mathcal {N}}}|D^2\varphi |$ and the Cauchy–Schwarz inequality that

$$\begin{aligned} \lambda \int _{\Omega }\!|\nabla \varphi |^{2p}&\le C_1\lambda \int _{\Omega }\!|\varphi ||\nabla \varphi |^{2p-2}|D^2\varphi |\le C_1\lambda \bigg (\int _{\Omega }\!|\nabla \varphi |^{2p-2}|D^2\varphi |^2\bigg )^\frac{1}{2} \bigg (\int _{\Omega }\!\varphi ^2|\nabla \varphi |^{2p-2}\bigg )^\frac{1}{2}. \end{aligned}$$

Moreover, utilizing Hölder’s inequality we may further estimate

$$\begin{aligned} \bigg (\int _{\Omega }\!\varphi ^2|\nabla \varphi |^{2p-2}\bigg )^\frac{1}{2} \le \bigg (\int _{\Omega }\!|\nabla \varphi |^{2p}\bigg )^{\frac{p-1}{2p}}\bigg (\int _{\Omega }\!|\varphi |^{2p}\bigg )^{\frac{1}{2p}}, \end{aligned}$$

so that with $I:=\int _{\Omega }\!|\nabla \varphi |^{2p}$ and $J:=\int _{\Omega }\!|\nabla \varphi |^{2p-2}|D^2\varphi |^2$ we conclude from the above and two applications of Young’s inequality that

$$\begin{aligned} \lambda I&\le C_1\lambda J^\frac{1}{2}I^\frac{p-1}{2p}\bigg (\int _{\Omega }\!|\varphi |^{2p}\bigg )^\frac{1}{2p}\\&\le \frac{\lambda }{2}I+2^{\frac{p-1}{p+1}}\lambda C_1^{\frac{2p}{p+1}}J^{\frac{p}{p+1}}\bigg (\int _{\Omega }\!|\varphi |^{2p} \bigg )^\frac{1}{p+1}\!\le \frac{\lambda }{2} I+\frac{\xi }{2} J+\frac{(2C_1)^{2p}}{2\xi ^p}\lambda ^{p+1}\int _{\Omega }\!|\varphi |^{2p}, \end{aligned}$$

which verifies the asserted inequality with $C:=(2C_1)^{2p}$ after evident rearrangement. $\square$

Making use of the point-wise inequality from Lemma 5.8 and results on trace embeddings we can control the boundary integrals appearing in Lemma 3.3 in the following way. (Compare also [3, Lemma 3.5].)

Lemma 5.10

Let $p\in (1,2)$ and $\xi >0$. There is $C=C(p,\xi )>0$ such that for each $\kappa \ge 0$, $\mu >0$, $c_\star \ge 0$ and all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) fulfills

$$\begin{aligned}&p\int _{\partial \Omega }\!|\nabla c_\varepsilon |^{2p-2}\frac{\partial |\nabla c_\varepsilon |^2}{\partial \nu }-2p\int _{\partial \Omega }\!|\nabla c_\varepsilon |^{2p-2}g_\varepsilon (n_\varepsilon )c_\star \frac{\partial c_\varepsilon }{\partial \nu }\\&\quad \le \xi \int _{\Omega }\!\big |\nabla |\nabla c_\varepsilon |^p\big |^2+\xi \int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}|D^2c_\varepsilon |^2+C\int _{\Omega }\!|c_\star -c_\varepsilon |^{2p} \end{aligned}$$

on $(0,\infty )$.

Proof

Denoting with ${K_{\partial }}$ the maximal curvature on $\partial \Omega$, we find that by Lemma 5.8

$$\begin{aligned} \frac{\partial |\nabla c_\varepsilon |^2}{\partial \nu }\le 2\frac{\partial c_\varepsilon }{\partial \nu }\Delta c_\varepsilon +2{K_{\partial }}\Big |\frac{\partial c_\varepsilon }{\partial \nu }\Big |^2\quad \text {on }\partial \Omega \times (0,\infty )\text { for all }\varepsilon \in (0,1), \end{aligned}$$

where by continuity up to the boundary and (2.1b)

$$\begin{aligned} \Delta c_\varepsilon =c_{\varepsilon t}+u_\varepsilon \cdot \nabla c_\varepsilon +g_\varepsilon (n_\varepsilon )c_\varepsilon =g_\varepsilon (n_\varepsilon )c_\star \quad \text {on }\partial \Omega \times (0,\infty )\text { for all }\varepsilon \in (0,1), \end{aligned}$$

so that, due to $|\frac{\partial c_\varepsilon }{\partial \nu }|=|\nabla c_\varepsilon |$ on $\partial \Omega \times (0,\infty )$ for all $t>0$ and $\varepsilon \in (0,1)$, we find

$$\begin{aligned}&p\int _{\partial \Omega }\!|\nabla c_\varepsilon |^{2p-2}\frac{\partial |\nabla c_\varepsilon |^2}{\partial \nu }-2p\int _{\partial \Omega }\!|\nabla c_\varepsilon |^{2p-2}g_\varepsilon (n_\varepsilon )c_\star \frac{\partial c_\varepsilon }{\partial \nu }\\&\quad \le 2p{K_{\partial }}\int _{\partial \Omega }\!|\nabla c_\varepsilon |^{2p-2}\Big |\frac{\partial c_\varepsilon }{\partial \nu }\Big |^2\le 2p{K_{\partial }}\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{2\;\!}\!\left( \partial \Omega \right) }^2 \end{aligned}$$

for all $t>0$ and $\varepsilon \in (0,1)$. Moreover, recalling that for $r\in (0,\frac{1}{2})$ we have $W^{1,2}(\Omega )\hookrightarrow \hookrightarrow W^{r+\frac{1}{2},2}(\Omega )\hookrightarrow L^{2\;\!}\!\left( \Omega \right)$ [9], we find from trace embeddings [18] and Ehrling’s lemma that for any $\xi >0$ there is $C_1=C_1(\xi )>0$ such that

$$\begin{aligned} \big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{2\;\!}\!\left( \partial \Omega \right) }^2\le \xi \big \Vert \nabla |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^2+C_1\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^2\quad \text {for all }t>0\text { and }\varepsilon \in (0,1). \end{aligned}$$

Therefore, since $|\nabla c_\varepsilon |=|\nabla (c_\star -c_\varepsilon )|$ and $|D^2c_\varepsilon |=|D^2(c_\star -c_\varepsilon )|$ on $\Omega \times (0,\infty )$ for all $\varepsilon \in (0,1)$, we readily obtain from invoking Lemma 5.9 for $\varphi =c_\star -c_\varepsilon$ that there is $C_2=C_2(p,\xi )>0$ satisfying

$$\begin{aligned}&p\int _{\partial \Omega }\!|\nabla c_\varepsilon |^{2p-2}\frac{\partial |\nabla c_\varepsilon |^2}{\partial \nu } -2p\int _{\partial \Omega }\!|\nabla c_\varepsilon |^{2p-2}g_\varepsilon (n_\varepsilon )c_\star \frac{\partial c_\varepsilon }{\partial \nu }\\&\quad \le \xi \int _{\Omega }\!\big |\nabla |\nabla c_\varepsilon |^p\big |^2+\xi \int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}|D^2c_\varepsilon |^2+C_2\int _{\Omega }\!|c_\star -c_\varepsilon |^{2p} \end{aligned}$$

for all $t>0$ and $\varepsilon \in (0,1)$. $\square$

Since our eventual smallness results cover $L^r$ norms for $u_\varepsilon$ with r slightly larger than the space dimension ${\mathcal {N}}$, we can consider these quantities to be of a sufficiently good nature to appear on the right hand side of our estimations. As such, we can adjust the arguments of [53, Lemma 5.1] to estimate the mixed term $\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}|u_\varepsilon |^2$ against terms appearing in Lemma 3.3 with beneficial sign and unproblematic terms we know how to control.

Lemma 5.11

Let $p\in (1,2)$ and $r>{\mathcal {N}}$. For each $\lambda >0$ and any $\xi >0$ one can find $C=C(p,r,\xi ,\lambda )>0$ such that for each $\kappa \ge 0$, $\mu >0$, $c_\star \ge 0$ and all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) satisfies

$$\begin{aligned} \lambda \int _{\Omega }\!|\nabla c_\varepsilon |^{2p}|u_\varepsilon |^2\le&\xi \int _{\Omega }\!\big |\nabla |\nabla c_\varepsilon |^p\big |^2+\xi \int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2} |D^2c_\varepsilon |^2\\&+C\Big (\Vert u_\varepsilon \Vert _{L^{r}(\Omega )}^{\frac{2r(p+1)}{r-{\mathcal {N}}}} +\Vert u_\varepsilon \Vert _{L^{r}(\Omega )}^{2(p+1)}\Big )\int _{\Omega }\!|c_\star -c_\varepsilon |^{2p} \end{aligned}$$

on $(0,\infty )$.

Proof

Since $r>2$ we can make use of Hölder’s inequality to find that

$$\begin{aligned} \lambda \int _{\Omega }\!|\nabla c_\varepsilon |^{2p}|u_\varepsilon |^2\le \lambda \bigg (\int _{\Omega }\!|u_\varepsilon |^r\bigg )^\frac{2}{r}\bigg (\int _{\Omega }\!|\nabla c_\varepsilon |^{\frac{2rp}{r-2}}\bigg )^\frac{r-2}{r}=\lambda \big \Vert u_\varepsilon \big \Vert _{L^{r}(\Omega )}^2\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{\frac{2r}{r-2}}(\Omega )}^2 \end{aligned}$$

(5.53)

for all $t>0$ and $\varepsilon \in (0,1)$. To estimate further, we note that an application of the Gagliardo–Nirenberg inequality yields $C_1>0$ such that

$$\begin{aligned} \big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{\frac{2r}{r-2}}(\Omega )}^2\le C_1\big \Vert \nabla |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^\frac{2{\mathcal {N}}}{r}\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^\frac{2(r-{\mathcal {N}})}{r}+C_1\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^2 \end{aligned}$$

for all $t>0$ and $\varepsilon \in (0,1)$. Thus, with $\xi >0$ fixed and using $r>{\mathcal {N}}$, we invoke Young’s inequality combined with the above in (5.53) to obtain $C_2=C_2(\xi )>0$ satisfying

$$\begin{aligned} \lambda \int _{\Omega }\!|\nabla c_\varepsilon |^{2p}|u_\varepsilon |^2\le \xi \big \Vert \nabla |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^2+\Big (C_2\lambda ^\frac{r}{r-{\mathcal {N}}}\big \Vert u_\varepsilon \big \Vert _{L^{r}(\Omega )}^\frac{2r}{r-{\mathcal {N}}}+C_1\lambda \big \Vert u_\varepsilon \big \Vert _{L^{r}(\Omega )}^2\Big )\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^2 \end{aligned}$$

for all $t>0$ and $\varepsilon \in (0,1)$. The latter term can be further estimated by again relying on the facts that $|\nabla c_\varepsilon |=|\nabla (c_\star -c_\varepsilon )|$ and $|D^2c_\varepsilon |=|D^2(c_\star -c_\varepsilon )|$ on $\Omega \times (0,\infty )$ for all $\varepsilon \in (0,1)$ and Lemma 5.9, which provides $C_3=C_3(p)>0$ such that

$$\begin{aligned} \lambda \int _{\Omega }\!|\nabla c_\varepsilon |^{2p}|u_\varepsilon |^2&\le \xi \int _{\Omega }\!\big |\nabla |\nabla c_\varepsilon |^p\big |^2+ \xi \int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}|D^2c_\varepsilon |^2\\&\quad +\frac{C_3}{\xi ^p}\Big (C_2\lambda ^\frac{r}{r-{\mathcal {N}}}\big \Vert u_\varepsilon \big \Vert _{L^{r}(\Omega )}^\frac{2r}{r-{\mathcal {N}}}+C_1\lambda \big \Vert u_\varepsilon \big \Vert _{L^{r}(\Omega )}^2\Big )^{p+1}\int _{\Omega }\!|c_\star -c_\varepsilon |^{2p} \end{aligned}$$

for all $t>0$ and $\varepsilon \in (0,1)$. Letting $C=C(p,\xi ,\lambda ):=\frac{2^{p+1}C_3}{\xi ^p}\big (\max \big \{C_2\lambda ^{\frac{r}{r-{\mathcal {N}}}},C_1\lambda \big \}\big )^{p+1}$ we can conclude the proof. $\square$

The treatment of $\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}n_\varepsilon ^2c_\varepsilon ^2$ appearing on the right hand side of the inequality in Lemma 5.7 is still open and our asymptotic information on $n_\varepsilon$ and $\nabla c_\varepsilon$ appears to be too weak to treat this directly. Thus, we will combine the functional inequality for $\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}$ with an suitably prepared functional for a quantity of $n_\varepsilon$ to combat these difficulties.

The function $\psi =\psi (s)$ we will introduce now has already been utilized in [53, Section 5.1] to establish similar estimations in a Keller–Segel–Navier–Stokes system with Neumann boundary condition for the chemical concentration. For $\kappa \ge 0$ and $\mu >0$ we set

$$\begin{aligned} \gamma :=\frac{\kappa }{\mu } \end{aligned}$$

(5.54)

and for $p\in (1,2)$ we define $\psi =\psi _{p,\gamma }\in W_{loc}^{2,\infty }((-\gamma ,\infty ))\cap C^{2}\!\left( {\mathbb {R}}{\setminus }\{0,\gamma \}\right)$ as

$$\begin{aligned} \psi _{p,\gamma }(s):={\left\{ \begin{array}{ll} 0&{}\text {if }s\le 0,\\ \frac{p}{2}\gamma ^{p-2}s^2\qquad &{}\text {if }s\in (0,\gamma ),\\ s^p-\frac{2-p}{2}\gamma ^p&{}\text {if }s\ge \gamma , \end{array}\right. } \end{aligned}$$

(5.55)

which in case of $\gamma =0$, i.e. $\kappa =0$, equals

$$\begin{aligned} \psi _{p,0}(s):=s_+^p\quad \text {for all }s\in {\mathbb {R}}. \end{aligned}$$

(5.56)

The key feature of this function is the following point-wise upper estimate for a combined expression of $\psi '$ and $\psi ''$. The underlying proof, which basically consists of straightforward calculations using the expressions for $\psi '$ and $\psi ''$ in their corresponding cases, can be found in [53, Lemma 5.2].

Lemma 5.12

Let $p\in (1,2)$, $\gamma \ge 0$ and $\psi =\psi _{p,\gamma }$ as defined by (5.55). Then

$$\begin{aligned} \big (\psi '(s)\big )^{\frac{2-p}{p-1}}\psi ''(s)\le p^\frac{1}{p-1}\quad \text {for all }s\in {\mathbb {R}}{\setminus }\{0,\gamma \}. \end{aligned}$$

Now, instead of just $n_\varepsilon$ we will consider a version of (2.1a), which is shifted by the quantity $\gamma$. In fact, we introduce the difference function

$$\begin{aligned} N_\varepsilon (x,t):=n_\varepsilon (x,t)-\gamma ,\quad x\in {\overline{\Omega }},\ t\ge 0 \end{aligned}$$

(5.57)

and note that hence

$$\begin{aligned} N_{\varepsilon t}=\Delta N_\varepsilon -\nabla \cdot \big ((N_\varepsilon +\gamma )\rho _\varepsilon f_\varepsilon (N_\varepsilon +\gamma )\nabla c_\varepsilon \big )-u_\varepsilon \cdot \nabla N_\varepsilon -\kappa N_\varepsilon -\mu N_\varepsilon ^2,\quad \text {on }\Omega \times (0,\infty ). \end{aligned}$$

(5.58)

According to the above equation for $N_\varepsilon$ and the specific form of $\psi$ we then obtain the following differential inequality. (Compare the similar arguments in [53, Lemma 5.3].)

Lemma 5.13

Assume $p\in (1,2)$ and $\xi >0$. There is $C=C(p,\xi )>0$ such that for each $\kappa \ge 0$, $\mu >0$, $c_\star \ge 0$ and all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) and the quantities $\gamma ,N_\varepsilon$ and $\psi =\psi _{p,\gamma }$ provided by (5.54), (5.57) and (5.55), respectively, satisfy

$$\begin{aligned}&\frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!\psi (N_\varepsilon )+\frac{1}{2}\int _{\Omega }\!\psi ''(N_\varepsilon )|\nabla N_\varepsilon |^2+\mu \int _{\Omega }\!\psi '(N_\varepsilon )N_\varepsilon ^2\\&\quad \le 2p(p-1)\int _{\{N_\varepsilon >\gamma \}}N_\varepsilon ^p|\nabla c_\varepsilon |^2+\xi \int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}|D^2c_\varepsilon |^2+C\int _{\Omega }\!|c_\star -c_\varepsilon |^{2p}+C\gamma ^{\frac{p^2}{p-1}} \end{aligned}$$

on $(0,\infty )$.

Proof

Since $n_\varepsilon$ is positive in ${\overline{\Omega }}\times (0,\infty )$, we know that $N_\varepsilon >-\gamma$ in ${\overline{\Omega }}\times (0,\infty )$ and that accordingly the mapping $t\mapsto \int _{\Omega }\!\psi \big (N_\varepsilon (\cdot ,t)\big )$ is of class $C^{0}\!\left( [0,\infty )\right) \cap C^{1}\!\left( (0,\infty )\right)$ for each $\varepsilon \in (0,1)$, due to the fact that $\psi \in W_{loc}^{2,\infty }\left( (-\gamma ,\infty )\right)$. Thus, we may employ $\psi '(N_\varepsilon )$ as a test-function in (5.58) and compute

$$\begin{aligned} \frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!\psi (N_\varepsilon )&=\int _{\Omega }\!\psi '(N_\varepsilon )\Big (\Delta N_\varepsilon -\nabla \cdot \big ((N_\varepsilon +\gamma )\rho _\varepsilon f_\varepsilon (N_\varepsilon +\gamma )\nabla c_\varepsilon \big )-u_\varepsilon \cdot \nabla N_\varepsilon -\kappa N_\varepsilon -\mu N_\varepsilon ^2\Big )\\&=-\int _{\Omega }\!\psi ''(N_\varepsilon )|\nabla N_\varepsilon |^2+\int _{\Omega }\!\psi ''(N_\varepsilon )(N_\varepsilon +\gamma )\rho _\varepsilon f_\varepsilon (N_\varepsilon +\gamma )(\nabla c_\varepsilon \cdot \nabla N_\varepsilon )\\ {}&\qquad -\,\kappa \int _{\Omega }\!\psi '(N_\varepsilon )N_\varepsilon -\mu \int _{\Omega }\!\psi '(N_\varepsilon )N_\varepsilon ^2\quad \text {for all }t>0\text { and }\varepsilon \in (0,1), \end{aligned}$$

where we also relied on the divergence-free property of $u_\varepsilon$ and the boundary conditions prescribed in (2.1) to find that all boundary integrals disappear. Additionally, since $\kappa \ge 0$, $\psi '\ge 0$ on $(0,\infty )$ and $\psi '\equiv 0$ on $(-\infty ,0)$, we have $\kappa \psi '(s)s\ge 0$ on ${\mathbb {R}}$, so that the third term on the right hand side can also be dropped. For the second term, we recall $\rho _\varepsilon (x)f_\varepsilon (s)\le 1$ for all $x\in \Omega$ and $s\ge 0$ and apply Young’s inequality to obtain

$$\begin{aligned} \frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\int _{\Omega }\!\psi (N_\varepsilon )+\frac{1}{2}\int _{\Omega }\!\psi ''(N_\varepsilon )|\nabla N_\varepsilon |^2+\mu \int _{\Omega }\!\psi '(n_\varepsilon )N_\varepsilon ^2&\le \frac{1}{2}\int _{\Omega }\!\psi ''(N_\varepsilon )(N_\varepsilon +\gamma )^2|\nabla c_\varepsilon |^2 \end{aligned}$$

(5.59)

for all $t>0$ and $\varepsilon \in (0,1)$, in light of the nonnegativity of $\psi ''$. To further estimate, we subdivide the domain according to the cases specified in (5.55), which, upon replacing $\psi ''$ correspondingly, entails

$$\begin{aligned}&\frac{1}{2}\int _{\Omega }\!\psi ''(N_\varepsilon )(N_\varepsilon +\gamma )^2|\nabla c_\varepsilon |^2\nonumber \\&\quad =\frac{1}{2}\int _{\{N_\varepsilon \in [0,\gamma ]\}}\!\!\psi ''(N_\varepsilon )(N_\varepsilon +\gamma )^2|\nabla c_\varepsilon |^2+\frac{1}{2}\int _{\{N_\varepsilon>\gamma \}}\!\!\psi ''(N_\varepsilon )(N_\varepsilon +\gamma )^2|\nabla c_\varepsilon |^2\nonumber \\&\quad \le \frac{1}{2}\int _{\{N_\varepsilon \in [0,\gamma ]\}}\!p\gamma ^{p-2}(2\gamma )^2 |\nabla c_\varepsilon |^2+\frac{1}{2}\int _{\{N_\varepsilon>\gamma \}}\! p(p-1)N_\varepsilon ^{p-2}(2N_\varepsilon )^2|\nabla c_\varepsilon |^2\nonumber \\&\quad \le 2p\gamma ^p\int _{\Omega }\!|\nabla c_\varepsilon |^2+2p(p-1)\int _{\{N_\varepsilon >\gamma \}}N_\varepsilon ^p|\nabla c_\varepsilon |^2 \end{aligned}$$

(5.60)

for all $t>0$ and $\varepsilon \in (0,1)$. Finally, in order to prepare the absorption of $2p\gamma ^p\int _{\Omega }\!|\nabla c_\varepsilon |^2$ into the terms present on the left hand-side of the differential inequality for $\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}$ in Lemma 5.7, we draw once more on Young’s inequality to estimate

$$\begin{aligned} 2p\gamma ^p\int _{\Omega }\!|\nabla c_\varepsilon |^2\le 2p\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}+2p|\Omega |\gamma ^{\frac{p^2}{p-1}}\quad \text {for all }t>0\text { and }\varepsilon \in (0,1), \end{aligned}$$

and invoke Lemma 5.9 to find that there is $C_1=C_1(p)>0$ such for any given $\xi >0$ we have

$$\begin{aligned} 2p\gamma ^p\int _{\Omega }\!|\nabla c_\varepsilon |^2\le \xi \int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}|D^2c_\varepsilon |^2+\frac{C_1 (2p)^{p+1}}{\xi ^p}\int _{\Omega }\!|c_\star -c_\varepsilon |^{2p}+2p|\Omega |\gamma ^{\frac{p^2}{p-1}} \end{aligned}$$

(5.61)

for all $t>0$ and $\varepsilon \in (0,1)$. The collection of (5.59)–(5.61) entails the asserted inequality of the lemma. $\square$

Next, we turn our attention to the two remaining troublesome terms in the inequalities from Lemmas 5.7 and 5.13. We start with the one present in Lemma 5.13 and work along the lines of [53, Lemma 5.3].

Lemma 5.14

Let $p\in (\tfrac{{\mathcal {N}}+6}{6},2)$, then for any $\xi >0$ there is $C=C(p,\xi )>0$ such that for each $\kappa \ge 0$, $\mu >0$, $c_\star \ge 0$ and all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) and the quantities $\gamma$, $N_\varepsilon$ and $\psi =\psi _{p,\gamma }$ provided by (5.54), (5.57) and (5.55), respectively, satisfy

$$\begin{aligned}&2p(p-1)\int _{\{N_\varepsilon >\gamma \}}N_\varepsilon ^p|\nabla c_\varepsilon |^2\\&\quad \le \xi \int _{\Omega }\!\big |\nabla |\nabla c_\varepsilon |^p\big |^2+\xi \int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}|D^2c_\varepsilon |^2+\frac{1}{2}\int _{\Omega }\!\psi ''(N_\varepsilon )|\nabla N_\varepsilon |^2+C\int _{\Omega }\!|c_\star -c_\varepsilon |^{2p}\\&\qquad +C\left( \int _{\Omega }\!\psi (N_\varepsilon )\right) ^\frac{6p-{\mathcal {N}}}{6(p-1)-{\mathcal {N}}}+C\left( \int _{\Omega }\!n_\varepsilon \right) ^\frac{p^2}{p-1}+C\gamma ^\frac{p(6p-{\mathcal {N}})}{6(p-1)-{\mathcal {N}}} \end{aligned}$$

on $(0,\infty )$.

Proof

Because of $p>1$, the Hölder inequality implies

$$\begin{aligned} \int _{\{N_\varepsilon>\gamma \}}N_\varepsilon ^p|\nabla c_\varepsilon |^2&\le \left\| N_\varepsilon ^\frac{p}{2}\right\| ^2_{L^{\frac{6p}{3p-1}\;\!}\!\left( \{N_\varepsilon>\gamma \}\right) }\Big \Vert \nabla c_\varepsilon \Big \Vert ^2_{L^{6p\;\!}\!\left( \{N_\varepsilon>\gamma \}\right) }\nonumber \\&\le \left\| N_\varepsilon ^\frac{p}{2}\right\| _{L^{\frac{6p}{3p-1}\;\!}\!\left( \{N_\varepsilon >\gamma \}\right) }^2\Big \Vert \nabla c_\varepsilon \Big \Vert _{L^{6p}(\Omega )}^2 \end{aligned}$$

(5.62)

on $(0,\infty )$ for all $\varepsilon \in (0,1)$. To estimate the first factor in this product further, we draw on the specific form of $\psi$ provided by (5.55) to find that for $N_\varepsilon >\gamma$ we have $N_\varepsilon ^\frac{p}{2}=p^{-\frac{p}{2(p-1)}}\big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)}$ so that

$$\begin{aligned} \left\| N_\varepsilon ^\frac{p}{2}\right\| _{L^{\frac{6p}{3p-1}\;\!}\!\left( \{N_\varepsilon>\gamma \}\right) }^2\le p^\frac{-p}{p-1}\left\| \big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)}\right\| _{L^{\frac{6p}{3p-1}\;\!}\!\left( \Omega \right) }^2\quad \text {for all }t>0\text { and }\varepsilon \in (0,1). \end{aligned}$$

Hence, invoking the Gagliardo–Nirenberg inequality we obtain $C_1>0$ such that

$$\begin{aligned} \left\| N_\varepsilon ^\frac{p}{2}\right\| _{L^{\frac{6p}{3p-1}\;\!}\!\left( \{N_\varepsilon >\gamma \}\right) }^2\!\le&C_1\left\| \nabla \Big (\big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)}\Big )\right\| ^{\frac{{\mathcal {N}}}{3p}}_{L^{2}(\Omega )}\left\| \big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)}\right\| _{L^{2}(\Omega )}^{\frac{6p-{\mathcal {N}}}{3p}}\!\nonumber \\&+C_1\left\| \big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)}\right\| _{L^{\frac{2}{p}}(\Omega )}^2 \end{aligned}$$

(5.63)

for all $t>0$ and $\varepsilon \in (0,1)$. Similarly, for the second factor in the product on the right hand side of (5.62) the Gagliardo–Nirenberg inequality provides $C_2>0$ satisfying

$$\begin{aligned} \left\| \nabla c_\varepsilon \right\| _{L^{6p}(\Omega )}^2=\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{6}(\Omega )}^\frac{2}{p}\le C_2\left( \big \Vert \nabla |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^\frac{2{\mathcal {N}}}{3}\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^\frac{2(3-{\mathcal {N}})}{3}+\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^2\right) ^\frac{1}{p} \end{aligned}$$

for all $t>0$ and $\varepsilon \in (0,1)$. Therefore, in the case of ${\mathcal {N}}=2$ relying on Young’s inequality and in the case of ${\mathcal {N}}=3$ already trivially contained in the estimate above, we can find $C_3=C_3(p)>0$ such that

$$\begin{aligned} \left\| \nabla c_\varepsilon \right\| _{L^{6p}(\Omega )}^2\le C_3\left( \big \Vert \nabla |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^2+\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^2\right) ^\frac{1}{p}\quad \text {for all }t>0\text { and }\varepsilon \in (0,1). \end{aligned}$$

(5.64)

Plugging (5.63) and (5.64) back into (5.62) and drawing on Young’s inequality for fixed $\xi >0$ we obtain $C_4=C_4(p,\xi )>0$ satisfying

$$\begin{aligned}&2p(p-1)\int _{\{N_\varepsilon >\gamma \}}N_\varepsilon ^p|\nabla c_\varepsilon |^2\le \xi \left( \big \Vert \nabla |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^2+\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^2\right) \\&\qquad +C_4\left( \left\| \nabla \Big (\big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)}\Big )\right\| ^{\frac{{\mathcal {N}}}{3p}}_{L^{2}(\Omega )}\left\| \big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)}\right\| _{L^{2}(\Omega )}^{\frac{6p-{\mathcal {N}}}{3p}}+\left\| \big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)}\right\| _{L^{\frac{2}{p}}(\Omega )}^2\right) ^\frac{p}{p-1}\\&\quad \le \xi \left( \big \Vert \nabla |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^2+\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^2\right) \\&\qquad +2^{\frac{p}{p-1}}C_4\left\| \nabla \Big (\big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)} \Big )\right\| ^{\frac{{\mathcal {N}}}{3(p-1)}}_{L^{2}(\Omega )}\left\| \big (\psi '(N_\varepsilon ) \big )^\frac{p}{2(p-1)}\right\| _{L^{2}(\Omega )}^{\frac{6p-{\mathcal {N}}}{3(p-1)}}\\&\qquad +2^\frac{p}{p-1}C_4\left\| \big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)}\right\| _{L^{\frac{2}{p}}(\Omega )}^\frac{2p}{p-1} \end{aligned}$$

for all $t>0$ and $\varepsilon \in (0,1)$. Now, since in view of Lemma 5.12

$$\begin{aligned}&\left\| \nabla \Big (\big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)}\Big )\right\| _{L^{2}(\Omega )}^2\\&\quad = \frac{p^2}{4(p-1)^2}\int _{\Omega }\!\big (\psi '(N_\varepsilon )\big )^\frac{2-p}{p-1}\big (\psi ''(N_\varepsilon )\big )^2|\nabla N_\varepsilon |^2\le \frac{p^{\frac{2p-1}{p-1}}}{4(p-1)^2}\int _{\Omega }\!\psi ''(N_\varepsilon )|\nabla N_\varepsilon |^2 \end{aligned}$$

for all $t>0$ and $\varepsilon \in (0,1)$ and remarking that $\frac{{\mathcal {N}}+6}{6}<p$ entails $\frac{{\mathcal {N}}}{3(p-1)}<2$, we find from a second application of Young’s inequality that there is $C_5=C_5(p,\xi )>0$ such that

$$\begin{aligned} 2p(p-1)\int _{\{N_\varepsilon >\gamma \}}N_\varepsilon ^p|\nabla c_\varepsilon |^2&\le \xi \left( \int _{\Omega }\!\big |\nabla |\nabla c_\varepsilon |^p\big |^2+\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}\right) +\frac{1}{2}\int _{\Omega }\!\psi ''(N_\varepsilon )|\nabla N_\varepsilon |^2\nonumber \\&\quad +C_5\left\| \big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)}\right\| _{L^{2}(\Omega )}^{\frac{2(6p-{\mathcal {N}})}{6(p-1)-{\mathcal {N}}}}+C_5\left\| \big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)}\right\| _{L^{\frac{2}{p}}(\Omega )}^\frac{2p}{p-1} \end{aligned}$$

for all $t>0$ and $\varepsilon \in (0,1)$. Then, drawing on Lemma 5.9 again we obtain $C_6=C_6(p,\xi )>0$ such that

$$\begin{aligned}&2p(p-1)\int _{\{N_\varepsilon >\gamma \}}\!\!N_\varepsilon ^p|\nabla c_\varepsilon |^2\nonumber \\&\quad \le \xi \left( \int _{\Omega }\!\big |\nabla |\nabla c_\varepsilon |^p\big |^2+\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}|D^2c_\varepsilon |^2\right) +\frac{1}{2}\int _{\Omega }\!\psi ''(N_\varepsilon )|\nabla N_\varepsilon |^2\\&\qquad +C_6\int _{\Omega }\!|c_\star -c_\varepsilon |^{2p}+C_5\left\| \big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)}\right\| _{L^{2}(\Omega )}^{\frac{2(6p-{\mathcal {N}})}{6(p-1)-{\mathcal {N}}}}+C_5\left\| \big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)}\right\| _{L^{\frac{2}{p}}(\Omega )}^\frac{2p}{p-1}\nonumber \end{aligned}$$

(5.65)

for all $t>0$ and $\varepsilon \in (0,1)$. In order to eliminate the remaining appearances of $\psi '(N_\varepsilon )$ from this inequality, we split the domain again according to the cases in (5.55) so that

$$\begin{aligned} \left\| \big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)}\right\| _{L^{2}(\Omega )}^2&=\int _{\{N_\varepsilon \in [0,\gamma ]\}}\! p^\frac{p}{p-1}\gamma ^\frac{p(p-2)}{p-1}N_\varepsilon ^\frac{p}{p-1}+\int _{\{N_\varepsilon>\gamma \}}\! p^\frac{p}{p-1}N_\varepsilon ^p\\&\le p^\frac{p}{p-1}\gamma ^p|\Omega |+p^\frac{p}{p-1}\int _{\{N_\varepsilon>\gamma \}}\Big [\psi (N_\varepsilon )+\frac{2-p}{2}\gamma ^p\Big ]\\&\le p^\frac{p}{p-1}\frac{4-p}{2}\gamma ^p|\Omega |+p^\frac{p}{p-1}\int _{\Omega }\!\psi (N_\varepsilon )\quad \text {for all }t>0\text { and }\varepsilon \in (0,1), \end{aligned}$$

and hence, letting $C_7=C_7(p,\xi ):=2\cdot p^{\frac{p}{p-1}}\max \{\frac{4-p}{2}|\Omega |,1\}>0$, we can estimate

$$\begin{aligned} C_5\left\| \big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)}\right\| _{L^{2}(\Omega )}^\frac{2(6p-{\mathcal {N}})}{6(p-1)-{\mathcal {N}}}&\le C_5C_7^\frac{6p-{\mathcal {N}}}{6(p-1)-{\mathcal {N}}}\gamma ^\frac{p(6p-{\mathcal {N}})}{6(p-1)-{\mathcal {N}}}\nonumber \\&\quad +C_5C_7^\frac{6p-{\mathcal {N}}}{6(p-1)-{\mathcal {N}}}\left( \int _{\Omega }\!\psi (N_\varepsilon )\right) ^\frac{6p-{\mathcal {N}}}{6(p-1)-{\mathcal {N}}} \end{aligned}$$

(5.66)

for all $t>0$ and $\varepsilon \in (0,1)$. Arguing similarly, and using that $N_\varepsilon (x,t)=n_\varepsilon (x,t)-\gamma$ for all $(x,t)\in {\overline{\Omega }}\times [0,\infty )$, we also obtain

$$\begin{aligned} C_5\left\| \big (\psi '(N_\varepsilon )\big )^\frac{p}{2(p-1)}\right\| _{L^{\frac{2}{p}}(\Omega )}^\frac{2p}{p-1}&\le C_5\bigg (p^\frac{1}{p-1}\gamma |\Omega |+p^\frac{1}{p-1}\int _{\{N_\varepsilon>\gamma \}}N_\varepsilon \bigg )^\frac{p^2}{p-1}\nonumber \\&= C_5\left( p^\frac{1}{p-1}\gamma |\Omega |+p^\frac{1}{p-1}\int _{\{n_\varepsilon >2\gamma \}}(n_\varepsilon -\gamma )\right) ^\frac{p^2}{p-1}\nonumber \\&\le C_5p^\frac{p^2}{(p-1)^2}\left( \int _{\Omega }\!n_\varepsilon \right) ^\frac{p^2}{p-1} \end{aligned}$$

(5.67)

for all $t>0$ and $\varepsilon \in (0,1)$, whence collecting (5.65)–(5.67) concludes the proof. $\square$

For the final estimation step of this segment we return to $\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}n_\varepsilon ^2c_\varepsilon ^2$, where the influence of $n_\varepsilon$ can now be counteracted by the favorably signed $\mu \int _{\Omega }\!\psi '(N_\varepsilon )N_\varepsilon ^2$ on the left hand-side in Lemma 5.13.

Lemma 5.15

Let $p\in (\frac{{\mathcal {N}}}{2},2)$. Then there are $q=q(p)\ge p$, $a_j=a_j(p)>1$, $j\in \{1,2,3\}$, and $b_j=b_j(p)>1$, $j\in \{1,2,3,4\}$ such that for each $\lambda >0$ and any $\xi >0$ one can find $C=C(p,\xi ,\lambda )>0$ with the property that for each $\kappa \ge 0$, $\mu >0$, $c_\star \ge 0$ and all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) and the quantities $\gamma$, $N_\varepsilon$ and $\psi =\psi _{p,\gamma }$ provided by (5.54), (5.57) and (5.55), respectively, satisfy

$$\begin{aligned}&\lambda \int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}n_\varepsilon ^2c_\varepsilon ^2\nonumber \\&\quad \le \xi \int _{\Omega }\!\big |\nabla |\nabla c_\varepsilon |^p\big |^2+\xi \int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}|D^2c_\varepsilon |^2+\mu \int _{\Omega }\!\psi '(N_\varepsilon )N_\varepsilon ^{2}\nonumber \\&\qquad +\frac{Cc_\star ^{2a_1}}{\mu ^\frac{2a_1}{p+1}}\left( \int _{\Omega }\!|\nabla c_\varepsilon |^{2p}\right) ^{b_1}+\frac{Cc_\star ^{2a_2}}{\mu ^\frac{2a_2}{p+1}}\left( \int _{\Omega }\!|\nabla c_\varepsilon |^{2p}\right) ^{b_2}+\frac{C}{\mu ^{\frac{2a_3}{p+1}}}\left( \int _{\Omega }\!|\nabla c_\varepsilon |^{2p}\right) ^{b_3}\nonumber \\&\qquad +\frac{C}{\mu ^{\frac{2a_2}{p+1}}}\left( \int _{\Omega }\!|\nabla c_\varepsilon |^{2p}\right) ^{b_4}+\frac{C}{\mu ^\frac{2a_2}{p+1}}\int _{\Omega }\!|c_\star -c_\varepsilon |^{2q}+ C\gamma ^{p+1}\int _{\Omega }\!|c_\star -c_\varepsilon |^{2p}+Cc_\star ^{2p}\gamma ^{p+1} \end{aligned}$$

(5.68)

on $(0,\infty )$.

Proof

As $n_\varepsilon =N_\varepsilon +\gamma$ on $\Omega \times (0,\infty )$, in a first step we find that

$$\begin{aligned} \lambda \int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}n_\varepsilon ^2c_\varepsilon ^2=&\,\lambda \int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}(N_\varepsilon +\gamma )^2c_\varepsilon ^2\le 2\lambda \gamma ^2\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}c_\varepsilon ^2\nonumber \\&+2\lambda \int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}N_\varepsilon ^2c_\varepsilon ^2 \end{aligned}$$

(5.69)

for all $t>0$ and $\varepsilon \in (0,1)$. Then, recalling that by nonnegativity of $n_\varepsilon$ we have $N_\varepsilon \ge -\gamma$, we split the domain in the second integral to obtain

$$\begin{aligned} 2\lambda \int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}N_\varepsilon ^2c_\varepsilon ^2&=2\lambda \int _{\{N_\varepsilon \in [-\gamma ,\gamma ]\}}|\nabla c_\varepsilon |^{2p-2}N_\varepsilon ^2c_\varepsilon ^2 +2\lambda \int _{\{N_\varepsilon>\gamma \}}|\nabla c_\varepsilon |^{2p-2}N_\varepsilon ^2c_\varepsilon ^2\\&\le 2\lambda \gamma ^2\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}c_\varepsilon ^2\\&\quad +2\lambda \int _{\{N_\varepsilon>\gamma \}}|\nabla c_\varepsilon |^{2p-2}N_\varepsilon ^2c_\varepsilon ^2\quad \text {for all }t>0\text { and }\varepsilon \in (0,1), \end{aligned}$$

so that (5.69) turns into

$$\begin{aligned} \lambda \int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}n_\varepsilon ^2c_\varepsilon ^2\le 4\lambda \gamma ^2\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}c_\varepsilon ^2+2\lambda \int _{\{N_\varepsilon >\gamma \}}|\nabla c_\varepsilon |^{2p-2}N_\varepsilon ^2c_\varepsilon ^2 \end{aligned}$$

(5.70)

for all $t>0$ and $\varepsilon \in (0,1)$. Writing $\gamma ^2=\gamma ^{\frac{p-1}{p}}\cdot \gamma ^{\frac{p+1}{p}}$ we find that invoking Young’s inequality yields $C_1=C_1(\lambda )>0$ such that

$$\begin{aligned} 4\lambda \gamma ^2\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}c_\varepsilon ^2\le \gamma \int |\nabla c_\varepsilon |^{2p}+C_1\gamma ^{p+1}\int _{\Omega }\!c_\varepsilon ^{2p}\quad \text {for all }t>0\text { and }\varepsilon \in (0,1), \end{aligned}$$

so that drawing on Lemma 5.9 entails that for $\xi >0$ there is $C_2=C_2(p,\xi )>0$ such that

$$\begin{aligned}&4\lambda \gamma ^2\int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}c_\varepsilon ^2\nonumber \\&\quad \le \ \xi \int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}|D^2c_\varepsilon |^2+C_2\gamma ^{p+1}\int _{\Omega }\!|c_\star -c_\varepsilon |^{2p}+C_1\gamma ^{p+1}\int _{\Omega }\!(c_\varepsilon -c_\star +c_\star )^{2p}\nonumber \\&\quad \le \ \xi \int _{\Omega }\!|\nabla c_\varepsilon |^{2p-2}|D^2c_\varepsilon |^2+\big (C_2+4^{p}C_1\big )\gamma ^{p+1}\int _{\Omega }\!|c_\star -c_\varepsilon |^{2p}+4^p C_1|\Omega |c_\star ^{2p}\gamma ^{p+1} \end{aligned}$$

(5.71)

is valid for all $t>0$ and $\varepsilon \in (0,1)$, where we again relied on the facts that $|\nabla c_\varepsilon |=|\nabla (c_\star -c_\varepsilon )|$ and $|D^2c_\varepsilon |=|D^2(c_\star -c_\varepsilon )|$ on $\Omega \times (0,\infty )$. For the second integral in (5.70) we note that $2<p+1$ allows for an application of Young’s inequality to find $C_3=C_3(p,\lambda )>0$ satisfying

$$\begin{aligned} 2\lambda \int _{\{N_\varepsilon>\gamma \}}\!|\nabla c_\varepsilon |^{2p-2}N_\varepsilon ^2c_\varepsilon ^2\le p\mu \int _{\{N_\varepsilon>\gamma \}}\!N_\varepsilon ^{p+1}+\frac{C_3}{\mu ^{\frac{2}{p-1}}}\int _{\{N_\varepsilon >\gamma \}}\!|\nabla c_\varepsilon |^{2p+2}c_\varepsilon ^{\frac{2p+2}{p-1}} \end{aligned}$$

for all $t>0$ and $\varepsilon \in (0,1)$. Herein, since in the first integral we have $N_\varepsilon >\gamma$, we may rewrite $pN_\varepsilon ^{p+1}=\psi '(N_\varepsilon )N_\varepsilon ^2$ according to the definition of $\psi$ in (5.55), so that

$$\begin{aligned} 2\lambda \int _{\{N_\varepsilon>\gamma \}}|\nabla c_\varepsilon |^{2p-2}N_\varepsilon ^2c_\varepsilon ^2&\le \mu \int _{\{N_\varepsilon>\gamma \}}\psi '(N_\varepsilon )N_\varepsilon ^{2}+\frac{C_3}{\mu ^{\frac{2}{p-1}}}\int _{\{N_\varepsilon >\gamma \}}|\nabla c_\varepsilon |^{2p+2}c_\varepsilon ^{\frac{2p+2}{p-1}}\nonumber \\&\le \mu \int _{\Omega }\!\psi '(N_\varepsilon )N_\varepsilon ^{2}+\frac{C_3}{\mu ^{\frac{2}{p-1}}}\int _{\Omega }\!|\nabla c_\varepsilon |^{2p+2}c_\varepsilon ^{\frac{2p+2}{p-1}} \end{aligned}$$

(5.72)

for all $t>0$ and $\varepsilon \in (0,1)$. To treat the remaining integral in (5.72), we first estimate

$$\begin{aligned} \int _{\Omega }\!|\nabla c_\varepsilon |^{2p+2}c_\varepsilon ^\frac{2p+2}{p-1}&=\int _{\Omega }\!|\nabla c_\varepsilon |^{2(p+1)}(c_\varepsilon -c_\star +c_\star )^{\frac{2(p+1)}{p-1}}\nonumber \\&\le 4^{\frac{p+1}{p-1}}c_\star ^{\frac{2(p+1)}{p-1}}\int _{\Omega }\!|\nabla c_\varepsilon |^{2(p+1)}+4^\frac{p+1}{p-1}\int _{\Omega }\!|\nabla c_\varepsilon |^{2(p+1)}|c_\star -c_\varepsilon |^{\frac{2(p+1)}{p-1}} \end{aligned}$$

(5.73)

for all $t>0$ and $\varepsilon \in (0,1)$. Moreover, we note that the condition $p>\frac{{\mathcal {N}}}{2}$ implies $\frac{{\mathcal {N}}+2}{{\mathcal {N}}}p>p+1$ and that hence we can pick $\beta :=\beta (p)\in \big (p+1,\frac{({\mathcal {N}}+2)p}{{\mathcal {N}}}\big )$. Thus, an application of Hölder’s inequality entails that with $C_4=C_4(p,\lambda ):=4^{\frac{p+1}{p-1}}C_3|\Omega |^\frac{\beta -(p+1)}{\beta }$ we have

$$\begin{aligned} 4^{\frac{p+1}{p-1}}\frac{C_3}{\mu ^{\frac{2}{p-1}}}c_\star ^{\frac{2(p+1)}{p-1}}\int _{\Omega }\!|\nabla c_\varepsilon |^{2(p+1)}\le \frac{C_4}{\mu ^{\frac{2}{p-1}}}c_\star ^{\frac{2(p+1)}{p-1}}\Vert \nabla c_\varepsilon \Vert _{L^{2\beta }(\Omega )}^{2(p+1)}\quad \text {for all }t>0\text { and }\varepsilon \in (0,1) \end{aligned}$$

and, likewise, an application of Young’s inequality shows that

$$\begin{aligned} 4^{\frac{p+1}{p-1}}\frac{C_3}{\mu ^{\frac{2}{p-1}}}\int _{\Omega }\!|\nabla c_\varepsilon |^{2(p+1)}|c_\star -c_\varepsilon |^{\frac{2(p+1)}{p-1}}\le 4^{\frac{p+1}{p-1}}\frac{C_3}{\mu ^{\frac{2}{p-1}}}\int _{\Omega }\!|\nabla c_\varepsilon |^{2\beta }+4^{\frac{p+1}{p-1}}\frac{C_3}{\mu ^{\frac{2}{p-1}}}\int _{\Omega }\!|c_\star -c_\varepsilon |^{2q} \end{aligned}$$

for all $t>0$ and $\varepsilon \in (0,1)$, where we abbreviated $q=q(p):=\frac{(p+1)\beta }{(p-1)(\beta -(p+1))}$. Note that $\beta <\frac{({\mathcal {N}}+2)p}{{\mathcal {N}}}$ and $p<2$ imply $\beta <\frac{p(p+1)}{p-1}$ and thus

$$\begin{aligned} \frac{\beta }{\beta -(p+1)}\ge p, \end{aligned}$$

which entails that $q\ge p$ is valid in light of $\frac{p+1}{p-1}>1$. Hence, we find $C_5=C_5(p,\lambda )>0$ such that in (5.73) we have

$$\begin{aligned} \frac{C_3}{\mu ^{\frac{2}{p-1}}}\int _{\Omega }\!|\nabla c_\varepsilon |^{2p+2}c_\varepsilon ^\frac{2p+2}{p-1}\le&\frac{C_5}{\mu ^{\frac{2}{p-1}}}c_\star ^{\frac{2(p+1)}{p-1}}\Vert \nabla c_\varepsilon \Vert _{L^{2\beta }(\Omega )}^{2(p+1)}+\frac{C_5}{\mu ^{\frac{2}{p-1}}}\Vert \nabla c_\varepsilon \Vert _{L^{2\beta }(\Omega )}^{2\beta }\nonumber \\&+\frac{C_5}{\mu ^{\frac{2}{p-1}}}\Vert c_\star -c_\varepsilon \Vert _{L^{2q}(\Omega )}^{2q} \end{aligned}$$

(5.74)

for all $t>0$ and $\varepsilon \in (0,1)$. To control the terms involving the spatial gradient of $c_\varepsilon$ we let $r=r(p):=\frac{{\mathcal {N}}(\beta -p)}{2\beta }\in (0,1)$ and draw on the Gagliardo–Nirenberg inequality to find $C_6=C_6(p)>0$ such that

$$\begin{aligned} \Vert \nabla c_\varepsilon \Vert _{L^{2\beta }(\Omega )}^{2}&=\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{{\nicefrac {2\beta }{p}}\;\!}\!\left( \Omega \right) }^\frac{2}{p}\le C_6\big \Vert |\nabla |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^\frac{2r}{p}\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^\frac{2(1-r)}{p}+C_6\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^\frac{2}{p} \end{aligned}$$

for all $t>0$ and $\varepsilon \in (0,1)$, whence we conclude from (5.74) that there is $C_7=C_7(p,\lambda )>0$ satisfying

$$\begin{aligned}&\frac{C_3}{\mu ^{\frac{2}{p-1}}}\int _{\Omega }\!|\nabla c_\varepsilon |^{2p+2}c_\varepsilon ^\frac{2p+2}{p-1}\\&\quad \le \frac{C_7}{\mu ^{\frac{2}{p-1}}}c_\star ^{\frac{2(p+1)}{p-1}}\left( \big \Vert |\nabla |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^\frac{2r(p+1)}{p}\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^\frac{2(p+1)(1-r)}{p}+\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^\frac{2(p+1)}{p}\right) \\&\qquad +\frac{C_7}{\mu ^{\frac{2}{p-1}}}\left( \big \Vert |\nabla |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^\frac{2r\beta }{p}\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^\frac{2(1-r)\beta }{p}+\big \Vert |\nabla c_\varepsilon |^p\big \Vert _{L^{2}(\Omega )}^\frac{2\beta }{p}\right) +\frac{C_7}{\mu ^{\frac{2}{p-1}}}\Vert c_\star -c_\varepsilon \Vert _{L^{2q}(\Omega )}^{2q} \end{aligned}$$

for all $t>0$ and $\varepsilon \in (0,1)$. Now, since our choice of $p+1<\beta <\frac{({\mathcal {N}}+2)p}{{\mathcal {N}}}$ implies that $\frac{2r(p+1)}{p}<\frac{2r\beta }{p}<2$, we can invoke Young’s inequality twice to conclude the existence of $C_8=C_8(p,\xi ,\lambda )>0$ such that

$$\begin{aligned}&C_3\int _{\Omega }\!|\nabla c_\varepsilon |^{2p+2}c_\varepsilon ^\frac{2p+2}{p-1}\nonumber \\&\quad \le \xi \int _{\Omega }\!\big |\nabla |\nabla c_\varepsilon |^p\big |^2 +\frac{C_8c_\star ^{2a_1}}{\mu ^{\frac{2a_1}{p+1}}}\left( \int _{\Omega }\!|\nabla c_\varepsilon |^{2p}\right) ^{b_1}+\frac{C_7c_\star ^{2a_2}}{\mu ^{\frac{2a_2}{p+1}}}\left( \int _{\Omega }\!|\nabla c_\varepsilon |^{2p}\right) ^{b_2}\nonumber \\&\qquad +\frac{C_8}{\mu ^{\frac{2a_3}{p+1}}}\left( \int _{\Omega }\!|\nabla c_\varepsilon |^{2p}\right) ^{b_3}+\frac{C_7}{\mu ^{\frac{2a_2}{p+1}}}\left( \int _{\Omega }\!|\nabla c_\varepsilon |^{2p}\right) ^{b_4}+\frac{C_7}{\mu ^{\frac{2a_2}{p+1}}}\int _{\Omega }\!|c_\star -c_\varepsilon |^{2q} \end{aligned}$$

(5.75)

holds for all $t>0$ and $\varepsilon \in (0,1)$, where $a_1=\frac{(p+1)p}{(p-1)(p-r(p+1))}>1$, $b_1=\frac{(p+1)(1-r)}{p-r(p+1)}>1$, $a_2=\frac{p+1}{p-1}>1$, $b_2=\frac{p+1}{p}>1$, $a_3=\frac{p(p+1)}{(p-1)(p-r\beta )}>1$, $b_3=\frac{(1-r)\beta }{p-r\beta }>1$ and $b_4=\frac{\beta }{p}>1$ due to $p>1$ our choice for $\beta \in \big (p+1,\frac{({\mathcal {N}}+2)p}{{\mathcal {N}}}\big )$ and $r=\frac{{\mathcal {N}}(\beta -p)}{2\beta }$. Collecting (5.70)–(5.72) and (5.75) we arrive at (5.68). $\square$

5.4 Eventual smallness of the quantities $\int _{\Omega }\!n_\varepsilon ^p$ and $\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}$ for some $p>\tfrac{{\mathcal {N}}+6}{6}$

Combining all the inequalities from the previous section we arrive at the announced ODE with superlinear forcing terms. To be precise we have the following.

Lemma 5.16

Let $p\in (\frac{{\mathcal {N}}+6}{6},2)$, $r>{\mathcal {N}}$, $\mu _0>0$ and $c_\star \ge 0$. There are $q_{\diamond }=q_{\diamond }(p)\ge p$, $b_{\diamond }^{(j)}=b_{\diamond }^{(j)}(p)>1$, $j\in \{1,\dots ,5\}$ and $K_{\diamond }^{(1)}=K_{\diamond }^{(1)}(p,r,\mu _0,c_\star )>0$ as well as $K_{\diamond }^{(2)}=K_{\diamond }^{(2)}(p,r)>0$ with the following property: For each $\kappa \ge 0$, $\mu \ge \mu _0$ and all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) and the quantities $\gamma$, $N_\varepsilon$ and $\psi =\psi _{p,\gamma }$ provided by (5.54), (5.57) and (5.55), respectively, satisfy

$$\begin{aligned}&\frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\bigg (\int _{\Omega }\!\psi (N_\varepsilon )+\int _{\Omega }\!|\nabla c_\varepsilon |^{2p} \bigg )\\&\quad \le \ K_{\diamond }^{(1)} \sum _{j=1}^5\left( \int _{\Omega }\!\psi (N_\varepsilon )+\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}\right) ^{b_{\diamond }^{(j)}} +K_{\diamond }^{(1)}\Vert c_\star -c_\varepsilon \Vert _{L^{2q}(\Omega )}^{2q}+K_{\diamond }^{(2)}\bigg (\int _{\Omega }\!n_\varepsilon \bigg )^{\frac{p^2}{p-1}}\\&\qquad +K_{\diamond }^{(2)}\Big (3+\gamma ^{p+1}+\Vert u_\varepsilon \Vert _{L^{r}(\Omega )}^{\frac{2r(p+1)}{r-{\mathcal {N}}}}+\Vert u_\varepsilon \Vert _{L^{r}(\Omega )}^{2(p+1)}\Big )\Vert c_\star -c_\varepsilon \Vert _{L^{2q}(\Omega )}^{2p}\\&\qquad +K_{\diamond }^{(2)}\Big (c_\star ^{2p}\gamma ^{p+1}+\gamma ^{\frac{p(6p-{\mathcal {N}})}{6(p-1)-{\mathcal {N}}}}+\gamma ^{\frac{p^2}{p-1}}\Big ) \end{aligned}$$

on $(0,\infty )$.

Proof

Take any $p\in (\frac{{\mathcal {N}}+6}{6},2)$, which in particular also implies that $p>\frac{{\mathcal {N}}}{2}$, and define $\lambda _0=\lambda _0(p):=2p\big (2(p-1)+{\mathcal {N}}\big )$ and $\xi _0=\xi _0(p):=\min \big \{\frac{p-1}{2p},\frac{p}{5}\big \}$. Then let $C_j=C_j(p,\xi _0)>0$, $j\in \{1,2,3\}$ be the constants provided by Lemmas 5.10,  5.13 and  5.14, respectively, and let the constant obtained in Lemma 5.11 be denoted by $C_4(p,r,\xi _0,\lambda _0)>0$. Moreover, let $q=q(p)\ge p$, $a_j=a_j(p)>1$, $j\in \{1,2,3\}$ and $b_j=b_j(p)>1$, $j\in \{1,\dots ,4\}$ as well as $C_5(p,\xi _0,\lambda _0)>0$ be given by Lemma 5.15 and additionally introduce $b_5=b_5(p):=\frac{6p-{\mathcal {N}}}{6(p-1)-{\mathcal {N}}}>1$. If we now set

$$\begin{aligned} C_0=C_0(p,r):=\max \{C_1,\dots ,C_5\}\quad \text {and}\quad \underline{\mu _0}:=\min \{\mu _0^{a_1},\mu _0^{a_2},\mu _0^{a_3}\}^{\frac{2}{p+1}}, \end{aligned}$$

and make use of the assumption $\mu \ge \mu _0$, we find upon combination of the estimates prepared in Lemmas 5.7, 5.10, 5.11, 5.13, 5.14 and 5.15 when employed for $\xi =\xi _0$ and $\lambda =\lambda _0$ that

$$\begin{aligned}&\frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\bigg (\int _{\Omega }\!\psi (N_\varepsilon )+\int _{\Omega }\!|\nabla c_\varepsilon |^{2p} \bigg )\\&\quad \le \ \frac{C_0c_\star ^{2a_1}}{\underline{\mu _0}}\left( \int _{\Omega }\!|\nabla c_\varepsilon |^{2p}\right) ^{b_1} +\frac{C_0c_\star ^{2a_2}}{\underline{\mu _0}}\left( \int _{\Omega }\!|\nabla c_\varepsilon |^{2p}\right) ^{b_2}\nonumber +\frac{C_0}{\underline{\mu _0}}\left( \int _{\Omega }\!|\nabla c_\varepsilon |^{2p}\right) ^{b_3}\\&\qquad +\frac{C_0}{\underline{\mu _0}}\left( \int _{\Omega }\!|\nabla c_\varepsilon |^{2p}\right) ^{b_4}+C_0\bigg (\int _{\Omega }\!\psi (N_\varepsilon )\bigg )^{b_5}+\frac{C_0}{\underline{\mu _0}}\int _{\Omega }\!(c_\star -c_\varepsilon )^{2q}+C_0\bigg (\int _{\Omega }\!n_\varepsilon \bigg )^{\frac{p^2}{p-1}}\\&\qquad +C_0\Big (3+\gamma ^{p+1}+\Vert u_\varepsilon \Vert _{L^{r}(\Omega )}^{\frac{2r(p+1)}{r-{\mathcal {N}}}}+\Vert u_\varepsilon \Vert _{L^{r}(\Omega )}^{2(p+1)}\Big )\int _{\Omega }\!(c_\star -c_\varepsilon )^{2p}\\&\qquad +C_0\Big (c_\star ^{2p}\gamma ^{p+1}+\gamma ^{\frac{p(6p-{\mathcal {N}})}{6(p-1)-{\mathcal {N}}}}+\gamma ^{\frac{p^2}{p-1}}\Big ) \end{aligned}$$

for all $t>0$ and $\varepsilon \in (0,1)$. Herein, we make use of the fact that $p\ge q$ and the Hölder inequality to find that with $C_6:=\max \{1,|\Omega |\}$ we can estimate the last integral on the right hand side in both of the cases $p>q$ and $p=q$ by

$$\begin{aligned} \Vert c_\star -c_\varepsilon \Vert _{L^{2p}(\Omega )}^{2p}\le |\Omega |^{\frac{q-p}{q}}\Vert c_\star -c_\varepsilon \Vert _{L^{2q}(\Omega )}^{2p}\le C_6\Vert c_\star -c_\varepsilon \Vert _{L^{2q}(\Omega )}^{2p} \end{aligned}$$

for all $t>0$. Therefore, the asserted differential inequality follows upon letting $q_{\diamond }=q_{\diamond }(p):=q$, $b_{\diamond }^{(j)}=b_{\diamond }^{(j)}(p):=b_j$, $K_{\diamond }^{(1)}=K_{\diamond }^{(1)}(p,r,\mu _0,c_\star ):=\frac{1}{\underline{\mu _0}}\max \{C_0c_\star ^{2a_1},C_0c_\star ^{2a_2},C_0\}$ and $K_{\diamond }^{(2)}=K_{\diamond }^{(2)}(p,r):=C_6C_0$ due to $\int _{\Omega }\!\psi (N_\varepsilon )$ and $\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}$ both being nonnegative. $\square$

Making use of the fact that an optimal combination of the relaxation results from Sects. 5.1 and 5.2 entail the existence of some time $t_\varepsilon >0$ such that $\int _{\Omega }\!\psi \big (N_\varepsilon (\cdot ,t_\varepsilon )\big )+\int _{\Omega }\!|\nabla c_\varepsilon (\cdot ,t_\varepsilon )|^{2p}$ is small, we can now follow a reasoning similar to [53, Lemma 5.4] to exploit the ODE from Lemma 5.16 to obtain smallness also for larger times and, making use of the asymptotics from Lemma 5.1, even for $\int _{\Omega }\!n_\varepsilon ^p$ instead of just the shifted version $\int _{\Omega }\!\psi (N_\varepsilon )$.

Proposition 5.17

Let $\omega >0$, $\mu _0>0$ and $c_\star \ge 0$. Then one can find $\theta _{\diamond }^{(6)}=\theta _{\diamond }^{(6)}(\omega )>0$ such that for all $\delta >0$ there is $\eta _{\diamond }^{(6)}(\omega ,\mu _0,\delta ,c_\star )>0$ with the property that whenever $\kappa \ge 0$ and $\mu \ge \mu _0$ satisfy (1.6) for some $\eta <\eta _{\diamond }^{(6)}(\omega ,\mu _0,\delta ,c_\star )$, there is $t_{\diamond }^{(6)}=t_{\diamond }^{(6)}(\omega ,\mu _0,\delta ,\eta ,n_0,c_0,u_0,c_\star )>0$ such that for any $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) satisfies

$$\begin{aligned} \int _{\Omega }\!n_\varepsilon ^{p}(\cdot ,t)+\int _{\Omega }\!|\nabla c_\varepsilon (\cdot ,t)|^{2p}<\delta \quad \text {for all }t\ge t_{\diamond }^{(6)}, \end{aligned}$$

where $p=p(\omega ):=\frac{{\mathcal {N}}+6}{6}+\theta _{\diamond }^{(6)}$.

Proof

Given $\omega >0$ denote with $\theta _1=\theta _1(\omega ):=\theta _{\diamond }^{(1)}(\omega )\in (0,\min \{1-\frac{{\mathcal {N}}}{6},\omega \})$, $\theta _3=\theta _3(\omega ):=\theta _{\diamond }^{(3)}(\omega )>0$ and $\theta _5=\theta _5(\omega ):=\theta _{\diamond }^{(5)}(\omega )>2-\frac{2}{3} {\mathcal {N}}$ the numbers provided by Lemmas 5.2, 5.4 and 5.6, respectively. Accordingly, remarking that $\frac{{\mathcal {N}}-3}{3}+\frac{\theta _5}{2}>0$ for both ${\mathcal {N}}=2$ and ${\mathcal {N}}=3$, we introduce $\theta _6=\theta _{\diamond }^{(6)}(\omega ):=\min \{\theta _1,\frac{{\mathcal {N}}-3}{3}+\frac{\theta _5}{2}\}>0$, $p=p(\omega )=\frac{{\mathcal {N}}+6}{6}+\theta _6$ and $r=r(\omega ):={\mathcal {N}}+\theta _3$ and then let $q=q(\omega ):=q_{\diamond }(p)\ge 1$, $b_j=b_j(\omega ):=b_{\diamond }^{(j)}(p)>1$, $j\in \{1,\dots ,5\}$ and $K_1=K_1(\omega ,\mu _0,c_\star ):=K_{\diamond }^{(1)}(p,r,\mu _0,c_\star )>0$ as well as $K_2=K_2(\omega ):=K_{\diamond }^{(2)}(p,r)>0$ be given by Lemma 5.16.

Let us proceed with the preparation of suitably small parameters $\delta _i>$, $i\in \{1,\dots ,4\}$ and $\gamma _0>0$. To this end let $\delta >0$ be given. We then pick a small $\delta _1=\delta _1(\omega ,\mu _0,\delta ,c_\star )>0$ such that

$$\begin{aligned} 2^p\delta _1<\frac{\delta }{2}\qquad \text {and such that}\qquad K_1\sum _{j=1}^5\delta _1^{b_j-1}\le \frac{1}{8}, \end{aligned}$$

(5.76)

which is possible due to $b_j>1$ for $j\in \{1,\dots ,5\}$. Then we fix $\delta _2=\delta _2(\omega )>0$ such that

$$\begin{aligned} \delta _2^\frac{2r(p+1)}{r-{\mathcal {N}}}+\delta _2^{2(p+1)}<\frac{1}{2} \end{aligned}$$

(5.77)

and $\gamma _0=\gamma _0(\omega ,\mu _0,\delta ,c_\star )>0$ such that

$$\begin{aligned} \gamma _0^{p+1}<\frac{1}{2} \end{aligned}$$

(5.78)

and

$$\begin{aligned}&2^{p-1}(4-p)|\Omega |\gamma _0^p<\frac{\delta }{2}, \end{aligned}$$

(5.79)

$$\begin{aligned}&K_2(2|\Omega |\gamma _0)^\frac{p^2}{p-1}<\frac{\delta _1}{32}, \end{aligned}$$

(5.80)

as well as

$$\begin{aligned} K_2\Big (c_\star ^{2p}\gamma _0^{p+1}+\gamma _0^\frac{p(6p-{\mathcal {N}})}{6(p-1)-{\mathcal {N}}}+\gamma _0^\frac{p^2}{p-1}\Big )<\frac{\delta _1}{32} \end{aligned}$$

(5.81)

are valid. Likewise, we pick $\delta _3=\delta _3(\omega ,\mu _0,\delta ,c_\star )>0$ satisfying

$$\begin{aligned} K_1\delta _3^q<\frac{\delta _1}{32}\quad \text {and}\quad 4K_2\delta _3^p<\frac{\delta _1}{32} \end{aligned}$$

(5.82)

and $\delta _4=\delta _4(\omega ,\mu _0,\delta ,c_\star )>0$ fulfilling

$$\begin{aligned} (2\delta _4)^p<\frac{\delta _1}{4}. \end{aligned}$$

(5.83)

These numbers at hand, we take $\eta _1=\eta _1(\omega ,\mu _0,\delta ,c_\star ):=\eta _{\diamond }^{(1)}(\omega ,\delta _4)>0$, $\eta _3=\eta _3(\omega ):=\eta _{\diamond }^{(3)}(\omega ,\delta _2)>0$, $\eta _4=\eta _4(\omega ,\mu _0,\delta ,c_\star ):=\eta _{\diamond }^{(4)}(\omega ,q,\delta _3,c_\star )>0$ and $\eta _5=\eta _5(\omega ,\mu _0,\delta ,c_\star ):=\eta _{\diamond }^{(5)}(\omega ,\delta _4,c_\star )>0$ as provided by Lemmas 5.2, 5.4, 5.5 and 5.6, respectively. We can then define

$$\begin{aligned} \eta _6=\eta _{\diamond }^{(6)}(\omega ,\mu _0,\delta ,c_\star ):=\min \big \{\eta _1,\eta _3,\eta _{4},\eta _5,\gamma _0\big \}>0 \end{aligned}$$

(5.84)

and claim that with this $\eta _6$ our assertion holds. To this end, we henceforth assume that $\kappa \ge 0$ and $\mu \ge \mu _0$ satisfy (1.6) for some $\eta <\eta _6$. Since $\eta <\eta _1$ and $p\le \frac{{\mathcal {N}}+6}{6}+\theta _1$, we find that with $t_1=t_1(\omega ,\mu _0,\delta ,\eta ,c_\star ):=t_{\diamond }^{(1)}(\omega ,\mu _0,\delta _4,\eta )>0$ from Lemma 5.2 we have

$$\begin{aligned} \int _{t-1}^t\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s <\delta _4^2\quad \text {for all }t\ge t_1+1\text { and }\varepsilon \in (0,1). \end{aligned}$$

(5.85)

Similarly, $\eta <\eta _3$ and $r={\mathcal {N}}+\theta _3$ in combination with Lemma 5.4 entail that we can find $t_3=t_3(\omega ,\mu _0,\delta ,\eta ,n_0,u_0):=t_{\diamond }^{(3)}(\omega ,\mu _0,\delta _2,\eta ,n_0,u_0)>0$ for which

$$\begin{aligned} \Vert u_\varepsilon (\cdot ,t)\Vert _{L^{r}(\Omega )}\le \delta _2\quad \text {for all }t\ge t_3\text { and }\varepsilon \in (0,1), \end{aligned}$$

(5.86)

whereas, stressing once more that p and q only depend on $\omega$, we infer from $\eta <\eta _{4}$ and Lemma 5.5 that there is some $t_4=t_4(\omega ,\mu _0,\delta ,\eta ,c_0,c_\star ):=t_{\diamond }^{(4)}(\omega ,\mu _0,\delta _3,c_\star )>0$ such that

$$\begin{aligned} \big \Vert \big (c_\star -c_\varepsilon \big )(\cdot ,t)\big \Vert _{L^{2q}(\Omega )}^{2}<\delta _3\quad \text {for all }t\ge t_4\text { and }\varepsilon \in (0,1). \end{aligned}$$

(5.87)

In addition, due to $\eta <\eta _5$ and $2p=\frac{{\mathcal {N}}+6}{3}+2\theta _6\le \frac{{\mathcal {N}}+6}{3}+\frac{2{\mathcal {N}}-6}{3}+\theta _5={\mathcal {N}}+\theta _5$, an application of Lemma 5.6 entails that there is $t_5=t_5(\omega ,\mu _0,\delta ,\eta ,n_0,c_0,u_0,c_\star ):=t_{\diamond }^{(5)}(\omega ,\mu _0,\delta _4,\eta ,n_0,c_0,u_0,c_\star )>0$ such that

$$\begin{aligned} \int _{t-1}^t\Vert \nabla c_\varepsilon (\cdot ,s)\Vert _{L^{2p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s<\delta _4\quad \text {for all }t\ge t_5+1\text { and }\varepsilon \in (0,1) \end{aligned}$$

(5.88)

and, moreover, we conclude from $\eta <\gamma _0$ and the assumption (1.6) that $\gamma :=\frac{\kappa }{\mu }$ fulfills

$$\begin{aligned} \gamma <\gamma _0, \end{aligned}$$

which entails that ${\hat{\kappa }}:=\mu \gamma _0$ satisfies ${\hat{\kappa }}>\kappa$. Hence, we can employ Lemma 5.1 for this ${\hat{\kappa }}$ to obtain that

$$\begin{aligned} \int _{\Omega }\!n_\varepsilon (\cdot ,t)\le 2|\Omega |\gamma _0\quad \text {for all }t\ge \frac{\ln (2)}{\mu _0\gamma _0}\text { and }\varepsilon \in (0,1), \end{aligned}$$

(5.89)

where we used that ${\hat{\kappa }}\ge \mu _0\gamma _0$ and hence $\frac{\ln (2)}{{\hat{\kappa }}}\le \frac{\ln (2)}{\mu _0\gamma _0}$. Accordingly, introducing

$$\begin{aligned} t_6=t_{\diamond }^{(6)}(\omega ,\mu _0,\delta ,\eta ,n_0,c_0,u_0,c_\star ):=\max \Big \{t_1+1,t_3,t_4,t_5+1,\frac{\ln (2)}{\mu _0\gamma _0}\Big \}>0, \end{aligned}$$

we find from (5.85), (5.88) and an application of Hölder’s inequality that

$$\begin{aligned}&\int _{t-1}^t\!\Big (\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{p}(\Omega )}+\Vert \nabla c_\varepsilon (\cdot ,s)\Vert _{L^{2p}(\Omega )}^2\Big ){{\,\mathrm{d\!}\,}}s\\&\quad \le \bigg (\int _{t-1}^t\!\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{p}(\Omega )}^2\bigg )^{\frac{1}{2}}\!+\int _{t-1}^t\!\Vert \nabla c_\varepsilon (\cdot ,s)\Vert _{L^{2p}(\Omega )}^2{{\,\mathrm{d\!}\,}}s\le 2\delta _4 \end{aligned}$$

for all $t\ge t_6$ and $\varepsilon \in (0,1)$, from which we conclude that for each $\varepsilon \in (0,1)$ and $t\ge t_6$ there is $t_\varepsilon =t_\varepsilon (t)\in (t-1,t)$ with

$$\begin{aligned} \Vert n_\varepsilon (\cdot ,t_\varepsilon )\Vert _{L^{p}(\Omega )}+\Vert \nabla c_\varepsilon (\cdot ,t_\varepsilon )\Vert _{L^{2p}(\Omega )}^2\le 2\delta _4, \end{aligned}$$

so that by (5.83) we have

$$\begin{aligned} \Vert n_\varepsilon (\cdot ,t_\varepsilon )\Vert _{L^{p}(\Omega )}^p+\Vert \nabla c_\varepsilon (\cdot ,t_\varepsilon )\Vert _{L^{2p}(\Omega )}^{2p}<(2\delta _4)^p\le \frac{\delta _1}{4}. \end{aligned}$$

Then, since $\psi =\psi _{p,\gamma }$ as defined in (5.55) satisfies $\psi (\sigma )\le \sigma ^p$ for all $\sigma \ge 0$ and $\psi (\sigma )=0$ for $\sigma <0$, we find from the above that $y_\varepsilon \in C^{0}\!\left( [t_6,\infty )\right) \cap C^{1}\!\left( [t_\varepsilon ,\infty )\right)$ given by

$$\begin{aligned} y_\varepsilon (s):=\int _{\Omega }\!\psi \big (N_\varepsilon (\cdot ,s)\big )+\int _{\Omega }\!|\nabla c_\varepsilon (\cdot ,s)|^{2p},\qquad s\ge t_6, \end{aligned}$$

fulfills

$$\begin{aligned} y_\varepsilon (t_\varepsilon )\le \int _{\Omega }\!N_\varepsilon ^p(\cdot ,t_\varepsilon )+\int _{\Omega }\!|\nabla c_\varepsilon (\cdot ,t_\varepsilon )|^{2p}<\frac{\delta _1}{4} \end{aligned}$$

and that thus we can introduce the non-empty set

$$\begin{aligned} S_\varepsilon :=\left\{ T_0\in [t_\varepsilon ,t]\ \Big |\ y_\varepsilon (s)<\delta _1\text { for all }s\in [t_\varepsilon ,T_0]\right\} \end{aligned}$$

(5.90)

and its well-defined supremum $T_{\varepsilon }:=\sup S_\varepsilon \in (t_\varepsilon ,t]$. Now, if in fact $T_{\varepsilon }=t$, we may conclude that $y_\varepsilon (t)\le \delta _1$ holds for all $t\ge t_6$. To verify this claim we will be aided by the differential inequality we prepared in Lemma 5.16. First, however, let us remark that in light of $t_\varepsilon \ge t_6$, (5.89), (5.87) and (5.86) we have

$$\begin{aligned}&\sup _{s>t_\varepsilon }\int _{\Omega }\!n_\varepsilon (\cdot ,s)\le 2|\Omega |\gamma _0\quad \text {and}\quad \sup _{s>t_\varepsilon }\big \Vert \big (c_\star -c_\varepsilon \big )(\cdot ,s)\big \Vert _{L^{2q}(\Omega )}^2<\delta _3\quad \text {as well as}\quad \\&\sup _{s>t_\varepsilon }\Vert u_\varepsilon (\cdot ,s)\Vert _{L^{r}(\Omega )}<\delta _2. \end{aligned}$$

Then, using $\gamma <\gamma _0$, we conclude from Lemma 5.16 that

$$\begin{aligned}&\frac{{{\,\mathrm{d\!}\,}}}{{{\,\mathrm{d\!}\,}}t}\bigg (\int _{\Omega }\!\psi (N_\varepsilon )+\int _{\Omega }\!|\nabla c_\varepsilon |^{2p} \bigg )\\&\quad \le \ K_1\sum _{j=1}^5\left( \int _{\Omega }\!\psi \big (N_\varepsilon (\cdot ,t)\big )+\int _{\Omega }\!|\nabla c_\varepsilon |^{2p}\right) ^{b_j}+K_1\delta _3^q+K_2\big (2|\Omega |\gamma _0\big )^{\frac{p^2}{p-1}}\\&\qquad +K_2\Big (3+\gamma _0^{p+1}+\delta _2^{\frac{2r(p+1)}{r-{\mathcal {N}}}}+\delta _2^{2(p+1)}\Big )\delta _3^p+K_2\Big (c_\star ^{2p}\gamma _0^{p+1}+\gamma _0^{\frac{p(6p-{\mathcal {N}})}{6(p-1)-{\mathcal {N}}}}+\gamma _0^{\frac{p^2}{p-1}}\Big ) \end{aligned}$$

holds for all $t>t_\varepsilon$. Consequently, in agreement with (5.80), (5.78), (5.77), (5.81) and (5.82) the function $y_\varepsilon$ satisfies

$$\begin{aligned} y_\varepsilon '(s)&\le K_1\sum _{j=1}^5y_\varepsilon ^{b_j}(s)+\frac{\delta _1}{32}+\frac{\delta _1}{32}+4K_2\delta _3^p+\frac{\delta _1}{32}\\&<K_1\sum _{j=1}^5y_\varepsilon ^{b_j}(s)+\frac{\delta _1}{32}+\frac{\delta _1}{32}+\frac{\delta _1}{32}+\frac{\delta _1}{32}=K_1\sum _{j=1}^5y_\varepsilon ^{b_j}(s)+\frac{\delta _1}{8} \end{aligned}$$

on $(t_\varepsilon ,\infty )$. Making additional use of the definition of the set $S_{\varepsilon }$ in (5.90), our choice of $T_{\varepsilon }$ and the second estimate in (5.76) we obtain

$$\begin{aligned} y_\varepsilon '(s)&< \delta _1\cdot \Big (K_1\sum _{j=1}^5\delta _1^{b_j-1}(s)+\frac{1}{8}\Big )\le \delta _1\Big (\frac{1}{8}+\frac{1}{8}\Big )=\frac{\delta _1}{4}\quad \text {on }(t_\varepsilon ,T_{\varepsilon }). \end{aligned}$$

Therefore, the integration of this inequality across $(t_\varepsilon ,s)$ entails

$$\begin{aligned} y_\varepsilon (s)<y_\varepsilon (t_\varepsilon )+\frac{\delta _1}{4}(s-t_\varepsilon )<y_\varepsilon (t_\varepsilon )+\frac{\delta _1}{4}(T_{\varepsilon }-t_\varepsilon )\le \frac{\delta _1}{4}+\frac{\delta _1}{4}=\frac{\delta _1}{2}\quad \text {for all }s\in (t_\varepsilon ,T_{\varepsilon }), \end{aligned}$$

due to $T_{\varepsilon }\le t<t_\varepsilon +1$, which, in light of the continuity of $y_\varepsilon$, excludes the possibility of $T_{\varepsilon }<t$ and thereby verifies that indeed

$$\begin{aligned} y_\varepsilon (t)\le \delta _1\quad \text {for all }t\ge t_6. \end{aligned}$$

(5.91)

To see that this eventual smallness of $y_\varepsilon$ also provides information towards the conjectured eventual smallness properties of $n_\varepsilon$ in $L^{p}(\Omega )$ and $\nabla c_\varepsilon$ in $L^{2p}(\Omega )$, we return to the specific form of $\psi$ to estimate $\int _{\Omega }\!\psi (N_\varepsilon )$ from below. In fact, recalling from (5.55) that $\psi (\sigma )\ge 0$ on ${\mathbb {R}}$ and $\psi (\sigma )=\sigma ^p-\frac{2-p}{2}\gamma ^p$ for $\sigma \ge \gamma$, we remove parts of the domain to obtain that

$$\begin{aligned} \int _{\Omega }\!\psi (N_\varepsilon )\ge \int _{\{N_\varepsilon>\gamma \}}\psi (N_\varepsilon )=\int _{\{N_\varepsilon>\gamma \}}N_\varepsilon ^p-\int _{\{N_\varepsilon>\gamma \}}\frac{2-p}{2}\gamma ^p\ge \int _{\{N_\varepsilon >\gamma \}}N_\varepsilon ^p-\frac{2-p}{2}|\Omega |\gamma ^p \end{aligned}$$

for all $t>0$. Moreover, as $N_\varepsilon \equiv n_\varepsilon -\gamma$ we can estimate $\int _{\{N_\varepsilon >\gamma \}}N_\varepsilon ^p$ against $\int _{\Omega }\!n_\varepsilon ^p$ by means of

$$\begin{aligned} \int _{\{N_\varepsilon>\gamma \}}N_\varepsilon ^p&=\int _{\{n_\varepsilon>2\gamma \}}(n_\varepsilon -\gamma )^p>\int _{\{n_\varepsilon>2\gamma \}}\Big (\frac{n_\varepsilon }{2}\Big )^p=\frac{1}{2^p}\int _{\Omega }\!n_\varepsilon ^p-\frac{1}{2^p}\int _{\{n_\varepsilon \le 2\gamma \}}n_\varepsilon ^p\\&\ge \frac{1}{2^p}\int _{\Omega }\!n_\varepsilon ^p-|\Omega |\gamma ^p\quad \text {for all }t>0 \end{aligned}$$

leading to

$$\begin{aligned} \int _{\Omega }\!n_\varepsilon ^p\le 2^p\int _{\Omega }\!\psi (N_\varepsilon )+\frac{(2-p)2^p}{2}|\Omega |\gamma ^p+2^p|\Omega |\gamma ^p\le 2^p\int _{\Omega }\!\psi (N_\varepsilon )+2^{p-1}(4-p)|\Omega |\gamma ^p \end{aligned}$$

(5.92)

for all $t>0$. A combination of (5.92) with (5.91), our condition $\gamma <\gamma _0$, the first inequality in (5.76) and (5.79) finally shows

$$\begin{aligned} \int _{\Omega }\!n_\varepsilon ^p(\cdot ,t)+\int _{\Omega }\!\big |\nabla c_\varepsilon (\cdot ,t)\big |^{2p}&\le 2^py_\varepsilon (t)+2^{p-1}(4-p)|\Omega |\gamma ^p\\&\le 2^p\delta _1+2^{p-1}(4-p)|\Omega |\gamma _0^p\\&\le \frac{\delta }{2}+\frac{\delta }{2}=\delta \qquad \text {for all }t\ge t_6, \end{aligned}$$

concluding the proof. $\square$

6 Eventual regularity of the weak solution

Recalling that $q={\mathcal {N}}$ is to some extent a critical number with respect to regularity enforced from bounds on $\nabla c_\varepsilon$ in $L^q$ in Keller–Segel-type systems and noticing that the previous proposition provides a corresponding bound in slightly better spaces, we are now in a position to derive eventual regularity estimates of higher order. In view of the Arzelà–Ascoli theorem, we can, finally, transfer these bounds to our limit object obtained in Proposition 4.1, from which we then may ultimately conclude that the weak solution provided by Theorem 1.1 satisfies enhanced regularity for large times under the assumptions of Theorem 1.2.

6.1 Eventual boundedness properties

The first step of our regularity improvements consists of exploiting $2p>{\mathcal {N}}$ in Proposition 5.17 together with well-known semigroup arguments to obtain an $L^\infty$ bound on $n_\varepsilon$ for large times, which, in a second step can then be refined into eventual Hölder regularity for the fluid-component, finally, making the equations of our PDE-systems accessible for parabolic Schauder theory. We start with the aforementioned eventual $L^\infty$ bound for $n_\varepsilon$, the acquisition of which is quite similar to [53, Lemma 6.1].

Lemma 6.1

Let $\omega >0$, $\mu _0>0$ and $c_\star \ge 0$. There is $\eta _{\diamond }^{(7)}=\eta _{\diamond }^{(7)}(\omega ,\mu _0,c_\star )>0$ such that whenever $\kappa \ge 0$ and $\mu \ge \mu _0>0$ satisfy (1.6) for some $\eta <\eta _{\diamond }^{(7)}(\omega ,\mu _0,c_\star )$, there are $t_{\diamond }^{(7)}=t_{\diamond }^{(7)}(\omega ,\mu _0,\eta ,n_0,c_0,u_0,c_\star )>0$ and $C=C(\omega ,\kappa ,\mu _0,\eta ,c_\star )>0$ such that for each $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) fulfills

$$\begin{aligned} \Vert n_\varepsilon (\cdot ,t)\Vert _{L^{\infty }(\Omega )}\le C\quad \text {for all }t\ge t_{\diamond }^{(7)}. \end{aligned}$$

Proof

We let $\theta _3=\theta _3(\omega ):=\theta _{\diamond }^{(3)}(\omega )$, $\eta _3=\eta _3(\omega ):=\eta _{\diamond }^{(3)}(\omega ,1)$ be provided by Lemma 5.4, take $\theta _6=\theta _6(\omega ):=\theta _{\diamond }^{(6)}(\omega )$, $\eta _6=\eta _6(\omega ,\mu _0,c_\star ):=\eta _{\diamond }^{(6)}(\omega ,\mu _0,1,c_\star )$ from Proposition 5.17 and, correspondingly, set $\theta _7=\theta _{\diamond }^{(7)}(\omega ):=\min \{\frac{\theta _3}{2},\theta _6\}>0$ and $\eta _7=\eta _{\diamond }^{(7)}(\omega ,\mu _0,c_\star ):=\min \{\eta _3,\eta _6\}>0$. Moreover, we again write $p=p(\omega ):=\frac{{\mathcal {N}}+6}{6}+\theta _7$, note that $2p>{\mathcal {N}}$ when ${\mathcal {N}}\in \{2,3\}$ and introduce $r=r(\omega ):={\mathcal {N}}+\theta _7$. Then, assuming that (1.6) is satisfied for some $\eta <\eta _7$ we find from an application of Lemma 5.4 that with $t_3=t_3(\omega ,\mu _0,\eta ,n_0,u_0):=t_{\diamond }^{(3)}(\omega ,\mu _0,1,\eta ,n_0,u_0)>0$ we have

$$\begin{aligned} \Vert u_\varepsilon (\cdot ,t)\Vert _{L^{r}(\Omega )}\le 1\quad \text {for all }t\ge t_3\ \text { and }\varepsilon \in (0,1) \end{aligned}$$

and Proposition 5.17 yields $t_6=t_6(\omega ,\mu _0,\eta ,n_0,c_0,u_0,c_\star ):=t_{\diamond }^{(6)}(\omega ,\mu _0,1,\eta ,n_0,c_0,u_0,c_\star )>0$ such that

$$\begin{aligned} \Vert \nabla c_\varepsilon (\cdot ,t)\Vert _{L^{2p}(\Omega )}\le 1\quad \text {for all }t\ge t_6\ \text { and }\varepsilon \in (0,1). \end{aligned}$$

Furthermore, letting $C_1=C_1(\omega ,\mu _0,\eta ,c_\star ):=\max \big \{1,2|\Omega |\eta \big \}>0$ and ${\hat{\kappa }}={\hat{\kappa }}(\mu ,\eta ):=\eta \mu$ we obtain from (1.6) that ${\hat{\kappa }}\ge \kappa$ and therefore Lemma 5.1 entails that

$$\begin{aligned} \Vert n_\varepsilon (\cdot ,t)\Vert _{L^{1}(\Omega )}\le \frac{2|\Omega |{\hat{\kappa }}}{\mu }= 2|\Omega |\eta \le C_1\quad \text {for all }t\ge \frac{\ln (2)}{{\hat{\kappa }}}=\frac{\ln (2)}{\eta \mu }\text { and }\varepsilon \in (0,1), \end{aligned}$$

so that with $t_7=t_{\diamond }^{(7)}(\omega ,\mu _0,\eta ,n_0,c_0,u_0,c_\star ):=\max \big \{t_3,t_6,\frac{\ln (2)}{\eta \mu _0}\big \}+1$ we conclude that for all $\varepsilon \in (0,1)$

$$\begin{aligned}&\Vert u_\varepsilon (\cdot ,t)\Vert _{L^{r}(\Omega )}\le C_1,\ \quad \Vert \nabla c_\varepsilon (\cdot ,t)\Vert _{L^{2p}(\Omega )}\le C_1\ \text {and}\nonumber \\&\Vert n_\varepsilon (\cdot ,t)\Vert _{L^{1}(\Omega )}\le C_1\quad \ \text {for all }t\ge t_7-1, \end{aligned}$$

(6.1)

due to $\mu \ge \mu _0$. Now, to estimate the $L^\infty$-norm of $n_\varepsilon$, we denote by $(e^{\sigma \Delta })_{\sigma \ge 0}$ the Neumann heat semigroup in $\Omega$ and draw on the maximum principle in (2.1a) to find that by means of a corresponding variation of constants representation we have

$$\begin{aligned}&\big \Vert n_\varepsilon (\cdot ,t)\big \Vert _{L^{\infty }(\Omega )}\le \big \Vert e^{\Delta }n_\varepsilon (\cdot ,t-1)\big \Vert _{L^{\infty }(\Omega )}\nonumber \\&\quad +\int _{t-1}^t\Big \Vert e^{(t-s)\Delta }\nabla \cdot \big ((\rho _\varepsilon f_\varepsilon (n_\varepsilon )\nabla c_\varepsilon +u_\varepsilon )n_\varepsilon \big )(\cdot ,s)\Big \Vert _{L^{\infty }(\Omega )}{{\,\mathrm{d\!}\,}}s\nonumber \\&\quad +\int _{t-1}^t \Big \Vert e^{(t-s)\Delta }\big (\kappa n_\varepsilon -\mu n_\varepsilon ^2\big )_{+}(\cdot ,s)\Big \Vert _{L^{\infty }(\Omega )}{{\,\mathrm{d\!}\,}}s \end{aligned}$$

(6.2)

for $t\ge t_7$ and $\varepsilon \in (0,1)$. Herein, a well-known smoothing property of the Neumann heat semigroup (e.g. [44, Lemma 1.3]) implies that there is $C_2>0$ satisfying

$$\begin{aligned} \big \Vert e^{\Delta }n_\varepsilon (\cdot ,t-1)\big \Vert _{L^{\infty }(\Omega )}\le C_2\big \Vert n_\varepsilon (\cdot ,t-1)\big \Vert _{L^{1}(\Omega )}\le C_1C_2\quad \text {for all }t\ge t_7\text { and }\varepsilon \in (0,1) \end{aligned}$$

(6.3)

and the point-wise estimates $0\le (\kappa \sigma -\mu \sigma ^2)_+\le \frac{\kappa ^2}{4\mu }\le \frac{\kappa ^2}{4\mu _0}$ for all $\sigma \ge 0$ entail that the third summand may be estimated by

$$\begin{aligned} \int _{t-1}^t\Big \Vert e^{(t-s)\Delta }\big (\kappa n_\varepsilon -\mu n_\varepsilon ^2\big )_{+}(\cdot ,s)\Big \Vert _{L^{\infty }(\Omega )}{{\,\mathrm{d\!}\,}}s\le \frac{\kappa ^2}{4\mu _0}\quad \text {for all }t\ge t_7\text { and }\varepsilon \in (0,1). \end{aligned}$$

(6.4)

For the second summand in (6.2) we again note $|\rho _\varepsilon (x)f_\varepsilon (s)|\le 1$ for $x\in \Omega$ and $s\ge 0$, pick $p_0=p_0(\omega )\in ({\mathcal {N}},2p)$ as well as $r_0=r_0(\omega )\in ({\mathcal {N}},r)$ and employ a smoothing estimate for the Neumann heat semigroup covered by e.g. [13, Lemma 3.3] or [26, Lemma 3.1], to find $C_3=C_3(\omega )>0$ such that

$$\begin{aligned}&\int _{t-1}^t\Big \Vert e^{(t-s)\Delta }\nabla \cdot \big ((\rho _\varepsilon f_\varepsilon (n_\varepsilon )\nabla c_\varepsilon +u_\varepsilon )n_\varepsilon \big )(\cdot ,s)\Big \Vert _{L^{\infty }(\Omega )}{{\,\mathrm{d\!}\,}}s\nonumber \\&\quad \le C_3\int _{t-1}^t(t-s)^{-\frac{1}{2}-\frac{{\mathcal {N}}}{2p_0}}\big \Vert (\nabla c_\varepsilon n_\varepsilon )(\cdot ,s)\big \Vert _{L^{p_0}(\Omega )}{{\,\mathrm{d\!}\,}}s\nonumber \\&\qquad +C_3\int _{t-1}^t(t-s)^{-\frac{1}{2}-\frac{{\mathcal {N}}}{2r_0}}\big \Vert (u_\varepsilon n_\varepsilon )(\cdot ,s)\big \Vert _{L^{r_0}(\Omega )}{{\,\mathrm{d\!}\,}}s \end{aligned}$$

(6.5)

for all $t\ge t_7$ and $\varepsilon \in (0,1)$. Additionally, introducing the obviously finite numbers

$$\begin{aligned} M_\varepsilon (T):=\sup _{t\in (t_7,T)}\big \Vert n_\varepsilon (\cdot ,t)\big \Vert _{L^{\infty }(\Omega )},\quad T\ge t_7,\ \varepsilon \in (0,1), \end{aligned}$$

as well as $a_1=a_1(\omega ):=1-\frac{2p-p_0}{2p\,p_0}\in (0,1)$ and $a_2=a_2(\omega ):=1-\frac{r-r_0}{r\,r_0}$, we obtain by means of Hölder’s inequality and (6.1) that

$$\begin{aligned} \big \Vert (\nabla c_\varepsilon n_\varepsilon )(\cdot ,s)\big \Vert _{L^{p_0}(\Omega )}&\le \big \Vert \nabla c_\varepsilon (\cdot ,s)\big \Vert _{L^{2p}(\Omega )}\big \Vert n_\varepsilon (\cdot ,s)\big \Vert _{L^{\frac{2p\,p_0}{2p-p_0}}(\Omega )}\nonumber \\&\le C_1 \big \Vert n_\varepsilon (\cdot ,s)\big \Vert _{L^{\infty }(\Omega )}^{a_1}\big \Vert n_\varepsilon (\cdot ,s)\big \Vert _{L^{1}(\Omega )}^{1-a_1}\nonumber \\&\le C_1^{2-a_1} M_\varepsilon ^{a_1}(T)\qquad \text {for all }\ s\in (t_7,T)\ \text { and }\ \varepsilon \in (0,1) \end{aligned}$$

(6.6)

and, similarly,

$$\begin{aligned} \big \Vert (u_\varepsilon n_\varepsilon )(\cdot ,s)\big \Vert _{L^{r_0}(\Omega )}\le \big \Vert u_\varepsilon (\cdot ,s)\big \Vert _{L^{r}(\Omega )}\big \Vert n_\varepsilon (\cdot ,s)\big \Vert _{L^{\frac{r\,r_0}{r-r_0}}(\Omega )}\le C_1^{2-a_2} M_\varepsilon ^{a_2}(T) \end{aligned}$$

(6.7)

for all $s\in (t_7,T)$ and $\varepsilon \in (0,1)$. Because of $p_0>{\mathcal {N}}$ and $r_0>{\mathcal {N}}$ we can moreover fix $C_4=C_4(\omega )>0$ satisfying $\int _0^1\sigma ^{-\frac{1}{2}-\frac{{\mathcal {N}}}{2p_0}}{{\,\mathrm{d\!}\,}}\sigma +\int _0^1\sigma ^{-\frac{1}{2}-\frac{{\mathcal {N}}}{2r_0}}{{\,\mathrm{d\!}\,}}\sigma \le C_4$, so that gathering the estimates (6.2)–(6.7) yields the existence of $C_i=C_i(\omega ,\kappa ,\mu _0,\eta ,c_\star )>0$, $i\in \{5,6\}$, such that for all $T>t_7$ and $\varepsilon \in (0,1)$ we have

$$\begin{aligned} \big \Vert n_\varepsilon (\cdot ,t)\big \Vert _{L^{\infty }(\Omega )}&\le C_5+C_5 M_\varepsilon ^{a_1}(T)+C_5 M_\varepsilon ^{a_2}(T)\\&\le C_6+ C_6 M_\varepsilon ^{a}(T)\quad \text {for all }t\in (t_7,T)\text { and }\varepsilon \in (0,1), \end{aligned}$$

with $a:=\max \{a_1,a_2\}$ according to Young’s inequality, which, due to $a<1$ also implies that

$$\begin{aligned} \sup _{t\in (t_7,T)}\big \Vert n_\varepsilon (\cdot ,t)\big \Vert _{L^{\infty }(\Omega )}=M_\varepsilon (T)\le \max \Big \{1, (2C_6)^\frac{1}{1-a}\Big \}\qquad \text {for all }T>t_7\ \text { and }\ \varepsilon \in (0,1) \end{aligned}$$

and thereby completes the proof upon taking $T\nearrow \infty$. $\square$

The eventual $L^\infty$ bound for $n_\varepsilon$ at hand, we can now turn our attention to estimates for the Stokes semigroup and embedding properties of $D(A_p^\varrho )$ for suitable $p>{\mathcal {N}}$ and appropriate values of $\varrho \in (\frac{{\mathcal {N}}}{2p},\frac{1}{2})$ to derive Hölder bounds on $u_\varepsilon$. (Refer also to [53, Lemma 7.1].)

Lemma 6.2

Let $\omega >0$, $\mu _0>0$ and $c_\star \ge 0$. There is $\eta _{\diamond }^{(8)}=\eta _{\diamond }^{(8)}(\omega ,\mu _0,c_\star )>0$ with the following property: Whenever $\kappa \ge 0$ and $\mu \ge \mu _0$ fulfill (1.6) for some $\eta <\eta _{\diamond }^{(8)}(\omega ,\mu _0,c_\star )$, one can find $\alpha =\alpha (\omega )\in (0,1)$, $t_{\diamond }^{(8)}=t_{\diamond }^{(8)}(\omega ,\mu _0,\eta ,n_0,c_0,u_0,c_\star )>0$ and $C=C(\omega ,\kappa ,\mu _0,\eta ,c_\star )>0$ such that for all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) satisfies

$$\begin{aligned} \Vert u_\varepsilon \Vert _{C^{\alpha ,\frac{\alpha }{2}}\!\,({\overline{\Omega }}\times [t,t+1])}\le C\quad \text {for all }t\ge t_{\diamond }^{(8)}. \end{aligned}$$

Proof

The Hölder bounds for $u_\varepsilon$ will be obtained by means of well-established embedding results for the domains of fractional powers of the Stokes operator in $L^{p}$ for $p>{\mathcal {N}}$. For this, however, we first need to fix some parameters. We take $\theta _3=\theta _3(\omega ):=\theta _{\diamond }^{(3)}(\omega )>0$ and $\eta _3=\eta _3(\omega ):=\eta _{\diamond }^{(3)}(\omega ,1)>0$ from Lemma 5.4 and let Lemma 6.1 provide $\eta _7=\eta _7(\omega ,\mu _0,c_\star ):=\eta _{\diamond }^{(7)}(\omega ,\mu _0,c_\star )>0$. We then set $\eta _8=\eta _{\diamond }^{(8)}(\omega ,\mu _0,c_\star ):=\min \{\eta _3,\eta _7\}>0$ and assume that $\kappa \ge 0$ and $\mu \ge \mu _0$ are such that (1.6) is satisfied for some $\eta <\eta _8$. Moreover, we pick $\varrho _\circ =\varrho _\circ (\omega )>\frac{{\mathcal {N}}-2}{4}$ satisfying $\varrho _\circ \le \frac{{\mathcal {N}}-2}{4}+\theta _3$, which especially entails that $2\varrho _\circ -\frac{{\mathcal {N}}}{2}>-1$ and hence we may fix $q=q(\omega )>{\mathcal {N}}$ such that

$$\begin{aligned} 2\varrho _\circ -\frac{{\mathcal {N}}}{2}>-\frac{{\mathcal {N}}}{q}. \end{aligned}$$

(6.8)

Additionally, we pick $p=p(\omega )\in ({\mathcal {N}},q)$ and then also fix $r=r(\omega )\in ({\mathcal {N}},p)$ fulfilling

$$\begin{aligned} r\le {\mathcal {N}}+\theta _3\qquad \text {as well as}\qquad r<\frac{qp}{q-p} \end{aligned}$$

and $\varrho =\varrho (\omega )\in (0,1)$ such that

$$\begin{aligned} \frac{{\mathcal {N}}}{2p}<\varrho <\frac{1}{2}. \end{aligned}$$

(6.9)

In light of $\eta <\eta _3$, $\varrho _\circ \le \frac{{\mathcal {N}}-2}{4}+\theta _3$ and $r\le {\mathcal {N}}+\theta _3$ we find from an application of Lemma 5.4 that with $t_3=t_3(\omega ,\mu _0,\eta ,n_0,u_0):=t_{\diamond }^{(3)}(\omega ,\mu _0,1,\eta ,n_0,u_0)>0$ we have

$$\begin{aligned} \big \Vert A^{\varrho _\circ }u_\varepsilon (\cdot ,t)\big \Vert _{L^{2}(\Omega )}<1\qquad \text {and}\qquad \Vert u_\varepsilon (\cdot ,t)\Vert _{L^{r}(\Omega )}<1\quad \text {for all }t\ge t_3\text { and }\varepsilon \in (0,1). \end{aligned}$$

(6.10)

Similarly, we can make use of the restriction $\eta <\eta _7$ to take $C_1=C_1(\omega ,\kappa ,\mu _0,\eta ,c_\star )>0$ and $t_7=t_7(\omega ,\mu _0,\eta ,n_0,c_0,u_0,c_\star ):=t_{\diamond }^{(7)}(\omega ,\mu _0,\eta ,n_0,c_0,u_0,c_\star )>0$ from Lemma 6.1, which satisfy

$$\begin{aligned} \Vert n_\varepsilon (\cdot ,t)\Vert _{L^{p}(\Omega )}\le C_1\quad \text {for all }t\ge t_7\text { and }\varepsilon \in (0,1). \end{aligned}$$

(6.11)

Now, setting $t_8=t_{\diamond }^{(8)}(\omega ,\mu _0,\eta ,n_0,c_0,u_0,c_\star ):=\max \{t_3,t_7\}+1$ and making use of the variation of constants representation of $u_\varepsilon (\cdot ,t)$ obtained from (2.1c), we find after application of the fractional power $A^\varrho$ that

$$\begin{aligned} \big \Vert A^\varrho u_\varepsilon (\cdot ,t)\big \Vert _{L^{p}(\Omega )}\le&\big \Vert A^\varrho e^{-A}u_\varepsilon (\cdot ,t-1)\big \Vert _{L^{p}(\Omega )}\nonumber \\&+\int _{t-1}^t\Big \Vert A^\varrho e^{-(t-s)A}{\mathcal {P}}\big (\nabla \cdot (Y_\varepsilon u_\varepsilon \otimes u_\varepsilon )(\cdot ,s)\big )\Big \Vert _{L^{p}(\Omega )}{{\,\mathrm{d\!}\,}}s\nonumber \\&+\int _{t-1}^t\Big \Vert A^\varrho e^{-(t-s)A}{\mathcal {P}}\big (n_\varepsilon (\cdot ,s)\nabla \phi \big )\Big \Vert _{L^{p}(\Omega )}{{\,\mathrm{d\!}\,}}s\qquad \text {for all }t\ge t_8, \end{aligned}$$

(6.12)

where we made use of the fact that $u_\varepsilon$ is solenoidal for rewriting $(Y_\varepsilon u_\varepsilon \cdot \nabla )u_\varepsilon =\nabla \cdot (Y_\varepsilon u_\varepsilon \otimes u_\varepsilon )$ on $\Omega \times (0,\infty )$. Here, standard regularity estimates for the Stokes semigroup (e.g. [15, p.201], c.f. also [48, Lemma 3.1]) provide $C_2=C_2(\omega )>0$ such that

$$\begin{aligned} \big \Vert A^\varrho e^{-A}u_\varepsilon (\cdot ,t&-1)\big \Vert _{L^{p}(\Omega )}\le C_2\Vert u_\varepsilon (\cdot ,t-1)\Vert _{L^{r}(\Omega )}\quad \text {for all }t\ge t_8. \end{aligned}$$

(6.13)

Drawing on semigroup estimates for the Stokes semigroup (see e.g. [15] or [48, Lemma 3.1]) and the boundedness of ${\mathcal {P}}$ in $L^{p\;\!}\!\left( \Omega ;{\mathbb {R}}^{\mathcal {N}}\right)$ we find $C_3>0$ fulfilling

$$\begin{aligned}&\int _{t-1}^t\Big \Vert A^\varrho e^{-(t-s)A}{\mathcal {P}}\big (\nabla \cdot (Y_\varepsilon u_\varepsilon \otimes u_\varepsilon )(\cdot ,s)\big ) \Big \Vert _{L^{p}(\Omega )}{{\,\mathrm{d\!}\,}}s\\&\quad \le C_3 \int _{t-1}^t (t-s)^{-\varrho -\frac{1}{2}}\Big \Vert \big (Y_\varepsilon u_\varepsilon \otimes u_\varepsilon \big ) (\cdot ,s)\Big \Vert _{L^{p}(\Omega )}{{\,\mathrm{d\!}\,}}s \end{aligned}$$

for all $t\ge t_8$. Employing Hölder’s inequality and relying on the embedding $D\big (A_2^{\varrho _\circ }\big )\hookrightarrow L^{q\;\!}\!\left( \Omega ;{\mathbb {R}}^{\mathcal {N}}\right)$ implied by the condition (6.8), we further obtain $C_4=C_4(\omega )>0$ and $a:=1-\frac{(q-p)r}{qp}$ satisfying

$$\begin{aligned} \Big \Vert \big (Y_\varepsilon u_\varepsilon \otimes u_\varepsilon \big )(\cdot ,s)\Big \Vert _{L^{p}(\Omega )}&\le \big \Vert Y_\varepsilon u_\varepsilon (\cdot ,s)\big \Vert _{L^{q}(\Omega )}\big \Vert u_\varepsilon (\cdot ,s)\big \Vert _{L^{\frac{qp}{q-p}}(\Omega )}\\&\le C_4\big \Vert A^{\varrho _\circ } Y_\varepsilon u_\varepsilon (\cdot ,s)\big \Vert _{L^{2}(\Omega )}\big \Vert u_\varepsilon (\cdot ,s)\Vert _{L^{\infty }(\Omega )}^a\big \Vert u_\varepsilon (\cdot ,s)\big \Vert _{L^{r}(\Omega )}^{1-a} \end{aligned}$$

for all $s>0$. Since $Y_\varepsilon$ and $A^{\varrho _\circ }$ commute and $Y_\varepsilon$ is nonexpansive on $L_\sigma ^2(\Omega )$, we hence find from (6.10) and (6.9) that there is $C_5=C_5(\omega )>0$ such that

$$\begin{aligned}&\int _{t-1}^t\Big \Vert A^\varrho e^{-(t-s)A}{\mathcal {P}}\big (\nabla \cdot (Y_\varepsilon u_\varepsilon \otimes u_\varepsilon )(\cdot ,s)\big )\Big \Vert _{L^{p}(\Omega )}{{\,\mathrm{d\!}\,}}s \le C_3C_4\int _{t-1}^t(t-s)^{-\varrho -\frac{1}{2}}\big \Vert u_\varepsilon (\cdot ,s)\big \Vert _{L^{\infty }(\Omega )}^a{{\,\mathrm{d\!}\,}}s\nonumber \\&\qquad \le C_5\sup _{s\in (t-1,t)}\big \Vert u_\varepsilon (\cdot ,s)\big \Vert _{L^{\infty }(\Omega )}^a\quad \text {for all }t\ge t_8. \end{aligned}$$

(6.14)

Similarly, the boundedness of $\nabla \phi$ contained in (1.4) and (6.11) entail the existence of $C_6>0$ and $C_7=C_7(\omega ,\kappa ,\mu _0,\eta ,c_\star )>0$ satisfying

$$\begin{aligned} \int _{t-1}^t\Big \Vert A^\varrho e^{-(t-s)A}{\mathcal {P}}\big (n_\varepsilon (\cdot ,s)\nabla \phi \big )\Big \Vert _{L^{p}(\Omega )}{{\,\mathrm{d\!}\,}}s&\le C_6\int _{t-1}^t(t-s)^{-\varrho }\big \Vert n_\varepsilon (\cdot ,s)\big \Vert _{L^{p}(\Omega )}{{\,\mathrm{d\!}\,}}s\nonumber \\&\le C_6 C_1\int _{t-1}^t(t-s)^{-\varrho }{{\,\mathrm{d\!}\,}}s\le \frac{C_7}{1-\varrho } \end{aligned}$$

(6.15)

for all $t\ge t_8$. Taking an arbitrary $T>t_8$ we introduce

$$\begin{aligned} M_\varepsilon (T):=\sup _{s\in (t_8,T)}\big \Vert A^\varrho u_\varepsilon (\cdot ,t)\big \Vert _{L^{p}(\Omega )},\quad T>t_0,\ \varepsilon \in (0,1), \end{aligned}$$

and then find from collecting (6.12)–(6.15) that there is $C_8=C_8(\omega ,\kappa ,\mu _0,\eta ,c_\star )>0$ such that

$$\begin{aligned} \big \Vert A^\varrho u_\varepsilon (\cdot ,t)\big \Vert _{L^{p}(\Omega )}\le C_8+C_8 M_\varepsilon ^a(T)\quad \text {for all }t\in (t_8,T), \end{aligned}$$

which, since our choice of r ensures $a<1$, implies that for $C_{9}=C_{9}(\omega ,\kappa ,\mu _0,\eta ,c_\star ):=\max \big \{1,(2C_8)^\frac{1}{1-a}\big \}$ we have

$$\begin{aligned} M_\varepsilon (T)\le C_{9}\quad \text {for all }t>t_8. \end{aligned}$$

Here, noticing that the assumption $\varrho >\frac{{\mathcal {N}}}{2p}$ in (6.9) provides the existence of $\alpha _1=\alpha _1(\omega )>0$ such that $D\big (A_p^\varrho \big )\hookrightarrow C^{\alpha _1}\!\left( {\overline{\Omega }};{\mathbb {R}}^3\right)$ (e.g. [20, pp. 35–36]) we thereby obtain $C_{10}=C_{10}(\omega ,\kappa ,\mu _0,\eta ,c_\star )>0$ fulfilling

$$\begin{aligned} \big \Vert u_\varepsilon (\cdot ,t)\big \Vert _{C^{\alpha _1}\!\left( {\overline{\Omega }}\right) }\le C_{10}\quad \text {for all }t\ge t_8. \end{aligned}$$

Following a similar estimation procedure (see also [41, Lemma 3.4] for details) we can find also $C_{11}=C_{11}(\omega ,\kappa ,\mu _0,\eta ,c_\star )>0$ satisfying

$$\begin{aligned} \big \Vert A^\varrho u_\varepsilon (\cdot ,t)-A^\varrho u_\varepsilon (\cdot ,T_0)\big \Vert _{L^{p}(\Omega )}\le C_{11}(t-T_0)^{1-\varrho }\quad \text {for all }T_0\ge t_8\text { and }t\in [T_0,T_0+1], \end{aligned}$$

which in combination with the Hölder bound above concludes the proof with $\alpha =\alpha (\omega )=\min \{\alpha _1,1-\varrho \}$ and $C=C(\omega ,\kappa ,\mu _0,\eta ,c_\star ):=\max \{C_{10},C_{11}\}$. $\square$

In the penultimate step we will obtain a suitably large $T_0>0$ such that for any $T>T_0$ there are $\varepsilon$-uniform bounds for $n_\varepsilon ,c_\varepsilon$ in the space $C^{2+\alpha ,1+\frac{\alpha }{2}}\!\left( {\overline{\Omega }}\times [T_0,T]\right)$ and for $u_\varepsilon$ in $C^{2+\alpha ,1+\frac{\alpha }{2}}\!\left( {\overline{\Omega }}\times [T_0,T];{\mathbb {R}}^{\mathcal {N}}\right)$. The arguments involved draw on standard regularity theory for parabolic equations and have previously been employed in various similar chemotaxis settings with signal absorption (e.g. [2, 27, 35, 50]).

Proposition 6.3

Let $\omega >0$, $\mu _0>0$ and $c_\star \ge 0$. Then there is $\eta _{\diamond }^{(9)}=\eta _{\diamond }^{(9)}(\omega ,\mu _0,c_\star )>0$ such that whenever $\kappa \ge 0$ and $\mu \ge \mu _0$ satisfy (1.6) for some $\eta <\eta _{\diamond }^{(9)}(\omega ,\mu _0,c_\star )$, one can find $\alpha =\alpha (\omega )\in (0,1)$ and $T_0=T_0(\omega ,\mu _0,\eta ,n_0,c_0,u_0,c_\star )>0$ with the property that for each $T>T_0$ there is $C=C(\omega ,\kappa ,\mu _0,\eta ,c_\star ,T)>0$ such that for all $\varepsilon \in (0,1)$ the solution $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ of (2.1) fulfills

$$\begin{aligned} \Vert n_\varepsilon \Vert _{C^{2+\alpha ,1+\frac{\alpha }{2}}\!\,({\overline{\Omega }}\times [T_0,T])}+\Vert c_\varepsilon \Vert _{C^{2+\alpha ,1+\frac{\alpha }{2}}\!\,({\overline{\Omega }}\times [T_0,T])}+\Vert u_\varepsilon \Vert _{C^{2+\alpha ,1+\frac{\alpha }{2}}\!\,({\overline{\Omega }}\times [T_0,T])}\le C. \end{aligned}$$

Proof

Let us start once more by fixing our parameters appropriately. We take $\theta _3=\theta _3(\omega ):=\theta _{\diamond }^{(3)}(\omega )>0$ and $\eta _3=\eta _3(\omega ):=\eta _{\diamond }^{(3)}(\omega ,1)>0$ as provided by Lemma 5.4, $\theta _6=\theta _6(\omega ):=\theta _{\diamond }^{(6)}>0$ and $\eta _6=\eta _6(\omega ,\mu _0,c_\star ):=\eta _{\diamond }^{(6)}(\omega ,\mu _0,1,c_\star )>0$ as in Proposition 5.17, $\eta _7=\eta _7(\omega ,\mu _0,c_\star ):=\eta _{\diamond }^{(7)}(\omega ,\mu _0,c_\star )>0$ as in Lemma 6.1 and let $\eta _8=\eta _8(\omega ,\mu _0,c_\star ):=\eta _{\diamond }^{(8)}(\omega ,\mu _0,c_\star )>0$ be given by Lemma 6.2. We set $\eta _9=\eta _{\diamond }^{(9)}(\omega ,\mu _0,c_\star ):=\min \{\eta _3,\eta _6,\eta _7,\eta _8\}$ and then assume for the remainder that (1.6) is satisfied for some $\eta <\eta _9$. Letting $p=p(\omega ):=\frac{{\mathcal {N}}+6}{6}+\theta _6$, $r=r(\omega ):={\mathcal {N}}+\theta _3$ and $q=q(\omega ):=\min \{2p,r\}$ we note that due to ${\mathcal {N}}\le 3$ we have $q\ge {\mathcal {N}}$. Moreover, we introduce

$$\begin{aligned}&h_{1,\varepsilon }(x,t):=\big (\rho _\varepsilon f_\varepsilon (n_\varepsilon )\nabla c_\varepsilon +u_\varepsilon \big )(x,t)\quad \text {and}\nonumber \\&h_{2,\varepsilon }(x,t):=\big (\kappa n_\varepsilon -\mu n_\varepsilon ^2\big )(x,t),\quad x\in \Omega ,\ t>0,\ \varepsilon \in (0,1) \end{aligned}$$

and then make use of the fact that $|\rho _\varepsilon (x)f_\varepsilon (s)|\le 1$ for all $x\in \Omega$ and $s\ge 0$ as well as a combination of Lemma 5.4, Proposition 5.17 and Lemma 6.1 to find that there are $T_1=T_1(\omega ,\mu _0,\eta ,n_0,c_0,u_0,c_\star )>0$ and $C_1(\omega ,\kappa ,\mu _0,\eta ,c_\star )>0$ such that

$$\begin{aligned}&\big \Vert n_\varepsilon (\cdot ,t)\big \Vert _{L^{\infty }(\Omega )}\le C_1,\qquad \big \Vert h_{1,\varepsilon }(\cdot ,t)\big \Vert _{L^{q}(\Omega )}\le C_1\qquad \text {and}\\&\big \Vert h_{2,\varepsilon }(\cdot ,t)\big \Vert _{L^{\infty }(\Omega )}\le C_1\quad \text {for all }t\ge T_1. \end{aligned}$$

Therefore, re-interpreting (2.1a) as

$$\begin{aligned} n_{\varepsilon t}=\Delta n_\varepsilon -\nabla \cdot \big (n_\varepsilon \, h_{1,\varepsilon }(x,t)\big )+h_{2,\varepsilon }(x,t),\quad x\in \Omega ,\ t>0,\ \varepsilon \in (0,1), \end{aligned}$$

we can draw on the fact that $q>{\mathcal {N}}$ and a Hölder regularity result for parabolic equations (e.g. [32, Theorem 1.3]) to conclude the existence of $\alpha _1=\alpha _1(\omega )>0$ such that for each $T>T_2=T_2(\omega ,\mu _0,\eta ,n_0,c_0,u_0,c_\star ):=T_1+1$ there is $C_2=C_2(\omega ,\kappa ,\mu _0,\eta ,c_\star ,T)>0$ satisfying

$$\begin{aligned} \Vert n_\varepsilon \Vert _{C^{\alpha _1,\frac{\alpha _1}{2}}\!\left( {\overline{\Omega }}\times [T_2,T]\right) }\le C_2\quad \text {for all }\varepsilon \in (0,1). \end{aligned}$$

Thus, making additional use of Lemma 6.2, we find $\alpha _2=\alpha _2(\omega )>0$, $T_3=T_3(\omega ,\mu _0,\eta ,n_0,c_0,u_0,c_\star )>T_2$ with the property that for each $T>T_3$ there is $C_3=C_3(\omega ,\kappa ,\mu _0,\eta ,c_\star ,T)$ fulfilling

$$\begin{aligned} \Vert n_\varepsilon \Vert _{C^{\alpha _2,\frac{\alpha _2}{2}}\!\left( {\overline{\Omega }}\times [T_3,T]\right) }+\Vert u_\varepsilon \Vert _{C^{\alpha _2,\frac{\alpha _2}{2}}\!\left( {\overline{\Omega }}\times [T_3,T]\right) }\le C_3\quad \text {for all }\varepsilon \in (0,1). \end{aligned}$$

Accordingly, the coefficients and the right-hand side of the homogeneous Dirichlet boundary value problem for ${\widehat{c}}_{\varepsilon }$ obtained from (5.25b) are all Hölder continuous with $\varepsilon$-independent bounds. Consequently, parabolic regularity theory (compare [25, IV.5.3 and III.5.1]) yields $\alpha _3=\alpha _3(\omega )\in (0,1)$ and $T_4=T_4(\omega ,\mu _0,\eta ,n_0,c_0,u_0,c_\star ):=T_3+1$ such that for each $T>T_4$ we can find $C_4=C_4(\omega ,\kappa ,\mu _0,\eta ,c_\star ,T)>0$ satisfying

$$\begin{aligned} \Vert {\widehat{c}}_{\varepsilon }\Vert _{C^{2+\alpha _3,1+\frac{\alpha _3}{2}}\!\left( {\overline{\Omega }}\times [T_4,T]\right) }\le C_4\quad \text {for all }\varepsilon \in (0,1), \end{aligned}$$

which due to $c_\star$ being constant and ${\widehat{c}}_{\varepsilon }\equiv c_\star -c_\varepsilon$ in ${\overline{\Omega }}\times (0,\infty )$ can of course easily be transferred to an $\varepsilon$-independent Hölder bound for $c_\varepsilon$. Returning to (2.1a), this time with

$$\begin{aligned} h_{3,\varepsilon }(x,t):=\big (\rho _\varepsilon f_\varepsilon (n_\varepsilon )\nabla c_\varepsilon +\rho _\varepsilon f_\varepsilon '(n_\varepsilon )n_\varepsilon \nabla c_\varepsilon +u_\varepsilon \big )(x,t),\quad x\in \Omega ,\ t>0,\ \varepsilon \in (0,1) \end{aligned}$$

and

$$\begin{aligned} h_{4,\varepsilon }(x,t)&:=\big (-\rho _\varepsilon f_\varepsilon (n_\varepsilon )n_\varepsilon \Delta c_\varepsilon -\rho _\varepsilon 'f_\varepsilon (n_\varepsilon )n_\varepsilon \nabla c_\varepsilon +\kappa n_\varepsilon -\mu n_\varepsilon ^2\big )(x,t),\\&\quad x\in \Omega ,\ t>0,\ \varepsilon \in (0,1), \end{aligned}$$

we write

$$\begin{aligned} n_{\varepsilon t}=\Delta n_\varepsilon -h_{3,\varepsilon }(x,t)\cdot \nabla n_\varepsilon +h_{4,\varepsilon }(x,t),\quad x\in \Omega ,\ t>0,\ \varepsilon \in (0,1), \end{aligned}$$

where now for some $\alpha _4=\alpha _4(\omega )\in (0,1)$ and $T>T_4$ there is $C_5=C_5(\omega ,\kappa ,\mu _0,\eta ,c_\star ,T)>0$ such that

$$\begin{aligned} \big \Vert h_{3,\varepsilon }\big \Vert _{C^{\alpha _4,\frac{\alpha _4}{2}}\!\left( {\overline{\Omega }}\times [T_4,T]\right) }+\big \Vert h_{4,\varepsilon }\big \Vert _{C^{\alpha _4,\frac{\alpha _4}{2}}\!\left( {\overline{\Omega }}\times [T_4,T]\right) }\le C_5\quad \text {for all }\varepsilon \in (0,1). \end{aligned}$$

Therefore, parabolic Schauder theory entails that there is $\alpha _5=\alpha _5(\omega )\in (0,1)$ such that for all $T>T_5=T_5(\omega ,\mu _0,\eta ,n_0,c_0,u_0,c_\star ):=T_4+1$ one can find $C_6=C_6(\omega ,\kappa ,\mu _0,\eta ,c_\star ,T)>0$ fulfilling

$$\begin{aligned} \Vert n_\varepsilon \Vert _{C^{2+\alpha _5,1+\frac{\alpha _5}{2}}\!\left( {\overline{\Omega }}\times [T_5,T]\right) }\le C_6\quad \text {for all }\varepsilon \in (0,1) \end{aligned}$$

and, similarly, while also considering maximal Sobolev estimates for (2.1c) to first attain Hölder bounds for $\nabla u_\varepsilon$ (see e.g. [27, Lemma 3.9 and Lemma 3.12] for details), we find $\alpha _6=\alpha _6(\omega )\in (0,1)$ and $T_6=T_6(\omega ,\mu _0,\eta ,n_0,c_0,u_0,c_\star )>0$ with the property that for each $T>T_6$ there is $C_7=C_7(\omega ,\kappa ,\mu _0,\eta ,c_\star ,T)>0$ such that

$$\begin{aligned} \Vert u_\varepsilon \Vert _{C^{2+\alpha _6,1+\frac{\alpha _6}{2}}\!\left( {\overline{\Omega }}\times [T_6,T]\right) }\le C_7\quad \text {for all }\varepsilon \in (0,1), \end{aligned}$$

which concludes the proof. $\square$

6.2 The proof of Theorem 1.2

Finally, we can combine our $\varepsilon$-uniform estimates with the limit procedure of Proposition 4.1 and the Arzelà–Ascoli theorem to obtain the outcome asserted by Theorem 1.2.

Proof of Theorem 1.2

We let $\eta _9=\eta _9(\omega ,\mu _0,c_\star ):=\eta _{\diamond }^{(9)}(\omega ,\mu _0,c_\star )>0$ be provided by Proposition 6.3 and assume that then $\kappa \ge 0$ and $\mu \ge \mu _0>0$ satisfy (1.6) for $\eta =\eta (\omega ,\mu _0,c_\star ):=\frac{\eta _9}{2}$. We denote by $(\varepsilon _j)_{j\in {\mathbb {N}}}$ and (n, c, u) the sequence and triple of limit functions obtained in Proposition 4.1. From the outcome of Proposition 6.3 and the Arzelà–Ascoli theorem we then find that there are $T_0=T_0(\omega ,\eta ,n_0,c_0,u_0,c_\star )>0$, a subsequence $(\varepsilon _{j_k})_{k\in {\mathbb {N}}}$ and functions ${\tilde{n}}\in C_{loc}^{2,1}\!\left( {\overline{\Omega }}\times [T_0,\infty )\right)$, ${\tilde{c}}\in C_{loc}^{2,1}\!\left( {\overline{\Omega }}\times [T_0,\infty )\right)$ and ${\tilde{u}}\in C_{loc}^{2,1}\!\left( {\overline{\Omega }}\times [T_0,\infty );{\mathbb {R}}^{\mathcal {N}}\right)$ such that

$$\begin{aligned}&n_\varepsilon \rightarrow \, {\tilde{n}}\qquad \text {in }\quad C_{loc}^{2,1}\!\left( {\overline{\Omega }}\times [T_0,\infty )\right) ,\\&c_\varepsilon \rightarrow \, {\tilde{c}}\qquad \text {in }\quad C_{loc}^{2,1}\!\left( {\overline{\Omega }}\times [T_0,\infty )\right) ,\\&u_\varepsilon \rightarrow \, {\tilde{u}}\qquad \text {in }\quad C_{loc}^{2,1}\!\left( {\overline{\Omega }}\times [T_0,\infty )\right) \end{aligned}$$

as $\varepsilon =\varepsilon _{j_k}\searrow 0$, where clearly $({\tilde{n}},{\tilde{c}},{\tilde{u}})$ and (n, c, u) have to coincide. Separately passing to the limit in each of the equations of (2.1) and constructing the associated pressure $P\in C_{loc}^{1,0}\!\left( {\overline{\Omega }}\times [T_0,\infty )\right)$ by means of standard procedures presented in [34, 36], we conclude that (n, c, u, P) solves (1.2) classically in ${\overline{\Omega }}\times [T_0,\infty )$. $\square$