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Reaction-Driven Relaxation in Three-Dimensional Keller–Segel–Navier–Stokes Interaction

Abstract

The Keller–Segel–Navier–Stokes system

$$\begin{aligned} \left\{ \begin{array}{rcll} n_t + u\cdot \nabla n &{}=&{} \Delta n - \chi \nabla \cdot (n\nabla c) + \rho n-\mu n^2,\\ c_t + u\cdot \nabla c &{}=&{} \Delta c-c+n, \\ u_t + (u\cdot \nabla )u &{}=&{} \Delta u + \nabla P + n \nabla \phi + f(x,t), \qquad \nabla \cdot u=0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$

is considered in a smoothly bounded convex domain \(\Omega \subset \mathbb {R}^3\), with \(\phi \in W^{2,\infty }(\Omega )\) and \(f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)\), and with \(\chi >0, \rho \in \mathbb {R}\) and \(\mu >0\). As recent literature has shown, for all reasonably mild initial data a corresponding no-flux/no-flux/Dirichlet initial-boundary value problem possesses a global generalized solution, but the knowledge on its regularity properties has not yet exceeded some information on fairly basic integrability features. The present study reveals that whenever \(\omega >0\), requiring that

$$\begin{aligned} \frac{\rho }{\min \{\mu ,\mu ^{\frac{3}{2}+\omega }\}} < \eta \end{aligned}$$

with some \(\eta =\eta (\omega )>0\), and that f satisfies a suitable assumption on ultimate smallness, is sufficient to ensure that each of these generalized solutions becomes eventually smooth and classical. Furthermore, under these hypotheses (\(\star \)) is seen to admit an absorbing set with respect to the topology in \(L^\infty (\Omega )\). By trivially applying to the case when \(\mu >0\) is arbitrary and \(\rho \le 0\), these results especially assert essentially unconditional statements on eventual regularity in taxis-reaction systems interacting with liquid environments, such as arising in contexts of models for broadcast spawning discussed in recent literature.

Introduction

Regularity issues form a central aspect in the literature concerned with the analysis of evolution systems accounting for taxis mechanisms. In fact, well-known findings on the occurrence of spontaneous singularity formation already in simple Keller-Segel type systems form quite unambiguous caveats which indicate that including chemotactic cross-diffusion as a model element may go along with substantial limitations of solution regularity ([22, 39, 48, 54]; cf. also the survey [34]). Corresponding mathematical questions naturally become yet more sophisticated when taxis processes are embedded into more intricate models, and understanding the singularity-supporting potential of chemotactic cross-diffusion in complex frameworks has accordingly attracted considerable attention in the past years ([1, 42]).

The present work addresses this problem area in the context of a model for the interaction of a chemotactically active population with a liquid environment, as found to be of relevance not only in experimental setups involving populations of swimming bacteria ([8, 37, 52]), but moreover also in descriptions of spatio-temporal evolution in processes of broadcast spawning during coral fertilization ([7, 25, 26, 38]). Specifically, we shall be concerned with the Keller–Segel–Navier–Stokes system

$$\begin{aligned} \left\{ \begin{array}{rcll} n_t + u\cdot \nabla n &{}=&{} \Delta n - \chi \nabla \cdot (n\nabla c) + \rho n - \mu n^2, \qquad &{} x\in \Omega , \ t>0,\\ c_t + u\cdot \nabla c &{}=&{} \Delta c-c+n, \qquad &{} x\in \Omega , \ t>0, \\ u_t + (u\cdot \nabla )u &{}=&{} \Delta u + \nabla P + n \nabla \phi + f(x,t), \qquad \nabla \cdot u=0, \qquad &{} x\in \Omega , \ t>0, \end{array} \right. \end{aligned}$$
(1.1)

for an unknown population density n in an N-dimensional domain \(\Omega \), and for a signal concentration c and an incompressible fluid represented through its velocity field u and an associated pressure P. By requiring \(\chi \) to be positive, (1.1) models attractive tactic motion of individuals toward increasing signal concentrations, additionally affected by transport of both these quantities through the surrounding fluid which, in turn, is influenced not only by an external force f but also by cells through buoyancy. Under the assumptions \(\mu >0\) and \(\rho \in \mathbb {R}\) considered here, (1.1) moreover accounts for quadratic degradation in the population density, and hence both addresses chemotaxis-growth processes in which populations undergo natural logistic-type proliferation and death ([18, 42]), and also covers situations in which quadratic absorption, then mainly accompanied by the choice \(\rho =0\) or even \(\rho <0\), is due to the inclusion of reaction mechanisms ([26]).

With regard to solution regularity, the interplay of chemotactic cross-diffusion with such zero-order dissipation seems quite delicate, though yet far from completely understood, already in contexts of corresponding fluid-free Keller-Segel systems. Indeed, in the resulting version of (1.1) with \(u\equiv 0\) any choice of \(\mu >0\) is sufficient to suppress any blow-up phenomenon in two-dimensional initial value problems in the sense that for widely arbitrary initial data, global bounded solutions always exist ([41, 45]); in associated three- and higher-dimensional counterparts, however, similar findings on exclusion of explosions to date seem to rely on the stronger hypothesis that \(\mu >\mu _0\) with some \(\mu _0=\mu _0(\Omega )>0\) ([53, 61]), while for small values of \(\mu >0\) only some weak solutions are known to exist globally ([32]). Although some studies concerned with simplified model variants have revealed some considerably strong singularity-counteracting effects of logistic damping in the sense of immediate regularization of strongly singular distributions ([33, 60]), not only results on possibly transient emergence of high population densities ([24, 56]), but moreover especially some detections of genuine blow-up both in high-dimensional systems with quadratic zero-order dissipation ([12]), and in three-dimensional models involving some subquadratic but yet superlinear absorption ([12]), indicate some persistence of taxis-driven destabilization also in the presence of such degradation mechanisms.

In light of these prerequisites, it seems far from surprising that the knowledge on corresponding issues in coupled chemotaxis-fluid systems of the form (1.1) is yet quite thin. After all, results on smooth global solvability could be established for the two-dimensional version of (1.1) whenever \(\mu \) is positive ([10, 50]), while a similar statement could be derived when \(N=3\) and \(\mu \ge 23\) at least for a Stokes simplification of (1.1) in which the nonlinear convective term \((u\cdot \nabla ) u\) is neglected ([49]). For the fully coupled three-dimensional Keller–Segel–Navier–Stokes system (1.1) with arbitrary \(\mu >0\), however, merely a statement on global existence of certain generalized solutions seems available, asserting quite poor regularity properties only (see Proposition 1.1 and Definition 9.2 below). In this sense, (1.1) seems much less understood than its well-studied relative

$$\begin{aligned} \left\{ \begin{array}{rcll} n_t + u\cdot \nabla n &{}=&{} \Delta n - \chi \nabla \cdot (n\nabla c), \qquad &{} x\in \Omega , \ t>0,\\ c_t + u\cdot \nabla c &{}=&{} \Delta c-nc, \qquad &{} x\in \Omega , \ t>0, \\ u_t + (u\cdot \nabla )u &{}=&{} \Delta u + \nabla P + n \nabla \phi , \qquad \nabla \cdot u=0, \qquad &{} x\in \Omega , \ t>0, \end{array} \right. \end{aligned}$$
(1.2)

and logistic variants thereof, in which the decisive difference in signal evolution, here accounting for signal consumption through individuals rather than production as in (1.1), has facilitated energy-based analytical approaches to establish results not only on global existence of solutions in fairly natural frameworks of weak solvability ([9, 31, 36, 57]), but also on qualitative aspects such as eventual regularization and large-time stabilization toward homogeneous states ([6, 31, 55, 58]; see also [28] and [4] for an analysis of small-data solutions).

In comparison to (1.2), the system (1.1) apparently lacks any similarly meaningful energy-like global dissipative features; in fact, the well-known gradient structure of the classical proliferation-free Keller-Segel system ([40]) seems to disappear already when only one of the extra model elements in (1.1) is added to the latter, that is when either logistic contributions are included without any fluid interplay, or when, alternatively, a coupling to the (Navier-)Stokes equations is considered in the absence of such zero-order terms. This can be viewed as indicating the possibility of dynamics considerably far from spatial homogeneity, and hence remarkably different from that in (1.2): Indeed, the combined action of self-enhancing chemoattraction with logistic proliferation is not only known to generate spatially structured equilibria ([29]), but may beyond this bring about some quite colorful dynamical facets, as detected partially in the course of numerical simulations ([19]), and partially even by means of rigorous analysis ([24, 56]). Apart from this, several findings in the recent analytical literature have revealed some nontrivial qualitative and even quantitative effects that fluid interaction may have on the solution behavior in various types of chemotaxis systems ([20, 25,26,27]).

Main results.    The purpose of the present study now consists in developing an approach that, despite the challenges accordingly resulting from a lack of favorable structural properties, is capable of identifying situations in which solutions to the fully coupled system (1.1) exhibit regular behavior at least eventually. To formulate this more precisely, let us assume henceforth that \(\Omega \subset \mathbb {R}^3\) be a bounded convex domain with smooth boundary, that \(\chi >0, \rho \in \mathbb {R}\) and \(\mu >0\), and that

$$\begin{aligned} \phi \in W^{2,\infty }(\Omega ) \qquad \text{ and } \qquad f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3). \end{aligned}$$
(1.3)

We shall then consider (1.1) along with the initial conditions

$$\begin{aligned} n(x,0)=n_0(x), \quad c(x,0)=c_0(x) \quad \text{ and } \quad u(x,0)=u_0(x), \qquad x\in \Omega , \end{aligned}$$
(1.4)

and under the boundary conditions

$$\begin{aligned} \frac{\partial n}{\partial \nu }=\frac{\partial c}{\partial \nu }=0 \quad \text{ and } \quad u=0 \qquad \text{ on } \partial \Omega , \end{aligned}$$
(1.5)

where our standing assumptions are that

$$\begin{aligned} \left\{ \begin{array}{l} n_0 \in C^0(\bar{\Omega }) \quad \text{ is } \text{ nonnegative } \text{ with } n_0\not \equiv 0, \quad \text{ that } \\ c_0 \in W^{1,\infty }(\Omega ) \quad \text{ is } \text{ nonnegative, } \quad \text{ and } \text{ that }\\ u_0 \in W^{2,2}(\Omega ;\mathbb {R}^3) \cap W_0^{1,2}(\Omega ;\mathbb {R}^3) \cap L^2_\sigma (\Omega ), \end{array} \right. \end{aligned}$$
(1.6)

with \(L^p_\sigma (\Omega ):=\{ \varphi \in L^p(\Omega ;\mathbb {R}^3) \ | \ \nabla \cdot \varphi =0 \}\) denoting the space of all solenoidal vector fields in \(L^p(\Omega ;\mathbb {R}^3)\) for \(p>1\).

In order to appropriately recall from [59] a basic result on existence and approximation, for \(\varepsilon \in (0,1)\) we furthermore introduce the regularized variant of (1.1), (1.4), (1.5) given by

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }ll} n_{\varepsilon t} + u_\varepsilon \cdot \nabla n_\varepsilon = \Delta n_\varepsilon - \chi \nabla \cdot (\frac{n_\varepsilon }{1+\varepsilon n_\varepsilon }\nabla c_\varepsilon ) + \rho n_\varepsilon - \mu n_\varepsilon ^2,&{} x\in \Omega , \ t>0,\\ c_{\varepsilon t} + u_\varepsilon \cdot \nabla c_\varepsilon = \Delta c_\varepsilon -c_\varepsilon +n_\varepsilon , \qquad &{} x\in \Omega , \ t>0, \\ u_{\varepsilon t} + (Y_\varepsilon u_\varepsilon \cdot \nabla )u_\varepsilon = \Delta u_\varepsilon + \nabla P_\varepsilon + n_\varepsilon \nabla \phi + f(x,t), \qquad \nabla \cdot u_\varepsilon =0,&{} x\in \Omega , \ t>0, \\ \frac{\partial n_\varepsilon }{\partial \nu }=0, \quad \frac{\partial c_\varepsilon }{\partial \nu }=0, \quad u_\varepsilon =0,&{} x\in \partial \Omega , \ t>0, \\ n_\varepsilon (x,0)=n_0(x), \quad c_\varepsilon (x,0)=c_0(x), \quad u_\varepsilon (x,0)=u_0(x), \qquad &{} x\in \Omega , \end{array} \right. \nonumber \\ \end{aligned}$$
(1.7)

with the family \((Y_\varepsilon )_{\varepsilon \in (0,1)}\) of Yosida approximations determined by

$$\begin{aligned} Y_\varepsilon v:=(1+\varepsilon A)^{-1} v, \qquad v\in L^2_\sigma (\Omega ), \ \varepsilon \in (0,1), \end{aligned}$$
(1.8)

where here and below, A represents the Stokes operator under homogeneous Dirichlet boundary conditions in \(\Omega \), with its respective realization in \(L^p(\Omega ;\mathbb {R}^3)\) for \(p>1\) defined in its domain \(D(A_p):=W^{2,p}(\Omega ;\mathbb {R}^3) \cap W_0^{1,p}(\Omega ;\mathbb {R}^3) \cap L^p_\sigma (\Omega )\), and with the corresponding fractional powers thereof denoted by \(A^\beta =A_p^\beta \), \(\beta \in \mathbb {R}\), in the sequel.

Within this setting, the following result on global existence and approximation has been obtained in [59].

Proposition 1.1

Let \(\Omega \subset \mathbb {R}^3\) be a bounded convex domain with smooth boundary, assume that \(\chi >0\), \(\rho \in \mathbb {R}\) and \(\mu >0\), and suppose that \(\phi \), f and \((n_0,c_0,u_0)\) comply with (1.3) and (1.6). Then there exist functions

$$\begin{aligned} \left\{ \begin{array}{l} n\in L^\infty ((0,\infty );L^1(\Omega )) \cap L^2_{loc}(\bar{\Omega }\times [0,\infty )) \cap L^\frac{16}{13}_{loc}([0,\infty );W^{1,\frac{16}{13}}(\Omega )), \\ c\in L^\infty ((0,\infty );L^6(\Omega )) \cap L^\frac{8}{5}_{loc}([0,\infty );W^{2,\frac{8}{5}}(\Omega )) \qquad \text{ and } \\ u\in L^\infty ((0,\infty );L^2_\sigma (\Omega )) \cap L^\frac{10}{3}_{loc}(\bar{\Omega }\times [0,\infty )) \cap L^2_{loc}([0,\infty );W_0^{1,2}(\Omega )), \end{array} \right. \end{aligned}$$
(1.9)

such that (ncu) forms a global generalized solution of (1.1), (1.4), (1.5) in terms of Definition 9.2 below. Furthermore, this solution can be obtained by approximation through the regularized problems (1.7) in the sense that for each \(\varepsilon \in (0,1)\) one can find functions

$$\begin{aligned} \left\{ \begin{array}{l} n_\varepsilon \in C^0(\overline{\Omega }\times [0,\infty )) \cap C^{2,1}(\overline{\Omega }\times (0,\infty )), \\ c_\varepsilon \in \bigcap _{q\ge 1} C^0([0,\infty );W^{1,q}(\Omega )) \cap C^{2,1}(\overline{\Omega }\times (0,\infty )), \\ u_\varepsilon \in \bigcap _{\beta \in (0,1)} C^0([0,\infty );D(A_2^\beta )) \cap C^{2,1}(\overline{\Omega }\times (0,\infty );\mathbb {R}^3) \qquad \text{ and } \\ P_\varepsilon \in C^{1,0}(\Omega \times (0,\infty )) \end{array} \right. \end{aligned}$$

such that \(n_\varepsilon >0\) and \(c_\varepsilon \ge 0\) in \(\overline{\Omega }\times (0,\infty )\), that \((n_\varepsilon ,c_\varepsilon ,u_\varepsilon ,P_\varepsilon )\) solves (1.7) in the classical sense, and that with some \((\varepsilon _j)_{j\in \mathbb {N}} \subset (0,1)\) fulfilling \(\varepsilon _j\searrow 0\) as \(j\rightarrow \infty \), we have

$$\begin{aligned} n_\varepsilon \rightarrow n, \quad c_\varepsilon \rightarrow c \quad \text{ and } \quad u_\varepsilon \rightarrow u \qquad \text{ a.e. } \text{ in } \Omega \times (0,\infty ) \end{aligned}$$

as \(\varepsilon =\varepsilon _j\searrow 0\).

In line with the above discussion, in general it seems unclear how far regularity properties beyond those documented in (1.9) can be expected, especially in view of the fact that the considered coupling to the full three-dimensional Navier-Stokes system apparently limits possible extensions of knowledge on boundedness features in the fluid-related part. The main results of the present manuscript now make sure that at least when the reproduction parameter \(\rho \) lies below a certain positive number, increasing with the degradation coefficient \(\mu \), all the solutions obtained in Proposition 1.1 become smooth and bounded after some individual relaxation time, and that in this case even a bounded absorbing set within a convenient topology can be identified:

Theorem 1.2

Let \(\Omega \subset \mathbb {R}^3\) be a bounded convex domain with smooth boundary, and let \(\chi >0\) and \(\phi \in W^{2,\infty }(\Omega )\). Then for all \(\omega >0\) there exist \(\eta =\eta (\omega )>0\) and \(\kappa =\kappa (\omega )>0\) with the following property: Suppose that \(\rho \in \mathbb {R}, \mu >0\) and \(f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)\) are such that

$$\begin{aligned} \rho < \eta \cdot \min \Big \{ \mu \, , \, \mu ^{\frac{3}{2}+\omega } \Big \} \end{aligned}$$
(1.10)

and

$$\begin{aligned} \limsup _{t\rightarrow \infty } \int _t^{t+1} \Vert f(\cdot ,s)\Vert _{L^{\frac{3}{2}+\omega }(\Omega )}^2 ds < \kappa , \end{aligned}$$
(1.11)

as well as

$$\begin{aligned} \sup _{t>0} \int _t^{t+1} \Vert f(\cdot ,s)\Vert _{L^\mathbf{p}(\Omega )}^\mathbf{q}ds < \infty \end{aligned}$$
(1.12)

hold with some \(\mathbf{p}>\frac{3}{2}\) and \(\mathbf{q}>\frac{2\mathbf{p}}{2\mathbf{p}-3}\), that \(n_0, c_0\) and \(u_0\) satisfy (1.6), and that (ncu) denotes the corresponding global generalized solution of (1.1), (1.4), (1.5) from Proposition 1.1. Then one can find \(t_0=t_0(\omega ,\eta ,f,n_0,c_0,u_0)>0\) such that

$$\begin{aligned} n\in C^{2,1}(\bar{\Omega }\times [t_0,\infty )), \quad c\in C^{2,1}(\bar{\Omega }\times [t_0,\infty )) \quad \text{ and } \quad u\in C^{2,1}(\bar{\Omega }\times [t_0,\infty );\mathbb {R}^3), \nonumber \\ \end{aligned}$$
(1.13)

and such that with some \(P\in C^{1,0}(\bar{\Omega }\times [t_0,\infty ))\), the quadruple (ncuP) is a classical solution of (1.1), (1.5) in \(\bar{\Omega }\times [t_0,\infty )\). Moreover, under these assumptions on f the problem (1.1) possesses a bounded absorbing set in \((L^\infty (\Omega ))^5\) in the sense that there exists \(C=C(\omega ,f)>0\) such that any such solution satisfies

$$\begin{aligned} \limsup _{t\rightarrow \infty } \Big \{ \Vert n(\cdot ,t)\Vert _{L^\infty (\Omega )} + \Vert c(\cdot ,t)\Vert _{L^\infty (\Omega )} + \Vert u(\cdot ,t)\Vert _{L^\infty (\Omega )} \Big \} \le C. \end{aligned}$$
(1.14)

Remark.    i) We emphasize that since (1.10) is trivially satisfied whenever \(\rho \le 0\), the conclusion of Theorem 1.2 fully covers situations in which the considered population does not spontaeously proliferate, as naturally present in contexts merely involving reaction-like quadratic degradation in the zero-order part, such as in the modeling framework addressed in [25] and [26] to describe broadcast spawning.

ii) Apart from those specified above, further possible dependencies on \(\Omega , \Phi \) and \(\chi \) influence the choice of the quantities \(\eta ,\kappa ,t_0\) and C in Theorem 1.2, as would, in a natural manner, the additional inclusion of non-normalized further system parameters such as diffusivities or rates of signal production and decay. In order to avoid further expansion of the already considerable technicalities in the arguments to be developed in the sequel, here and below we refrain from precisely tracking these explicitly.

iii) Already in the fluid-free case \(u\equiv 0\) trivially included, Theorem 1.2 provides some progress in comparison with the existing literature: In contrast to the precedent finding on eventual smoothness in the corresponding taxis-only version of (1.1) stated in [32, Theorem 1.1], namely, the above result reveals ultimate regularity under an assumption which, besides being independent of the initial data, relates the required smallness condition on \(\rho \) to \(\mu \) through (1.10) in an essentially explicit manner.

iv) A natural problem arising in the interpretation of Theorem 1.2 appears to consist in providing information about the large time behavior of solutions which is more detailed than that contained in (1.14). In fact, a preliminary result in this direction asserts that under the explicit assumption that \(\mu >\frac{\chi \sqrt{\rho _+}}{4}\), the solution from Proposition 1.1 satisfies \(\mathrm{ess} \lim _{t\rightarrow \infty } \big \{ \Vert n(\cdot ,t)-\frac{\rho _+}{\mu }\Vert _{L^1(\Omega )} + \Vert c(\cdot ,t)-\frac{\rho _+}{\mu }\Vert _{L^p(\Omega )} \big \} =0\) for all \(p\in [1,6)\), and that if furthermore we have \(\int _t^{t+1} \Vert f(\cdot ,s)\Vert _{L^\frac{6}{5}(\Omega )}^2 ds \rightarrow 0\) as \(t\rightarrow \infty \), then also \(\mathrm{ess} \lim _{t\rightarrow \infty } \Vert u(\cdot ,t)\Vert _{L^2(\Omega )}=0\) ([59]). Upon a straightforward interpolation, it evidently follows from Theorem 1.2 that if these requirements are fulfilled beyond those in (1.10)-(1.12), then actually for all \(p\in [1,\infty )\) we have

$$\begin{aligned} \Big \Vert n(\cdot ,t)-\frac{\rho _+}{\mu }\Big \Vert _{L^p(\Omega )} + \Big \Vert c(\cdot ,t)-\frac{\rho _+}{\mu }\Big \Vert _{L^p(\Omega )} + \Vert u(\cdot ,t)\Vert _{L^p(\Omega )} \rightarrow 0 \qquad \text{ as } t\rightarrow \infty ;\nonumber \\ \end{aligned}$$
(1.15)

apart from that, if even \(\rho =0\) then the results from [23] and [5] become directly applicable so as to assert the quantitative estimate

$$\begin{aligned} \frac{1}{C(t+1)} \le \frac{1}{|\Omega |} \Vert n(\cdot ,t)\Vert _{L^1(\Omega )} \le \Vert n(\cdot ,t)\Vert _{L^\infty (\Omega )} \le \frac{C}{t+1} \qquad \text{ for } \text{ all } t>t_0\quad \end{aligned}$$
(1.16)

with some appropriately large \(C>0\), and with \(t_0>0\) as in Theorem 1.2 (cf. also [23] for a bound on \(\sup _{t>t_0} (t+1)\Vert \nabla c(\cdot ,t)\Vert _{L^p(\Omega )}\) for arbitrary \(p\ge 1\)). In more general parameter frameworks, and especially for positive \(\rho \) and small values of \(\mu \), however, in view of known analytical results on multiplicity in corresponding steady state problems ([29]) and of simulation-based indications for the possibility of quite chaotic solution behavior ([19]) we do not expect simple asymptotics as in (1.15) to prevail the dynamics in (1.1).

v) Our approach will make essential use of the boundedness assumption on the physical domain \(\Omega \); in fact, inter alia due to a lack of comparison principles for (1.1) this requirement appears to be crucial already in the derivation of very basic boundedness features especially – but not exclusively – when \(\rho >0\) (cf. Sect. 2). We therefore have to leave open here the interesting question how far conclusions similar to those from Theorem 1.2 can be drawn in cases of unbounded domains, and particularly when \(\Omega =\mathbb {R}^3\).

Strategy.    Our analysis can be viewed as being composed of a first part in which the action of chemotaxis is yet essentially faded out, and a second level in which full tribute is paid to the whole complexity of (1.1). In particular, basic eventual smallness properties of \(n_\varepsilon \), as obtained for suitably small \(\rho \) from mere integration and subsequent interpolation, will be summarized in Sect. 2 and thereafter used in Sect. 3 to derive corresponding smallness features of \(A^\beta u_\varepsilon \) with respect to the norm in \(L^2(\Omega )\) for some \(\beta \) in the range \((\frac{1}{4},\infty )\) throughout which \(D(A_2^\beta )\) continuously embeds into \(L^3(\Omega ;\mathbb {R}^3)\). By means of a standard zero-order testing procedure and an application of maximal Sobolev regularity theory to the second Eq. in (1.7), in Sect. 4 this will in provide some basic knowledge on eventual smallness of \(\nabla c_\varepsilon \) in an appropriately integrated sense.

The second stage of our argument will then be entered in Sect. 5, where by inter alia explicitly using the first Eq. in (1.7) for a second time, a coupled quantity of the form \(\int _\Omega \psi (n_\varepsilon -\frac{\rho _+}{\mu }) + \int _\Omega |\nabla c_\varepsilon |^{2p}\) will be seen to enjoy some quasi-Lyapunov property for some \(p>\frac{3}{2}\) and a suitably designed function \(\psi \) on \(\mathbb {R}\) satisfying \(s^{-p}\psi (s)\rightarrow 1\) as \(s\rightarrow +\infty \); we remark already here that only from this point on, dependencies of the obtained estimates on \(\chi \) appear. In Sects. 6 and 7, the improved eventual smallness properties of \(\nabla c_\varepsilon \) thereby implied will be developed into \(L^\infty \) bounds for \(n_\varepsilon \) and a Hölder estimate for \(u_\varepsilon \), whereupon standard regularity theories for parabolic and Stokes evolution problems become applicable so as to confirm the statement from Theorem 1.2 in Sect. 8.

Without explicit further mentioning, throughout the sequel we shall suppose that \(\phi \in W^{2,\infty }(\Omega )\) is fixed, and given \(\chi >0,\rho \in \mathbb {R}\), \(\mu >0\), \(f\in C^1(\overline{\Omega }\times [0,\infty );\mathbb {R}^3)\) and initial data fulfilling (1.6), for \(\varepsilon \in (0,1)\) we let \((n_\varepsilon ,c_\varepsilon ,u_\varepsilon ,P_\varepsilon )\) denote the corresponding solution of (1.7) obtained in Proposition 1.1. Apart from that, let us announce already here that within each of our proofs, constants will be labeled as \(C_1, C_2,...\), and that in order to avoid an abundant globally consecutive numbering involving high indices, constants such as \(C_1\) may attain different values in different proofs.

Basic Eventual Bounds for \(n_\varepsilon \)

A starting point for our asymptotic analysis is formed by the following basic information on global and eventual bounds for the first solution component, as resulting from a simple integration of the first Eq. in (1.7) in a standard manner.

Lemma 2.1

i) We have

$$\begin{aligned} \int _\Omega n_\varepsilon (x,t)dx \le m:=\max \bigg \{ \int _\Omega n_0 \, , \, \frac{\rho _+ |\Omega |}{\mu } \bigg \} \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1)\qquad \end{aligned}$$
(2.1)

as well as

$$\begin{aligned} \int _t^{t+1} \int _\Omega n_\varepsilon ^2(x,s) dxds \le \frac{(\rho _+ +1)m}{\mu } \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1). \end{aligned}$$
(2.2)

ii) If \(\rho _0>0\) is such that \(\rho _0\ge \rho \), then

$$\begin{aligned} \int _\Omega n_\varepsilon (x,t)dx \le 2|\Omega | \cdot \frac{\rho _0}{\mu } \qquad \text{ for } \text{ all } t\ge \frac{\ln 2}{\rho _0} \text{ and } \varepsilon \in (0,1) \end{aligned}$$
(2.3)

and

$$\begin{aligned} \int _t^{t+1} \int _\Omega n_\varepsilon ^2(x,s) dxds \le 2|\Omega | \cdot \frac{\rho _0(\rho _0+1)}{\mu ^2} \qquad \text{ for } \text{ all } t\ge \frac{\ln 2}{\rho _0} \text{ and } \varepsilon \in (0,1). \nonumber \\ \end{aligned}$$
(2.4)

Proof

We fix any \(\rho _1\ge \rho _+\) and integrate the first Eq. in (1.7) over \(\Omega \) to see that

$$\begin{aligned} \frac{d}{dt} \int _\Omega n_\varepsilon = \rho \int _\Omega n_\varepsilon - \mu \int _\Omega n_\varepsilon ^2 \le \rho _1 \int _\Omega n_\varepsilon - \mu \int _\Omega n_\varepsilon ^2 \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1). \nonumber \\ \end{aligned}$$
(2.5)

Since \((\int _\Omega n_\varepsilon )^2 \le |\Omega | \cdot \int _\Omega n_\varepsilon ^2\) for all \(t>0\) and \(\varepsilon \in (0,1)\) by the Cauchy-Schwarz inequality, this implies that \(y_\varepsilon (t):=\int _\Omega n_\varepsilon (x,t)dx, \ t\ge 0\), \(\varepsilon \in (0,1)\), satisfies

$$\begin{aligned} y_\varepsilon '(t) \le \rho _1 y_\varepsilon (t) - \frac{\mu }{|\Omega |} y_\varepsilon ^2(t) \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1) \end{aligned}$$
(2.6)

and hence, by a comparison argument,

$$\begin{aligned} y_\varepsilon (t) \le \max \Big \{ y_\varepsilon (0) \, , \, \frac{\rho _1 |\Omega |}{\mu } \Big \} \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1). \end{aligned}$$
(2.7)

i) Choosing \(\rho _1:=\rho _+\) in (2.7), we immediately obtain (2.1), whereupon integrating (2.5) in time shows that

$$\begin{aligned} \int _\Omega n_\varepsilon (\cdot ,t+1) + \mu \int _t^{t+1} \int _\Omega n_\varepsilon ^2\le & {} \int _\Omega n_\varepsilon (\cdot ,t) + \rho _+ \int _t^{t+1} \int _\Omega n_\varepsilon \\\le & {} m+ \rho _+ m \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1) \end{aligned}$$

and thereby proves (2.2).

ii) Taking \(\rho _1:=\rho _0\) now, an explicit solution of the Bernoulli-type ODE associated with (2.6) yields the inequality

$$\begin{aligned} y_\varepsilon (t)\le & {} \bigg \{ \frac{\mu }{\rho _0 |\Omega |} \cdot \big (1-e^{-\rho _0 t}\big ) + \frac{1}{y_\varepsilon (0)} \cdot e^{-\rho _0 t} \bigg \}^{-1} \\\le & {} \Big \{ \frac{\mu }{\rho _0 |\Omega |} \cdot \big (1-e^{-\rho _0 t} \big ) \Big \}^{-1} \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1), \end{aligned}$$

which directly yields (2.3), because \(1-e^{-\rho _0 t} \ge \frac{1}{2}\) for all \(t\ge \frac{\ln 2}{\rho _0}\). Thereafter, once again integrating (2.5) we see that

$$\begin{aligned} \int _\Omega n_\varepsilon (\cdot ,t+1) + \mu \int _t^{t+1} \int _\Omega n_\varepsilon ^2\le & {} 2|\Omega | \cdot \frac{\rho _0}{\mu } + \rho _0 \cdot \Big (2|\Omega | \cdot \frac{\rho _0}{\mu }\Big ) \\= & {} 2|\Omega | \cdot \frac{\rho _0(\rho _0+1)}{\mu } \qquad \text{ for } \text{ all } t\ge \frac{\ln 2}{\rho _0} \text{ and } \varepsilon \in (0,1), \end{aligned}$$

and hence obtain (2.4). \(\quad \square \)

Through appropriate interpolation, the latter shows how the particular form of the assumption in (1.10) enters a further eventual boundedness property of \(n_\varepsilon \) which involves topological information somewhat weaker than that in (2.4), but which on the other hand apparently yields genuine ultimate smallness, and which does so also in some cases when in Lemma 2.1 the expression \(\frac{\rho _0}{\mu ^2}\) is large.

Lemma 2.2

Let \(\omega >0\). Then there exists \(\theta _1=\theta _1(\omega )\in (0,\frac{1}{2})\) with the following property: For all \(\delta >0\) one can find \(\eta _1=\eta _1(\omega ,\delta )>0\) such that whenever \(\rho \in \mathbb {R}\) and \(\mu >0\) are such that (1.10) holds with some \(\eta < \eta _1(\omega ,\delta )\), then there exists \(t_0=t_0(\eta )>0\) such that for each \(p\in [\frac{3}{2},\frac{3}{2}+\theta _1]\) and any \(\varepsilon \in (0,1)\) we have

$$\begin{aligned} \int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^p(\Omega )}^2 ds < \delta \qquad \text{ for } \text{ all } t>t_0. \end{aligned}$$
(2.8)

Proof

Given \(\omega >0\), we let \(\theta =\theta _1(\omega )>0\) be suitably small such that \(\theta <\frac{1}{2}\) and \(\theta <\omega \), noting that the latter ensures the inequality

$$\begin{aligned} \Big (\frac{3}{2}+\omega \Big ) \cdot \frac{4}{3+2\theta } = 2 \cdot \frac{3+2\omega }{3+2\theta }>2, \end{aligned}$$
(2.9)

and that the former restriction warrants that \(a=a(\omega ):=\frac{2+4\theta }{3+2\theta }\) satisfies \(a<1\).

Now writing

$$\begin{aligned} C_1=C_1(\omega ):=\max \Big \{ 1 \, , \, |\Omega |^\frac{8\theta }{9} \Big \}, \end{aligned}$$
(2.10)

for arbitrary \(\delta >0\) we choose \(\eta _1=\eta _1(\omega ,\delta )>0\) small enough fulfilling \(\eta _1 \le 1\) as well as

$$\begin{aligned} 4|\Omega |^{2-a} \eta _1^{2-a} < \frac{\delta }{C_1}, \end{aligned}$$
(2.11)

and henceforth assume that \(\rho \in \mathbb {R}\) and \(\mu >0\) are such that (1.10) holds with some positive \(\eta < \eta _1\). Then with

$$\begin{aligned} \rho _0=\rho _0(\eta ):=\eta \cdot \min \Big \{ \mu \, , \, \mu ^{\frac{3}{2}+\omega } \Big \} \end{aligned}$$
(2.12)

we clearly have \(\rho _0>\rho \) and moreover also \(\rho _0>0\), whence Lemma 2.1 applies to say that for any \(\varepsilon \in (0,1)\), both

$$\begin{aligned} \Vert n_\varepsilon (\cdot ,t)\Vert _{L^1(\Omega )} \le 2|\Omega | \cdot \frac{\rho _0}{\mu } \qquad \text{ for } \text{ all } t\ge t_0 \end{aligned}$$
(2.13)

and

$$\begin{aligned} \int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^2(\Omega )}^2 ds \le 2|\Omega | \cdot \frac{\rho _0(\rho _0+1)}{\mu ^2} \qquad \text{ for } \text{ all } t\ge t_0 \end{aligned}$$
(2.14)

are valid with \(t_0=t_0(\eta ):=\frac{\ln 2}{\rho _0}\). As \(\frac{3}{2}+\theta <2\), we may interpolate here by means of the Hölder inequality to infer from (2.13) and (2.14) that according to our definition of a we have

$$\begin{aligned}&\int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^{\frac{3}{2}+\theta }(\Omega )}^2 ds\nonumber \\&\quad \le \int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^2(\Omega )}^{2a} \cdot \Vert n_\varepsilon (\cdot ,s)\Vert _{L^1(\Omega )}^{2(1-a)} ds \nonumber \\&\quad \le \Big ( 2|\Omega | \cdot \frac{\rho _0}{\mu } \Big )^{2(1-a)} \int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^2(\Omega )}^{2a} ds \nonumber \\&\quad \le \Big ( 2|\Omega | \cdot \frac{\rho _0}{\mu } \Big )^{2(1-a)} \bigg ( \int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^2(\Omega )}^2 ds \bigg )^a \nonumber \\&\quad \le \Big ( 2|\Omega | \cdot \frac{\rho _0}{\mu } \Big )^{2(1-a)} \cdot \Big ( 2|\Omega | \cdot \frac{\rho _0(\rho _0+1)}{\mu ^2} \Big )^a \nonumber \\&\quad = (2|\Omega |)^{2-a} \rho _0^{2-a} (\rho _0+1)^a \mu ^{-2} \qquad \text{ for } \text{ all } t\ge t_0 \text{ and } \varepsilon \in (0,1). \nonumber \\ \end{aligned}$$
(2.15)

Here if \(\mu \le 1\), then (2.12) along with the restriction \(\eta \le \eta _1\le 1\) warrants that \(\rho _0+1 = \eta \mu ^{\frac{3}{2}+\omega } +1 \le \eta +1\le 2\), and that hence

$$\begin{aligned} (2|\Omega |)^{2-a} \rho _0^{2-a} (\rho _0+1)^a \mu ^{-2}\le & {} (2|\Omega |)^{2-a} \rho _0^{2-a} \cdot 2^a \cdot \mu ^{-2} \\= & {} 4|\Omega |^{2-a} \cdot \big (\eta \mu ^{\frac{3}{2}+\omega }\big )^{2-a} \mu ^{-2} \\= & {} 4|\Omega |^{2-a} \eta ^{2-a} \mu ^{(\frac{3}{2}+\omega )(2-a)-2}, \end{aligned}$$

so that computing \((\frac{3}{2}+\omega )(2-a)=(\frac{3}{2}+\omega ) \cdot { \frac{4}{3+2\theta } } \) and recalling (2.9) we find that in this case,

$$\begin{aligned} (2|\Omega |)^{2-a} \rho _0^{2-a} (\rho _0+1)^a \mu ^{-2} \le 4|\Omega |^{2-a} \eta ^{2-a} < \frac{\delta }{C_1} \end{aligned}$$

by (2.11).

Next, if \(\mu >1\) then (2.12) means that \(\rho _0=\eta \mu \), and thus in the case \(\rho _0 \le 1\) we can estimate

$$\begin{aligned} (2|\Omega |)^{2-a} \rho _0^{2-a} (\rho _0+1)^a \mu ^{-2}\le & {} (2|\Omega |)^{2-a} \rho _0^{2-a} \cdot 2^a \cdot \mu ^{-2} \\= & {} 4|\Omega |^{2-a} \big ( \eta \mu \big )^{2-a} \mu ^{-2} \\= & {} 4|\Omega |^{2-a} \eta ^{2-a} \mu ^{-a}, \end{aligned}$$

using that then we still have \(\rho _0+1\le 2\). As now \(\mu ^{-a} \le 1\), again invoking (2.11) we see that

$$\begin{aligned} (2|\Omega |)^{2-a} \rho _0^{2-a} (\rho _0+1)^a \mu ^{-2} <\frac{\delta }{C_1} \end{aligned}$$

also in this case.

Finally, if yet \(\mu >1\) but now \(\rho _0>1\), then estimating \(\rho _0+1\le 2\rho _0\) leads to the inequality

$$\begin{aligned} (2|\Omega |)^{2-a} \rho _0^{2-a} (\rho _0+1)^a \mu ^{-2}\le & {} (2|\Omega |)^{2-a} \rho _0^{2-a} \cdot (2\rho _0)^a \cdot \mu ^{-2} \\= & {} 4|\Omega |^{2-a} \eta ^2 \\\le & {} 4|\Omega |^{2-a} \eta ^{2-a} \\< & {} \frac{\delta }{C_1}, \end{aligned}$$

once more because of (2.11), and again due to the restriction \(\eta _1\le 1\).

In summary, in both cases \(\mu \le 1\) and \(\mu >1\) we infer from (2.15) that

$$\begin{aligned} \int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^{\frac{3}{2}+\theta }(\Omega )}^2 ds < \frac{\delta }{C_1} \qquad \text{ for } \text{ all } t\ge t_0 \text{ and } \varepsilon \in (0,1). \end{aligned}$$

Thus, if \(p\in [\frac{3}{2},\frac{3}{2}+\theta ]\) is arbitrary, then by the Hölder inequality we find that

$$\begin{aligned} \int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^p(\Omega )}^2 ds\le & {} |\Omega |^\frac{2(p_1-p)}{p_1 p} \int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^{\frac{3}{2}+\theta }(\Omega )}^2 ds \nonumber \\\le & {} |\Omega |^\frac{2(p_1-p)}{p_1 p} \cdot \frac{\delta }{C_1} \qquad \text{ for } \text{ all } t\ge t_0 \text{ and } \varepsilon \in (0,1)\qquad \end{aligned}$$
(2.16)

with \(p_1:=\frac{3}{2}+\theta \), where since

$$\begin{aligned} 0 \le \frac{2(p_1-p)}{p_1 p} \le \frac{2\theta }{(\frac{3}{2})^2} =\frac{8\theta }{9}, \end{aligned}$$

we can estimate

$$\begin{aligned} |\Omega |^\frac{2(p_1-p)}{p_1 p} \le \max \Big \{ 1 \, , \, |\Omega |^\frac{8\theta }{9} \Big \}. \end{aligned}$$

Recalling (2.10), we thereby see that (2.16) entails (2.8). \(\quad \square \)

Smallness of \(u_\varepsilon \) in \(D(A_2^\beta )\) for Some \(\beta >\frac{1}{4}\)

The purpose of this key section is to make sure that when applied to suitably small \(\delta >0\), the bounds obtained in Lemma 2.2 provide sufficient eventual smallness properties of the coupling-induced contribution \(n_\varepsilon \nabla \phi \) to the forcing term in the Navier-Stokes subsystem of (1.7), so as to warrant ultimate estimates for \(u_\varepsilon \) with respect to the norm in \(D(A_2^\beta )\) for some \(\beta \) exceeding the number \(\frac{1}{4}\) quite commonly encountered in regularity analysis of three-dimensional Navier-Stokes problems ([46]).

As a means to appropriately derive upper estimates for functions satisfying linearly damped ODIs involving sources for which certain averaged bounds are known, from [59, Lemma 3.4] let us recall the following observation that will be referred to not only in this section, but also in Lemma 4.1 below.

Lemma 3.1

Let \(t_0\in \mathbb {R}, T\in (t_0,\infty ]\) and \(a>0\), and suppose that \(y\in C^0([t_0,T)) \cap C^1((t_0,T))\) has the property that

$$\begin{aligned} y'(t) + ay(t) \le h(t) \qquad \text{ for } \text{ all } t\in (t_0,T) \end{aligned}$$
(3.1)

with some nonnegative \(h\in L^1(\mathbb {R})\) for which there exist \(\tau >0\) and \(b>0\) such that

$$\begin{aligned} \frac{1}{\tau } \int _t^{t+\tau } h(s)ds \le b \qquad \text{ for } \text{ all } t\in (t_0,T). \end{aligned}$$
(3.2)

Then

$$\begin{aligned} y(t) \le e^{-a(t-t_0)} y(t_0) + \frac{b\tau }{1-e^{-a\tau }} \qquad \text{ for } \text{ all } t\in [t_0,T), \end{aligned}$$
(3.3)

and in particular

$$\begin{aligned} y(t) \le y(t_0) + \frac{b\tau }{1-e^{-a\tau }} \qquad \text{ for } \text{ all } t\in [t_0,T). \end{aligned}$$
(3.4)

Through an analysis of the energy inequality associated with the approximate Navier-Stokes Eq. in (1.7), a first conclusion of Lemma 3.1 now asserts eventual smallness of a temporally averaged Dirichlet integral associated with the fluid velocity field, provided that the external force f satisfies a smallness assumption slightly weaker than that in (1.11).

Lemma 3.2

Let \(\omega >0\). Then for all \(\delta >0\) there exist \(\eta _2=\eta _2(\omega ,\delta )>0\) and \(\kappa _2=\kappa _2(\delta )>0\) with the property that if \(\rho \in \mathbb {R}\), \(\mu >0\) and \(f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)\) are such that (1.10) as well as

$$\begin{aligned} \limsup _{t\rightarrow \infty } \int _t^{t+1} \Vert f(\cdot ,s)\Vert _{L^\frac{6}{5}(\Omega )}^2 ds < \kappa \end{aligned}$$
(3.5)

hold with some \(\eta < \eta _2(\omega ,\delta )\) and \(\kappa < \kappa _2(\delta )\), then one can find \(t_0=t_0(\omega ,\eta ,\kappa ,n_0,u_0)>0\) such that for all \(\varepsilon \in (0,1)\) we have

$$\begin{aligned} \int _t^{t+1} \Vert \nabla u_\varepsilon (\cdot ,s)\Vert _{L^2(\Omega )}^2 < \delta \qquad \text{ for } \text{ all } t\ge t_0. \end{aligned}$$
(3.6)

Proof

Given \(\omega >0\), we let \(\theta =\theta _1(\omega ) \in (0,\frac{1}{2})\) be as provided by Lemma 2.2 and abbreviate \(p=p(\omega ):=\frac{3}{2}+\theta \) and \(C_1=C_1(\omega ):=|\Omega |^\frac{5p-6}{6p}\). Moreover, we let \(C_2:=\Vert \nabla \phi \Vert _{L^\infty (\Omega )}\) and invoke a Poincaré inequality and a Sobolev inequality to find \(C_3>0\) and \(C_4>0\) such that

$$\begin{aligned} C_3 \int _\Omega |\varphi |^2 \le \int _\Omega |\nabla \varphi |^2 \qquad \text{ for } \text{ all } \varphi \in W_0^{1,2}(\Omega ;\mathbb {R}^3) \end{aligned}$$
(3.7)

and

$$\begin{aligned} \Vert \varphi \Vert _{L^6(\Omega )} \le C_4 \Vert \nabla \varphi \Vert _{L^2(\Omega )} \qquad \text{ for } \text{ all } \varphi \in W_0^{1,2}(\Omega ;\mathbb {R}^3). \end{aligned}$$
(3.8)

Writing \(C_5:=1+(1-e^{-\frac{C_3}{2}})^{-1}\), for fixed \(\delta >0\) we thereupon pick \(\kappa _2=\kappa _2(\delta )>0\) and \(\delta _1=\delta _1(\omega ,\delta )>0\) such that

$$\begin{aligned} 2C_4^2 C_5 \kappa _2 <\frac{\delta }{8} \end{aligned}$$
(3.9)

as well as

$$\begin{aligned} 2C_1^2 C_2^2 C_4^2 C_5 \delta _1 < \frac{\delta }{8}. \end{aligned}$$
(3.10)

We then suppose that \(f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)\) has the property (3.5) with some \(\kappa <\kappa _2\), and that \(\rho \in \mathbb {R}\) and \(\mu >0\) are such that (1.10) holds with some \(\eta <\eta _2=\eta _2(\omega ,\delta ):=\eta _1(\omega ,\delta _1)\), where \(\eta _1(\cdot ,\cdot )\) is as provided by Lemma 2.2. Thus, (3.5) implies that there exists \(t_1=t_1(\kappa )>0\) such that

$$\begin{aligned} \int _t^{t+1} \Vert f(\cdot ,s)\Vert _{L^\frac{6}{5}(\Omega )}^2 ds \le \kappa \qquad \text{ for } \text{ all } t\ge t_1, \end{aligned}$$
(3.11)

whereas Lemma 2.2 yields \(t_2=t_2(\eta ,\kappa )>t_1\) such that for all \(\varepsilon \in (0,1)\), the corresponding solution has the property that that

$$\begin{aligned} \int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^p(\Omega )}^2 ds \le \delta _1 \qquad \text{ for } \text{ all } t\ge t_2. \end{aligned}$$
(3.12)

Furthermore, using Lemma 2.1 and the Hölder inequality, we see that with \(m:=\max \Big \{\int _\Omega n_0, \frac{\rho _+|\Omega |}{\mu }\Big \}\) we have

$$\begin{aligned}&\int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^p(\Omega )}^2 ds \nonumber \\&\quad \le C_6=C_6(\omega ,n_0):=|\Omega |^\frac{2-p}{p} \cdot \frac{(1+\rho _+)m}{\mu } \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1),\qquad \quad \end{aligned}$$
(3.13)

while by continuity of f on \(\bar{\Omega }\times [0,t_2]\) we can fix \(C_7=C_7(\eta ,\kappa )>0\) fulfilling

$$\begin{aligned} \int _t^{t+1} \Vert f(\cdot ,s)\Vert _{L^\frac{6}{5}(\Omega )}^2 ds \le C_7 \qquad \text{ for } \text{ all } t>0. \end{aligned}$$
(3.14)

We next abbreviate

$$\begin{aligned} C_8=C_8(\omega ,\eta ,\kappa ,n_0,u_0):= \int _\Omega |u_0|^2 + \frac{C_9}{1-e^{-\frac{C_3}{2}}}, \end{aligned}$$
(3.15)

where

$$\begin{aligned} C_9=C_9(\omega ,\eta ,\kappa ,n_0):=2C_2^2 C_4^2 C_6 + 2C_4^2 C_7. \end{aligned}$$
(3.16)

and choose \(t_0=t_0(\omega ,\eta ,\kappa ,n_0,u_0)>t_2\) large enough fulfilling

$$\begin{aligned} C_8 \, e^{-\frac{C_3}{2}(t_0-t_2)} \le \frac{\delta }{4}. \end{aligned}$$
(3.17)

Then testing the third Eq. in (1.7) by \(u_\varepsilon \) we obtain

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \int _\Omega |u_\varepsilon |^2 + \int _\Omega |\nabla u_\varepsilon |^2 = \int _\Omega n_\varepsilon u_\varepsilon \cdot \nabla \phi + \int _\Omega f\cdot u_\varepsilon \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1), \end{aligned}$$

where using the Hölder inequality, (3.8) as well as Young’s inequality show that by definition on \(C_2\) we have

$$\begin{aligned}&\int _\Omega n_\varepsilon u_\varepsilon \cdot \nabla \phi + \int _\Omega f\cdot u_\varepsilon \\&\quad \le \Vert u_\varepsilon \Vert _{L^6(\Omega )} \cdot \bigg \{ C_2 \Vert n_\varepsilon \Vert _{L^\frac{6}{5}(\Omega )} + \Vert f\Vert _{L^\frac{6}{5}(\Omega )} \bigg \} \\&\quad \le C_4 \Vert \nabla u_\varepsilon \Vert _{L^2(\Omega )} \cdot \bigg \{ C_2 \Vert n_\varepsilon \Vert _{L^\frac{6}{5}(\Omega )} + \Vert f\Vert _{L^\frac{6}{5}(\Omega )} \bigg \} \\&\quad \le \frac{1}{2} \int _\Omega |\nabla u_\varepsilon |^2 + \frac{C_4^2}{2} \cdot \bigg \{ C_2 \Vert n_\varepsilon \Vert _{L^\frac{6}{5}(\Omega )} + \Vert f\Vert _{L^\frac{6}{5}(\Omega )} \bigg \}^2 \\&\quad \le \frac{1}{2} \int _\Omega |\nabla u_\varepsilon |^2 + C_2^2 C_4^2 \Vert n_\varepsilon \Vert _{L^\frac{6}{5}(\Omega )}^2 + C_4^2 \Vert f\Vert _{L^\frac{6}{5}(\Omega )}^2 \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1). \end{aligned}$$

Therefore,

$$\begin{aligned}&\frac{d}{dt} \int _\Omega |u_\varepsilon |^2 + \int _\Omega |\nabla u_\varepsilon |^2 \le 2C_2^2 C_4^2 \Vert n_\varepsilon \Vert _{L^\frac{6}{5}(\Omega )}^2 \\&\qquad + 2 C_4^2 \Vert f\Vert _{L^\frac{6}{5}(\Omega )}^2 \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1), \end{aligned}$$

so that since according to (3.7) we have

$$\begin{aligned} \int _\Omega |\nabla u_\varepsilon |^2 \ge \frac{C_3}{2} \int _\Omega |u_\varepsilon |^2 + \frac{1}{2} \int _\Omega |\nabla u_\varepsilon |^2 \qquad \text{ for } \text{ all } t> \text{ and } \varepsilon \in (0,1), \end{aligned}$$

the functions \(y_\varepsilon (t):=\int _\Omega |u_\varepsilon (x,t)|^2 dx, \ t\ge 0\), and \(h_\varepsilon (t):=2C_2^2 C_4^2 \Vert n_\varepsilon (\cdot ,t)\Vert _{L^\frac{6}{5}(\Omega )}^2 + 2C_4^2 \Vert f(\cdot ,t)\Vert _{L^\frac{6}{5}(\Omega )}^2\), \(t>0\), satisfy

$$\begin{aligned} y_\varepsilon '(t) + \frac{C_3}{2} y_\varepsilon (t) + \frac{1}{2} \int _\Omega |\nabla u_\varepsilon |^2 \le h_\varepsilon (t) \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1). \end{aligned}$$
(3.18)

In order to firstly derive a temporally global bound for y from this, we recall (3.13), (3.14) and (3.16) to see that

$$\begin{aligned} \int _t^{t+1} h_\varepsilon (s) ds \le 2C_2^2 C_4^2 \cdot C_6 + 2C_4^2 \cdot C_7 = C_9 \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1), \end{aligned}$$

so that Lemma 3.1 turns (3.18) into the uniform estimate

$$\begin{aligned} y_\varepsilon (t) \le e^{-\frac{C_3}{2} t} y_\varepsilon (0) + \frac{C_9}{1-e^{-\frac{C_3}{2}}} \le C_8 \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1) \end{aligned}$$
(3.19)

by (3.15). In order to obtain a sharper bound for large times, we next use (3.12) and apply the Hölder inequality to see that by definition of \(C_1\),

$$\begin{aligned} \int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^\frac{6}{5}(\Omega )}^2 ds\le & {} \int _t^{t+1} \Big \{ \Vert n_\varepsilon (\cdot ,s)\Vert _{L^p(\Omega )} \cdot |\Omega |^\frac{5p-6}{6p} \Big \}^2 ds \\\le & {} C_1^2 \delta _1 \qquad \text{ for } \text{ all } t\ge t_2 \text{ and } \varepsilon \in (0,1), \end{aligned}$$

which combined with (3.11) and the fact that \(t_2>t_1\) implies that

$$\begin{aligned} \int _t^{t+1} h_\varepsilon (s)ds \le C_{10}:=2C_2^2 C_4^2 \cdot C_1^2 \delta _1 + 2C_4^2 \kappa \qquad \text{ for } \text{ all } t\ge t_2 \text{ and } \varepsilon \in (0,1).\nonumber \\ \end{aligned}$$
(3.20)

Consequently, Lemma 3.1 shows that (3.18) firstly entails the inequality

$$\begin{aligned} y_\varepsilon (t) \le e^{-\frac{C_3}{2}(t-t_2)} y_\varepsilon (t_2) + \frac{C_{10}}{1-e^{-\frac{C_3}{2}}} \qquad \text{ for } \text{ all } t\ge t_2 \text{ and } \varepsilon \in (0,1), \end{aligned}$$

whence thanks to (3.19) and (3.17),

$$\begin{aligned} y_\varepsilon (t)\le & {} e^{-\frac{C_3}{2}(t_0-t_2)} \cdot C_8 + \frac{C_{10}}{1-e^{-\frac{C_3}{2}}} \\\le & {} \frac{\delta }{4} + \frac{C_{10}}{1-e^{-\frac{C_3}{2}}} \qquad \text{ for } \text{ all } t\ge t_0 \text{ and } \varepsilon \in (0,1). \end{aligned}$$

Thereafter, an integration of (3.18) using (3.6) yields

$$\begin{aligned} \frac{1}{2} \int _t^{t+1} \int _\Omega |\nabla u_\varepsilon |^2\le & {} y_\varepsilon (t) + \int _t^{t+1} h_\varepsilon (s)ds \\\le & {} \frac{\delta }{4} + \frac{C_{10}}{1-e^{-\frac{C_3}{2}}} + C_{10} \\< & {} \frac{\delta }{4} + \frac{\delta }{4}=\frac{\delta }{2} \qquad \text{ for } \text{ all } t\ge t_0 \text{ and } \varepsilon \in (0,1), \end{aligned}$$

because (3.9) and (3.10) assert that

$$\begin{aligned} \frac{C_{10}}{1-e^{-\frac{C_3}{2}}} + C_{10} = 2C_1^2 C_2^2 C_4^2 C_5 \delta _1 + 2C_4^2 C_5 \kappa < \frac{\delta }{8}+\frac{\delta }{8}=\frac{\delta }{4}. \end{aligned}$$

The proof is thereby complete. \(\quad \square \)

In a next step we apply the latter to conveniently small \(\delta \) to see by an analysis based on classical smoothing properties of the Stokes semigroup that if now we make use of (1.11) in its full strength with regard to the topological framework therein, then we can achieve the announced goal of this section in the following flavor:

Lemma 3.3

Let \(\omega >0\). Then one can find \(\theta _3=\theta _3(\omega )>0\) such that for each \(\delta >0\) there exist \(\eta _3=\eta _3(\omega ,\delta )>0\) and \(\kappa _3=\kappa _3(\omega ,\delta )>0\) such that if \(f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)\), \(\rho \in \mathbb {R}\) and \(\mu >0\) have the property that (1.10) as well as (1.11) are valid with some \(\eta <\eta _3\) and \(\kappa <\kappa _3\), then one can choose \(t_0=t_0(\omega ,\eta ,\kappa ,n_0,u_0)>0\) such that for all \(\beta \in [\frac{1}{4},\frac{1}{4}+\theta _3]\), each \(r\in [3,3+\theta _3]\) and any \(\varepsilon \in (0,1)\) we have

$$\begin{aligned} \Vert A^\beta u_\varepsilon (\cdot ,t)\Vert _{L^2(\Omega )} < \delta \qquad \text{ for } \text{ all } t\ge t_0 \end{aligned}$$
(3.21)

and

$$\begin{aligned} \Vert u_\varepsilon (\cdot ,t)\Vert _{L^r(\Omega )} < \delta \qquad \text{ for } \text{ all } t\ge t_0. \end{aligned}$$
(3.22)

Proof

A substantial part of the proof will consist in creating an adequate setup for a testing procedure involving the third Eq. in (1.7), and properly preparing the choice of the constants \(\theta _3, \eta _3, \kappa _3\) and \(t_0\). To achieve this, for fixed \(\omega >0\) we take \(\theta _1=\theta _1(\omega )\in (0,\frac{1}{2})\) from Lemma 2.2 and let \(p=p(\omega ):=\frac{3}{2}+\min \{\theta _1,\omega \}\). Then since \(p>\frac{3}{2}\), we have \(\frac{5}{2}-\frac{3}{p}>\frac{1}{2}\) and \(\frac{5}{2}-\frac{3}{p}>\frac{3}{2}-\frac{3}{2p}\), and since \(p<2\) we furthermore know that \(\frac{3}{2}-\frac{3}{2p}<\frac{3}{4}<1\), which implies that it is possible to pick \(\beta _0=\beta _0(\omega )\in (\frac{1}{4},\frac{1}{2})\) fulfilling

$$\begin{aligned} \frac{3}{2}-\frac{3}{2p}< 2\beta _0 < \frac{5}{2}-\frac{3}{p}. \end{aligned}$$
(3.23)

Here the right inequality warrants that \(2\cdot \frac{1-2\beta _0}{2} - \frac{3}{p}>-\frac{3}{2}\) and that accordingly \(D(A_p^\frac{1-2\beta _0}{2}) \hookrightarrow L^2(\Omega )\) ([15, 21]), so that we can find \(C_1=C_1(\omega )>0\) such that

$$\begin{aligned} \Vert \varphi \Vert _{L^2(\Omega )} \le C_1\Vert A^\frac{1-2\beta _0}{2}\varphi \Vert _{L^p(\Omega )} \qquad \text{ for } \text{ all } \varphi \in D(A_p^\frac{1-2\beta _0}{2}), \end{aligned}$$

in particular implying that

$$\begin{aligned} \Vert A^{-\frac{1-2\beta _0}{2}}\varphi \Vert _{L^2(\Omega )} \le C_1\Vert \varphi \Vert _{L^p(\Omega )} \qquad \text{ for } \text{ all } \varphi \in L^p_\sigma (\Omega ). \end{aligned}$$
(3.24)

On the other hand, from the left inequality in (3.23) we obtain that \(2\beta _0-\frac{3}{2}>-\frac{3}{2p}\) and that \(2\cdot \frac{1+2\beta _0}{2} - \frac{3}{2}>1-\frac{3}{2p}\), respectively implying that \(D(A_2^{\beta _0})\hookrightarrow L^{2p}(\Omega )\) and \(D(A_2^\frac{1+2\beta _0}{2}) \hookrightarrow W^{1,2p}(\Omega )\); as a consequence, we can fix \(C_2=C_2(\omega )>0\) and \(C_3=C_3(\omega )>0\) fulfilling

$$\begin{aligned} \Vert \varphi \Vert _{L^{2p}(\Omega )} \le C_2 \Vert A^{\beta _0}\varphi \Vert _{L^2(\Omega )} \qquad \text{ for } \text{ all } \varphi \in D(A_2^{\beta _0}) \end{aligned}$$
(3.25)

and

$$\begin{aligned} \Vert \nabla \varphi \Vert _{L^{2p}(\Omega )} \le C_3 \Vert A^\frac{1+2\beta _0}{2}\varphi \Vert _{L^2(\Omega )} \qquad \text{ for } \text{ all } \varphi \in D(A_2^\frac{1+2\beta _0}{2}). \end{aligned}$$
(3.26)

Apart from this, the fact that \(\beta _0>\frac{1}{4}\) along with our restriction \(p>\frac{3}{2}\) enables us to pick \(\theta _3=\theta _3(\omega )>0\) such that

$$\begin{aligned} \frac{1}{4}+\theta _3\le \beta _0 \end{aligned}$$
(3.27)

and

$$\begin{aligned} 3+\theta _3 \le 2p. \end{aligned}$$
(3.28)

Here, (3.28) implies that \(L^{2p}(\Omega ) \subset L^{3+\theta _3}(\Omega )\), whence by (3.25), for later convenience we can also pick \(C_4=C_4(\omega )>0\) such that

$$\begin{aligned} \Vert \varphi \Vert _{L^{3+\theta _3}(\Omega )} \le C_4 \Vert A^{\beta _0}\varphi \Vert _{L^2(\Omega )} \qquad \text{ for } \text{ all } \varphi \in D(A_2^{\beta _0}). \end{aligned}$$
(3.29)

Next, recalling that \(\beta _0<\frac{1}{2}\), that \(\beta \le \beta _0\) by (3.27), and that the family \((A^{-\lambda })_{\lambda \in (0,1)}\) is bounded in the space of bounded linear operators on \(L^2(\Omega ;\mathbb {R}^3)\) ([43, Lemma 2.6.3]), it is clear that we can find \(C_5=C_5(\omega )>0\), \(C_6=C_6(\omega )>0\) and \(C_7=C_7(\omega )>0\) such that

$$\begin{aligned} C_5 \Vert A^{\beta _0} \varphi \Vert _{L^2(\Omega )} \le \Vert A^\frac{1+2\beta _0}{2}\varphi \Vert _{L^2(\Omega )} \qquad \text{ for } \text{ all } \varphi \in D(A_2^\frac{1+2\beta _0}{2}) \end{aligned}$$
(3.30)

and

$$\begin{aligned} \Vert A^{\beta _0}\varphi \Vert _{L^2(\Omega )} \le C_6 \Vert \nabla \varphi \Vert _{L^2(\Omega )} \qquad \text{ for } \text{ all } \varphi \in D(A_2^\frac{1}{2}) \end{aligned}$$
(3.31)

as well as

$$\begin{aligned} \Vert A^\beta \varphi \Vert _{L^2(\Omega )} \le C_7 \Vert A^{\beta _0}\varphi \Vert _{L^2(\Omega )} \qquad \text{ for } \text{ all } \varphi \in D(A^{\beta _0}), \end{aligned}$$
(3.32)

and that letting \(\mathcal{P}\) denote the Helmholtz projection on \(L^2(\Omega ;\mathbb {R}^3)\) ([46]) and recalling that \(\mathcal{P}\) actually is bounded on \(L^p(\Omega ;\mathbb {R}^3)\) ([14]), we can choose \(C_8=C_8(\omega )>0\) fulfilling

$$\begin{aligned} \Vert \mathcal{P}\varphi \Vert _{L^p(\Omega )} \le C_8\Vert \varphi \Vert _{L^p(\Omega )} \qquad \text{ for } \text{ all } \varphi \in L^p(\Omega ), \end{aligned}$$
(3.33)

whereupon we abbreviate

$$\begin{aligned} C_9=C_9(\omega ):=2C_1^2 C_2^2 C_3^2 C_8^2 \end{aligned}$$
(3.34)

and

$$\begin{aligned} C_{10}:=\Vert \nabla \phi \Vert _{L^\infty (\Omega )}. \end{aligned}$$
(3.35)

Now given \(\delta >0\), we fix \(\delta _1=\delta _1(\omega ,\delta )>0\) small enough such that with

$$\begin{aligned} C_{11}:=\max \Big \{ 1 \, , \, |\Omega |^\frac{\theta _3}{9} \Big \}, \end{aligned}$$
(3.36)

we have

$$\begin{aligned} C_4\sqrt{\delta _1} \le \frac{\delta }{C_{11}} \end{aligned}$$
(3.37)

and

$$\begin{aligned} \delta _1 \le \frac{1}{8C_9} \end{aligned}$$
(3.38)

as well as

$$\begin{aligned} C_7 \sqrt{\delta _1} \le \delta , \end{aligned}$$
(3.39)

then pick \(\delta _2=\delta _2(\omega ,\delta )>0\) and \(\delta _3=\delta _3(\omega ,\delta )>0\) satisfying

$$\begin{aligned} \frac{1}{1-e^{-\frac{C_5^2}{4}}} \cdot \delta _2 < \frac{\delta _1}{2} \end{aligned}$$
(3.40)

and

$$\begin{aligned} C_6^2 \delta _3 < \frac{\delta _1}{2}, \end{aligned}$$
(3.41)

and thereafter choose \(\delta _4=\delta _4(\omega ,\delta )>0\) and \(\tilde{\kappa }=\tilde{\kappa }(\omega ,\delta )>0\) with the properties that

$$\begin{aligned} 2C_1^2 C_8^2 C_{10}^2 \delta _4 < \frac{\delta _2}{2} \end{aligned}$$
(3.42)

as well as

$$\begin{aligned} 2C_1^2 C_8^2 C_{12} \tilde{\kappa }< \frac{\delta _2}{2}, \end{aligned}$$
(3.43)

where \(C_{12}=C_{12}(\omega ):=|\Omega |^\frac{2(q-p)}{pq}\) with \(q:=\frac{3}{2}+\omega \).

We now let \(\eta _1=\eta _1(\omega ,\delta _4)\) be as given by Lemma 2.2 and take \(\eta _2=\eta _2(\omega ,\delta _3)\) as well as \(\kappa _2=\kappa _2(\delta _3)\) from Lemma 3.2 to finally define \(\eta _3=\eta _3(\omega ,\delta ):=\min \{\eta _1,\eta _2\}\) and \(\kappa _3=\kappa _3(\omega ,\delta ):=\min \{\tilde{\kappa },\frac{\kappa _2}{C_{13}}\}\) with \(C_{13}=C_{13}(\omega ):=|\Omega |^\frac{5q-6}{3q}\).

Henceforth assuming that \(\rho \in \mathbb {R}\), \(\mu >0\) and \(f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)\) are such that (1.10) and (1.11) hold with some \(\eta <\eta _3\) and \(\kappa <\kappa _3\), we then obtain from Lemma 2.2 and the fact that \(p\le \frac{3}{2}+\theta _1\) that there exists \(t_1=t_1(\omega ,\eta ,\kappa ,n_0,u_0)>0\) such that for all \(\varepsilon \in (0,1)\) we have

$$\begin{aligned} \int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^p(\Omega )}^2 ds \le \delta _4 \qquad \text{ for } \text{ all } t\ge t_1 \text{ and } \varepsilon \in (0,1), \end{aligned}$$
(3.44)

whereas since \(p\le \frac{3}{2}+\omega \), (1.11) in combination with the Hölder inequality implies the existence of \(t_2=t_2(\omega ,\eta ,\kappa ,n_0,u_0)>t_1\) such that

$$\begin{aligned}&\int _t^{t+1} \Vert f(\cdot ,s)\Vert _{L^p(\Omega )}^2 ds \nonumber \\&\quad \le C_{12} \int _t^{t+1} \Vert f(\cdot ,s)\Vert _{L^{\frac{3}{2}+\omega }(\Omega )}^2 ds \le C_{12} \kappa \qquad \text{ for } \text{ all } t\ge t_2 \text{ and } \varepsilon \in (0,1)\qquad \qquad \end{aligned}$$
(3.45)

and

$$\begin{aligned}&\int _t^{t+1} \Vert f(\cdot ,s)\Vert _{L^\frac{6}{5}(\Omega )}^2 ds \nonumber \\&\quad \le C_{13} \int _t^{t+1} \Vert f(\cdot ,s)\Vert _{L^{\frac{3}{2}+\omega }(\Omega )}^2 ds \le C_{13}\kappa \le \kappa _2 \qquad \text{ for } \text{ all } t\ge t_2 \text{ and } \varepsilon \in (0,1).\nonumber \\ \end{aligned}$$
(3.46)

As \(\eta \le \eta _2\), the latter enables us to infer from Lemma 3.2 that there exists \(t_3=t_3(\omega ,\eta ,\kappa ,n_0,u_0)>t_2\) fulfilling

$$\begin{aligned} \int _{t_3}^{t_3+1} \Vert \nabla u_\varepsilon (\cdot ,s)\Vert _{L^2(\Omega )}^2 ds < \delta _3 \qquad \text{ for } \text{ all } \varepsilon \in (0,1), \end{aligned}$$
(3.47)

and we claim that this entails (3.22) if we let \(t_0=t_0(\omega ,\eta ,\kappa ,n_0,u_0):=t_3+1\).

To verify this, we first observe that (3.47) implies that for each \(\varepsilon \in (0,1)\) we can pick \(t_\varepsilon =t_\varepsilon (\omega ,\eta ,\kappa ,n_0,u_0)\in (t_3,t_3+1)\) such that \(\Vert \nabla u_\varepsilon (\cdot ,t_\varepsilon )\Vert _{L^2(\Omega )}^2 < \delta _3\), from which thanks to (3.31), (3.41) and (3.38) we obtain that

$$\begin{aligned} \int _\Omega |A^{\beta _0} u_\varepsilon (\cdot ,t_\varepsilon )|^2 \le C_6^2 \delta _3 < \frac{\delta _1}{2} \le \frac{1}{4C_9}. \end{aligned}$$
(3.48)

In particular, writing \(y(t):=\int _\Omega |A^{\beta _0} u_\varepsilon (x,t)|^2 dx, \ t\ge t_\varepsilon \), we see that

$$\begin{aligned} S:= \Big \{ T>t_\varepsilon \ \Big | \ y(t) \le \frac{1}{4C_9} \quad \text{ for } \text{ all } t\in [t_\varepsilon ,T) \Big \} \end{aligned}$$

is not empty and hence \(T:=\sup S \in (t_\varepsilon ,\infty ]\) well-defined. In order to make sure that actually \(T=\infty \), we use the third Eq. in (1.7) to compute

$$\begin{aligned}&\frac{1}{2} \frac{d}{dt} \int _\Omega |A^{\beta _0} u_\varepsilon |^2 + \int _\Omega |A^\frac{1+2\beta _0}{2}u_\varepsilon |^2 \nonumber \\&\quad = - \int _\Omega A^{2\beta _0} u_\varepsilon \cdot \mathcal{P}\big [ (Y_\varepsilon u_\varepsilon \cdot \nabla ) u_\varepsilon \big ] + \int _\Omega A^{2\beta _0} u_\varepsilon \cdot \mathcal{P}[n_\varepsilon \nabla \phi ] \nonumber \\&\qquad + \int _\Omega A^{2\beta _0} u_\varepsilon \cdot \mathcal{P}f \qquad \text{ for } \text{ all } t>0. \end{aligned}$$
(3.49)

Here by self-adjointness of A and its fractional powers, employing Young’s inequality, (3.24) and (3.33) we can estimate

$$\begin{aligned} \int _\Omega A^{2\beta _0} u_\varepsilon \cdot \mathcal{P}f= & {} \int _\Omega A^\frac{1+2\beta _0}{2} u_\varepsilon \cdot A^{-\frac{1-2\beta _0}{2}} \mathcal{P}f \nonumber \\\le & {} \frac{1}{4} \int _\Omega |A^\frac{1+2\beta _0}{2}u_\varepsilon |^2 + \int _\Omega |A^{-\frac{1-2\beta _0}{2}} \mathcal{P}f|^2 \nonumber \\\le & {} \frac{1}{4} \int _\Omega |A^\frac{1+2\beta _0}{2}u_\varepsilon |^2 + C_1^2 \Vert \mathcal{P}f\Vert _{L^p(\Omega )}^2 \nonumber \\\le & {} \frac{1}{4} \int _\Omega |A^\frac{1+2\beta _0}{2}u_\varepsilon |^2 + C_1^2 C_8^2 \Vert f\Vert _{L^p(\Omega )}^2 \qquad \text{ for } \text{ all } t>0, \end{aligned}$$
(3.50)

and likewise we obtain

$$\begin{aligned} \int _\Omega A^{2\beta _0} u_\varepsilon \cdot \mathcal{P}\big [ n_\varepsilon \nabla \phi \big ]\le & {} \frac{1}{4} \int _\Omega |A^\frac{1+2\beta _0}{2} u_\varepsilon |^2 + C_1^2 C_8^2 \Vert n_\varepsilon \nabla \phi \Vert _{L^p(\Omega )}^2 \nonumber \\\le & {} \frac{1}{4} \int _\Omega |A^\frac{1+2\beta _0}{2} u_\varepsilon |^2 + C_1^2 C_8^2 C_{10}^2 \Vert n_\varepsilon \Vert _{L^p(\Omega )}^2 \qquad \text{ for } \text{ all } t>0\nonumber \\ \end{aligned}$$
(3.51)

by definition of \(C_{10}\).

Proceeding similarly and then using the Cauchy-Schwarz inequality, (3.25) and (3.26), we see that the convective term can be controlled according to

$$\begin{aligned}&- \int _\Omega A^{2\beta _0} u_\varepsilon \cdot \mathcal{P}\big [ (Y_\varepsilon u_\varepsilon \cdot \nabla ) u_\varepsilon \big ] \nonumber \\&\quad \le \frac{1}{4} \int _\Omega |A^\frac{1+2\beta _0}{2}u_\varepsilon |^2 + C_1^2 C_8^2 \Big \Vert (Y_\varepsilon u_\varepsilon \cdot \nabla )u_\varepsilon \Big \Vert _{L^p(\Omega )}^2 \nonumber \\&\quad \le \frac{1}{4} \int _\Omega |A^\frac{1+2\beta _0}{2}u_\varepsilon |^2 + C_1^2 C_8^2 \Vert Y_\varepsilon u_\varepsilon \Vert _{L^{2p}(\Omega )}^2 \Vert \nabla u_\varepsilon \Vert _{L^{2p}(\Omega )}^2 \nonumber \\&\quad \le \frac{1}{4} \int _\Omega |A^\frac{1+2\beta _0}{2}u_\varepsilon |^2 + C_1^2 C_8^2 \cdot C_2^2 \Vert A^{\beta _0} Y_\varepsilon u_\varepsilon \Vert _{L^2(\Omega )}^2 \cdot C_3^2 \Vert A^\frac{1+2\beta _0}{2}u_\varepsilon \Vert _{L^2(\Omega )}^2\qquad \qquad \end{aligned}$$
(3.52)

for all \(t>0\). Since \(A^{\beta _0}\) and \(Y_\varepsilon \) commute on e.g. \(D(A_2)\), and since it can easily be checked that \(\Vert Y_\varepsilon \varphi \Vert _{L^2(\Omega )} \le \Vert \varphi \Vert _{L^2(\Omega )}\) for all \(\varphi \in L^2_\sigma (\Omega )\), by definition of T we herein have

$$\begin{aligned} \Vert A^{\beta _0} Y_\varepsilon u_\varepsilon \Vert _{L^2(\Omega )}^2 \le \Vert A^{\beta _0}u_\varepsilon \Vert _{L^2(\Omega )}^2 \le \frac{1}{4C_9} \qquad \text{ for } \text{ all } t\in (t_\varepsilon ,T), \end{aligned}$$

so that recalling the definition of \(C_9\), from (3.52) we obtain

$$\begin{aligned} - \int _\Omega A^{2\beta _0} u_\varepsilon \cdot \mathcal{P}\big [ (Y_\varepsilon u_\varepsilon \cdot \nabla ) u_\varepsilon \big ] \le \frac{3}{8} \int _\Omega |A^\frac{1+2\beta _0}{2}u_\varepsilon |^2 \qquad \text{ for } \text{ all } t\in (t_\varepsilon ,T). \end{aligned}$$

In conjunction with (3.49), (3.50) and (3.51), this shows that

$$\begin{aligned}&\frac{1}{2} \frac{d}{dt} \int _\Omega |A^{\beta _0}u_\varepsilon |^2 + \frac{1}{8} \int _\Omega |A^\frac{1+2\beta _0}{2}u_\varepsilon |^2 \\&\quad \le C_1^2 C_8^2 C_{10}^2 \Vert n_\varepsilon \Vert _{L^p(\Omega )}^2 + C_1^2 C_8^2 \Vert f\Vert _{L^p(\Omega )}^2 \qquad \text{ for } \text{ all } t\in (t_\varepsilon ,T), \end{aligned}$$

which in light of (3.30) implies that

$$\begin{aligned} y'(t) + \frac{C_5^2}{4} y(t) \le h(t) \qquad \text{ for } \text{ all } t\in (t_\varepsilon ,T) \end{aligned}$$
(3.53)

with \(h(t):=2C_1^2 C_8^2 C_{10}^2 \Vert n_\varepsilon (\cdot ,t)\Vert _{L^p(\Omega )}^2 + 2C_1^2 C_8^2 \Vert f(\cdot ,t)\Vert _{L^p(\Omega )}^2\), \(t>0\).

Since (3.44) and (3.45) combined with (3.42), (3.43) and the fact that \(\kappa \le \kappa _1\) guarantee that

$$\begin{aligned} \int _t^{t+1} h(s)ds\le & {} 2C_1^2 C_8^2 C_{10}^2 \delta _4 + 2C_1^2 C_8^2 C_{12}\kappa \\< & {} \frac{\delta _2}{2}+ \frac{\delta _2}{2}=\delta _2 \qquad \text{ for } \text{ all } t>t_\varepsilon , \end{aligned}$$

Lemma 3.1 says that (3.53) implies the inequality

$$\begin{aligned} y(t) \le e^{-\frac{C_5^2}{4}(t-t_\varepsilon )} y(t_\varepsilon ) + \frac{1}{1-e^{-\frac{C_5^2}{4}}} \cdot \delta _2 \qquad \text{ for } \text{ all } t\in [t_\varepsilon ,T), \end{aligned}$$

which according to (3.48) and (3.40) in particular warrants that

$$\begin{aligned} y(t)\le & {} \int _\Omega |A^{\beta _0}u_\varepsilon (\cdot ,t_\varepsilon )|^2 + \frac{1}{1-e^{-\frac{C_5^2}{4}}} \cdot \delta _2 \nonumber \\< & {} \frac{\delta _1}{2}+\frac{\delta _1}{2}=\delta _1 \qquad \text{ for } \text{ all } t\in [t_\varepsilon ,T). \end{aligned}$$
(3.54)

As therefore \(y(t)\le \frac{1}{8C_9}\) for all \(t\in (t_\varepsilon ,T)\) due to (3.38), by continuity of y this firstly shows that indeed T cannot be finite, and thereupon we secondly conclude from (3.54) that by (3.32) and (3.39),

$$\begin{aligned} \Vert A^\beta u_\varepsilon (\cdot ,t)\Vert _{L^2(\Omega )} \le C_7 \Vert A^{\beta _0}u_\varepsilon (\cdot ,t)\Vert _{L^2(\Omega )} < C_{7} \sqrt{\delta _1} \le \delta \qquad \text{ for } \text{ all } t \ge t_\varepsilon , \end{aligned}$$

and that by (3.29) and (3.37),

$$\begin{aligned} \Vert u_\varepsilon (\cdot ,t)\Vert _{L^{3+\theta _3}(\Omega )} \le C_4\sqrt{y(t)} < C_4 \sqrt{\delta _1}\le \frac{\delta }{C_{11}} \qquad \text{ for } \text{ all } t \ge t_\varepsilon . \end{aligned}$$

By means of the Hölder inequality, for arbitrary \(r\in [3,3+\theta _3]\) we hence obtain from the definition (3.36) of \(C_{11}\) that

$$\begin{aligned} \Vert u_\varepsilon (\cdot ,t)\Vert _{L^r(\Omega )} \le |\Omega |^\frac{3+\theta _3-r}{(3+\theta _3)r} \cdot \Vert u_\varepsilon (\cdot ,t)\Vert _{L^{3+\theta _3}(\Omega )} < C_{11} \cdot \frac{\delta }{C_{11}} =\delta \qquad \text{ for } \text{ all } t \ge t_\varepsilon , \end{aligned}$$

because \(0\le \frac{3+\theta _3-r}{(3+\theta _3)r} \le \frac{\theta _3}{9}\) for any such r. Since \(t_\varepsilon < t_3+1=t_0\), this completes the proof. \(\quad \square \)

Smallness of \(\int _t^{t+1} \Vert \nabla c_\varepsilon (\cdot ,s)\Vert _{L^q(\Omega )}^2 ds\) for Some \(q>3\)

We next address the taxis gradient as the quantity of apparently most crucial influence on regularity in the cross-diffusion interplay in (1.7), in this section aiming at the derivation of a spatio-temporal boundedness feature thereof.

In a preliminary step toward this, we pursue a standard \(L^2\) testing strategy for the equation determining \(c_\varepsilon \), hence obtaining some basic result on ultimate smallness which, apart from mere solenoidality, does not rely on any quantitative information about fluid regularity:

Lemma 4.1

Let \(\omega >0\). Then for all \(\delta >0\) there exists \(\eta _4=\eta _4(\omega ,\delta )>0\) with the property that if \(f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)\), \(\rho \in \mathbb {R}\) and \(\mu >0\) are such that (1.10) holds with some \(\eta <\eta _4\), then one can pick \(t_0=t_0(\delta ,\eta ,n_0,c_0)>0\) such that

$$\begin{aligned} \int _\Omega c_\varepsilon ^2(x,t)dx < \delta \qquad \text{ for } \text{ all } t\ge t_0 \end{aligned}$$
(4.1)

and any \(\varepsilon \in (0,1)\).

Proof

Given \(\omega >0\) and \(\delta >0\), we let \(\theta _1=\theta _1(\omega )\in (0,\frac{1}{2})\) be as given by Lemma 2.2 and choose \(\delta _1=\delta _1(\omega ,\delta )>0\) small enough such that

$$\begin{aligned} \frac{C_1 C_2 \delta _1}{1-e^{-1}} < \frac{\delta }{2} \end{aligned}$$
(4.2)

holds with \(C_1=C_1(\omega ):=|\Omega |^\frac{5p-6}{3p}\), where \(p=p(\omega ):=\frac{3}{2}+\theta _1\), and where \(C_2>0\) is a constant satisfying

$$\begin{aligned} \Vert \varphi \Vert _{L^6(\Omega )}^2 \le C_2 \cdot \bigg \{ \int _\Omega |\nabla \varphi |^2 + \int _\Omega \varphi ^2 \bigg \} \qquad \text{ for } \text{ all } \varphi \in W^{1,2}(\Omega ). \end{aligned}$$
(4.3)

Then with \(\eta _1:=\eta _1(\omega ,\delta _1)\) taken from Lemma 2.2, we claim that the desired conclusion holds if we let \(\eta _4=\eta _4(\omega ,\delta ):=\eta _1\).

To verify this, given \(\rho \in \mathbb {R}\) and \(\mu >0\) such that (1.10) is valid with some \(\eta <\eta _4\), we first apply Lemma 2.2 and the Hölder inequality to find \(t_1=t_1(\eta )>0\) such that

$$\begin{aligned}&\int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^\frac{6}{5}(\Omega )}^2 ds \le C_1 \int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^p(\Omega )}^2 ds \nonumber \\&\qquad < C_1 \delta _1 \qquad \text{ for } \text{ all } t\ge t_1 \text{ and } \varepsilon \in (0,1). \nonumber \\ \end{aligned}$$
(4.4)

We next recall that according to Lemma 2.1, writing \(m:=\max \Big \{ \int _\Omega n_0 \, , \, \frac{\rho _+|\Omega |}{\mu }\Big \}\) we have

$$\begin{aligned} \int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^2(\Omega )}^2 ds \le C_3=C_3(n_0):=\frac{(1+\rho _+)m}{\mu } \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1), \end{aligned}$$

which again by the Hölder inequality implies that

$$\begin{aligned} \int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^\frac{6}{5}(\Omega )}^2 ds \le C_4=C_4(n_0):=|\Omega |^\frac{2}{3} C_3 \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1).\nonumber \\ \end{aligned}$$
(4.5)

We now test the second Eq. in (1.7) against \(c_\varepsilon \) and apply the Hölder inequality along with (4.3) and Young’s inequality to find that

$$\begin{aligned}&\frac{1}{2} \frac{d}{dt} \int _\Omega c_\varepsilon ^2 + \int _\Omega |\nabla c_\varepsilon |^2 + \int _\Omega c_\varepsilon ^2 = \int _\Omega n_\varepsilon c_\varepsilon \\&\quad \le \Vert n_\varepsilon \Vert _{L^\frac{6}{5}(\Omega )} \Vert c_\varepsilon \Vert _{L^6(\Omega )} \\&\quad \le \Vert n_\varepsilon \Vert _{L^\frac{6}{5}(\Omega )} \cdot \sqrt{C_2} \cdot \bigg \{ \int _\Omega |\nabla c_\varepsilon |^2 + \int _\Omega c_\varepsilon ^2 \bigg \}^\frac{1}{2} \\&\quad \le \frac{1}{2} \bigg \{ \int _\Omega |\nabla c_\varepsilon |^2 + \int _\Omega c_\varepsilon ^2 \bigg \} + \frac{C_2}{2} \Vert n_\varepsilon \Vert _{L^\frac{6}{5}(\Omega )}^2 \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1), \end{aligned}$$

and that hence \(y_\varepsilon (t):=\int _\Omega c_\varepsilon ^2(x,t) dx\) and \(h_\varepsilon (t):=C_2 \Vert n_\varepsilon (\cdot ,t)\Vert _{L^\frac{6}{5}(\Omega )}^2\), \(t\ge 0\), \(\varepsilon \in (0,1)\), satisfy

$$\begin{aligned} y_\varepsilon '(t) + y_\varepsilon (t) \le h_\varepsilon (t) \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1). \end{aligned}$$
(4.6)

As

$$\begin{aligned} \int _t^{t+1} h_\varepsilon (s)ds \le C_2 C_4 \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1) \end{aligned}$$

by (4.5), in view of Lemma 3.1 this firstly implies that

$$\begin{aligned} y_\varepsilon (t)\le & {} e^{-t} y_\varepsilon (0) + \frac{C_2 C_4}{1-e^{-1}} \nonumber \\\le & {} C_5=C_5(n_0,c_0):= \int _\Omega c_0^2 + \frac{C_2 C_4}{1-e^{-1}} \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1). \end{aligned}$$
(4.7)

We thereupon fix \(t_0=t_0(\delta ,\eta ,n_0,c_0)>t_1\) large enough fulfilling

$$\begin{aligned} C_5 \, e^{-(t_0-t_1)} < \frac{\delta }{2} \end{aligned}$$
(4.8)

and again apply Lemma 3.1, where now using \(t_1\) as a starting point enables us to rely on the possibly stronger information (4.4), rather than (4.5), to conclude that since

$$\begin{aligned} \int _t^{t+1} h_\varepsilon (s)ds \le C_1 C_2 \delta _1 \qquad \text{ for } \text{ all } t>t_1 \text{ and } \varepsilon \in (0,1), \end{aligned}$$

we have

$$\begin{aligned} y_\varepsilon (t) \le e^{-(t-t_1)} y_\varepsilon (t_1) + \frac{C_1 C_2 \delta _1}{1-e^{-1}} \qquad \text{ for } \text{ all } t\ge t_1 \text{ and } \varepsilon \in (0,1). \end{aligned}$$

Thanks to (4.7), (4.8) and (4.2), this entails that

$$\begin{aligned} y_\varepsilon (t)\le & {} e^{-(t-t_1)} \cdot C_5 + \frac{C_1 C_2 \delta _1}{1-e^{-1}} \\\le & {} \frac{\delta }{2} + \frac{\delta }{2}=\delta \qquad \text{ for } \text{ all } t\ge t_0 \text{ and } \varepsilon \in (0,1), \end{aligned}$$

and thereby proves (4.1). \(\quad \square \)

With this information at hand, we can appropriately control the lower-order contributions to the linear inhomogeneous heat equation, as satisfied by \(c_\varepsilon \), in the course of an estimation procedure based on maximal Sobolev regularity theory for the latter, thereby obtaining higher order and especially gradient estimates.

Lemma 4.2

Let \(\omega >0\). Then there exists \(\theta _5=\theta _5(\omega )>0\) such that for any \(\delta >0\) one can find \(\eta _5=\eta _5(\omega ,\delta )>0\) and \(\kappa _5=\kappa _5(\omega ,\delta )>0\) with the property that if \(\rho \in \mathbb {R}\), \(\mu >0\) and \(f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)\) satisfy (1.10) and (1.11) with some \(\eta <\eta _5\) and \(\kappa <\kappa _5\), then there exists \(t_0=t_0(\omega ,\delta ,\eta ,n_0,c_0,u_0)>0\) such that for any choice of \(q\in [3,3+\theta _5]\) and \(\varepsilon \in (0,1)\) we have

$$\begin{aligned} \int _t^{t+1} \Vert \nabla c_\varepsilon (\cdot ,s)\Vert _{L^q(\Omega )}^2 ds < \delta \qquad \text{ for } \text{ all } t\ge t_0. \end{aligned}$$
(4.9)

Proof

With \(\theta _1=\theta _1(\omega )\in (0,\frac{1}{2})\) taken from Lemma 2.2, we let \(\theta =\theta (\omega ):=\min \{\theta _1,\omega \}\), \(p=p(\omega ):=\frac{3}{2}+\theta \) and \(\theta _5=\theta _5(\omega ):=\frac{3p}{3-p}-3\equiv \frac{12\theta }{3-2\theta }\), so that \(W^{2,p}(\Omega )\hookrightarrow W^{1,\frac{3p}{3-p}}(\Omega )\), and hence by using the Hölder inequality one can readily find \(C_1=C_1(\omega )>0\) such that for any \(q\in [3,3+\theta _5]\) we have

$$\begin{aligned} \Vert \nabla \varphi \Vert _{L^q(\Omega )} \le C_1\Vert \varphi \Vert _{W^{2,p}(\Omega )} \qquad \text{ for } \text{ all } \varphi \in W^{2,p}(\Omega ). \end{aligned}$$
(4.10)

As \(p>1\), well-known results on maximal Sobolev regularity properties of the Neumann heat semigroup \((e^{t(\Delta -1)})_{t\ge 0}\) ([17]) become applicable to provide \(C_2=C_2(\omega )>0\) such that whenever \(h\in C^0(\bar{\Omega }\times [0,2])\) and \(\varphi \in C^{2,1}(\bar{\Omega }\times [0,2])\) are such that

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

then

$$\begin{aligned} \int _0^2 \Vert \varphi (\cdot ,s)\Vert _{W^{2,p}(\Omega )}^2 ds \le C_2\int _0^2 \Vert h(\cdot ,s)\Vert _{L^p(\Omega )}^2 ds. \end{aligned}$$
(4.11)

Moreover abbreviating \(C_3=C_3(\omega ):=|\Omega |^\frac{2-p}{p}\), given \(\delta >0\) we can pick positive constants \(\delta _i=\delta _i(\omega ,\delta )>0\), \(i\in \{1,2,3\}\), small enough such that

$$\begin{aligned} 6C_2 \delta _1<\frac{\delta }{4C_1^2} \end{aligned}$$
(4.12)

and

$$\begin{aligned} 3C_1^2 C_2 \delta _2^2 \le \frac{1}{2} \end{aligned}$$
(4.13)

as well as

$$\begin{aligned} 24 C_2 C_3 \delta _3 < \frac{\delta }{4C_1^2}. \end{aligned}$$
(4.14)

We thereafter define \(\eta _5=\eta _5(\omega ,\delta ):=\min \{\eta _1,\eta _3,\eta _4\}\) and \(\kappa _5=\kappa _5(\omega ,\delta ):=\kappa _3\), where \(\eta _1:=\eta _1(\omega ,\delta _1)\) is as provided by Lemma 2.2, \(\eta _3:=\eta _3(\omega ,\delta _2)\) and \(\kappa _3:=\kappa _3(\omega ,\delta _2)\) are taken from Lemma 3.3 and \(\eta _4:=\eta _4(\omega ,\delta _3)\) is obtained by an application of Lemma 4.1.

Supposing henceforth that \(\rho \in \mathbb {R}, \mu >0\) and \(f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)\) are such that (1.10) and (1.11) hold with some \(\eta <\eta _5\) and \(\kappa <\kappa _5\), we then infer from Lemma 2.2 that since \(\eta <\eta _1\) and \(p=\frac{3}{2}+\theta \le \frac{3}{2}+\theta _1\), there exists \(t_1=t_1(\eta )>0\) such that for all \(\varepsilon \in (0,1)\),

$$\begin{aligned} \int _t^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^p(\Omega )}^2 ds < \delta _1 \qquad \text{ for } \text{ all } t\ge t_1 \text{ and } \varepsilon \in (0,1), \end{aligned}$$
(4.15)

and then use (1.11) to find \(t_2=t_2(\eta )>t_1\) fulfilling

$$\begin{aligned} \int _t^{t+1} \Vert f(\cdot ,s)\Vert _{L^{\frac{3}{2}+\omega }(\Omega )}^2 ds < \kappa \qquad \text{ for } \text{ all } t\ge t_2 \text{ and } \varepsilon \in (0,1). \end{aligned}$$

As \(\kappa <\kappa _3(\omega ,\delta _2)\) and \(\eta <\eta _3(\omega ,\delta _2)\), this allows us to invoke Lemma 3.3 which shows that with some \(t_3=t_3(\omega ,\eta ,\kappa ,n_0,u_0)>t_2\) we have

$$\begin{aligned} \Vert u_\varepsilon (\cdot ,t)\Vert _{L^3(\Omega )} < \delta _2 \qquad \text{ for } \text{ all } t\ge t_3 \text{ and } \varepsilon \in (0,1). \end{aligned}$$
(4.16)

Furthermore, the restriction \(\eta <\eta _4(\omega ,\delta _3)\) warrants that Lemma 4.1 becomes applicable so as to yield \(t_4=t_4(\omega ,\delta ,\eta ,n_0,c_0,u_0)>t_3\) such that for all \(\varepsilon \in (0,1)\),

$$\begin{aligned}&\int _t^{t+1} \Vert c_\varepsilon (\cdot ,s)\Vert _{L^p(\Omega )}^2 \nonumber \\&\quad \le C_3 \sup _{s\in (t,t+1)} \Vert c_\varepsilon (\cdot ,s)\Vert _{L^2(\Omega )}^2 < C_3 \delta _3 \qquad \text{ for } \text{ all } t\ge t_4 \text{ and } \varepsilon \in (0,1), \end{aligned}$$
(4.17)

where we have used the Hölder inequality and the fact that \(p\le \frac{3}{2}+\theta _1<2\).

Now given \(t>t_0=t_0(\omega ,\delta ,\eta ,n_0,c_0,u_0):=t_4+1\), we fix a cut-off function \(\zeta \in C^\infty ([t-1,t+1])\) such that \(\mathrm{supp} \, \zeta \subset (t-1,t+1]\), \(\zeta \equiv 1\) in \([t,t+1]\) and \(0\le \zeta ' \le 2\) in \([t-1,t+1]\), and let

$$\begin{aligned} z(x,t):=\zeta (t)\cdot c_\varepsilon (x,t), \qquad x\in \bar{\Omega }, \ t\in [t-1,t+1], \end{aligned}$$

for fixed \(\varepsilon \in (0,1)\). Then z is a solution of

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

whence applying (4.11) to \(\varphi (x,s):=z(x,t-1+s)\), \((x,s)\in \bar{\Omega }\times [0,2]\), shows that

$$\begin{aligned} \int _{t-1}^{t+1} \Vert z(\cdot ,s)\Vert _{W^{2,p}(\Omega )}^2 ds\le & {} C_2 \int _{t-1}^{t+1} \Big \Vert \zeta (s)n_\varepsilon (\cdot ,s) - u_\varepsilon (\cdot ,s)\cdot \nabla z(\cdot ,s)\nonumber \\&+ \zeta '(s)c_\varepsilon (\cdot ,s)\Big \Vert _{L^p(\Omega )}^2 ds \nonumber \\\le & {} 3C_2 \int _{t-1}^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^p(\Omega )}^2 ds \nonumber \\&+ 3C_2 \int _{t-1}^{t+1} \Big \Vert u_\varepsilon (\cdot ,s)\cdot \nabla z(\cdot ,s)\Big \Vert _{L^p(\Omega )}^2 ds \nonumber \\&+ 12C_2 \int _{t-1}^{t+1} \Vert c_\varepsilon (\cdot ,s)\Vert _{L^p(\Omega )}^2 ds, \end{aligned}$$
(4.18)

because \(0\le \zeta ^2 \le 1\) and \(\zeta '^2\le 4\). Here using (4.15) and (4.12) we can estimate

$$\begin{aligned} 3C_2 \int _{t-1}^{t+1} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^p(\Omega )}^2 ds \le 3C_2 \cdot 2\delta _1 < \frac{\delta }{4C_1^2}, \end{aligned}$$
(4.19)

while combining (4.17) with (4.14) yields

$$\begin{aligned} 12C_2 \int _{t-1}^{t+1} \Vert c_\varepsilon (\cdot ,s)\Vert _{L^p(\Omega )}^2 ds \le 12C_2 \cdot 2C_3\delta _3 < \frac{\delta }{4C_1^2}. \end{aligned}$$
(4.20)

In treating the second last integral in (4.18), we invoke the Hölder inequality along with (4.16), (4.10) and (4.13) to obtain

$$\begin{aligned} 3C_2 \int _{t-1}^{t+1}\Big \Vert u_\varepsilon (\cdot ,s)\cdot \nabla z(\cdot ,s)\Big \Vert _{L^p(\Omega )}^2 ds\le & {} 3C_2 \int _{t-1}^{t+1} \Vert u_\varepsilon (\cdot ,s)\Vert _{L^3(\Omega )}^2 \Vert \nabla z(\cdot ,s)\Vert _{L^\frac{3p}{3-p}(\Omega )}^2 ds\\\le & {} 3C_2 \cdot \delta _2^2 \int _{t-1}^{t+1} \Vert \nabla z(\cdot ,s)\Vert _{L^\frac{3p}{3-p}(\Omega )}^2 ds \\\le & {} 3C_2\delta _2^2 \cdot C_1^2 \int _{t-1}^{t+1} \Vert z(\cdot ,s)\Vert _{W^{2,p}(\Omega )}^2 ds \\\le & {} \frac{1}{2} \int _{t-1}^{t+1} \Vert z(\cdot ,s)\Vert _{W^{2,p}(\Omega )}^2 ds. \end{aligned}$$

In conjunction with (4.18), (4.19) and (4.20), this shows that

$$\begin{aligned}&\frac{1}{2C_1^2} \int _{t-1}^{t+1} \Vert \nabla z(\cdot ,s)\Vert _{L^q(\Omega )}^2 ds \\&\quad \le \frac{1}{2} \int _{t-1}^{t+1} \Vert z(\cdot ,s)\Vert _{W^{2,p}(\Omega )}^2 ds < \frac{\delta }{4C_1^2} + \frac{\delta }{4C_1^2}=\frac{\delta }{2C_1^2} \qquad \text{ for } \text{ all } t\ge t_0 \end{aligned}$$

and thereby proves the lemma, because \(z\equiv c_\varepsilon \) in \(\Omega \times (t,t+1)\). \(\quad \square \)

Smallness of \(\nabla c_\varepsilon \) in \(L^{2p}(\Omega )\) for Some \(p>\frac{3}{2}\)

In this section of key importance we shall next aim at deriving appropriate temporally uniform eventual smallness properties of \(n_\varepsilon \), and especially of \(\nabla c_\varepsilon \), with respect to norms which can be viewed supercritical in the sense that their control will quite directly imply \(L^\infty \) bounds for \(n_\varepsilon \). Indeed, as seen in Lemma 6.1 below, the space \(L^3(\Omega )\) will retain some threshold character with regard to taxis gradient regularity, quite elaborately analyzed in contexts of fluid-free Keller-Segel systems ([3]), at least to a certain extent also in the present setting, and accordingly the main objective of this section will be to ultimately bound \(\nabla c_\varepsilon \) with respect to the norm in \(L^{2p}(\Omega )\) for some \(p>\frac{3}{2}\). This will be accomplished on the basis of the observation that if, in dependence of the parameter \(\omega \) in (1.10) and (1.11) the number \(p>\frac{3}{2}\) is chosen suitably close to \(\frac{3}{2}\), then for some appropriately constructed function \(\psi =\psi (s)\) on \(\mathbb {R}\) vanishing at \(s=0\) and essentially growing like \(s^p\) as \(s\rightarrow +\infty \), the quantity

$$\begin{aligned} \int _\Omega \psi \Big (n_\varepsilon -\frac{\rho _+}{\mu }\Big ) + \int _\Omega |\nabla c_\varepsilon |^{2p} \end{aligned}$$
(5.1)

plays the role of a quasi-entropy functional in the sense of satisfying a superlinearly forced ODI with an eventually small source (cf. (5.52)), and hence remaining conveniently controllable beyond times at which this functional is small. This conclusion, to be drawn in Lemma 5.4, will be prepared by Lemma 5.1 and Lemma 5.3 which separately describe the time evolution of the summands appearing in (5.1).

Construction of a quasi-entropy functional coupling \(n_\varepsilon \) to \(\nabla c_\varepsilon \)

In the following, given \(\rho \in \mathbb {R}\) and \(\mu >0\) we write

$$\begin{aligned} \gamma :=\frac{\rho _+}{\mu }, \end{aligned}$$
(5.2)

and for \(p\in (1,2)\) we introduce \(\psi =\psi _{p,\gamma }\in W^{2,\infty }_{loc}((-\gamma ,\infty )) \cap C^2(\mathbb {R}\setminus \{0,\gamma \})\) by defining, in the case \(\gamma >0\),

$$\begin{aligned} \psi _{p,\gamma }(s):=\left\{ \begin{array}{ll} 0 &{} \text{ if } s\le 0, \\ \frac{p}{2} \gamma ^{p-2} s^2 \qquad &{} \text{ if } s\in (0,\gamma ), \\ s^p - \frac{2-p}{2} \gamma ^p \qquad &{} \text{ if } s\ge \gamma , \end{array} \right. \end{aligned}$$
(5.3)

and, in the case \(\gamma =0\),

$$\begin{aligned} \psi _{p,\gamma }(s):=s_+^p \qquad \text{ for } \text{ all } s\in \mathbb {R}. \end{aligned}$$
(5.4)

Moreover, for \(\varepsilon \in (0,1)\) we let

$$\begin{aligned} N_\varepsilon (x,t):=n_\varepsilon (x,t)-\gamma , \qquad x\in \bar{\Omega }, \ t\ge 0, \end{aligned}$$
(5.5)

and then obtain upon straightforward computation that in both cases \(\rho >0\) and \(\rho \le 0\) we have

$$\begin{aligned} N_{\varepsilon t}=\Delta N_\varepsilon - \chi \nabla \cdot \Big ((N_\varepsilon +\gamma )\nabla c_\varepsilon \Big ) - |\rho |N_\varepsilon - \mu N_\varepsilon ^2 - u_\varepsilon \cdot \nabla N_\varepsilon \qquad \text{ in } \Omega \times (0,\infty ).\nonumber \\ \end{aligned}$$
(5.6)

The following lemma, explicitly requiring convexity in order to avoid yet further technicalities, then relates the temporal growth of \(\int _\Omega |\nabla c_\varepsilon |^{2p}\) to quantities that contain \(N_\varepsilon \) in their essential part:

Lemma 5.1

Let \(p\in (\frac{3}{2},2)\) and \(r>3\). Then for all \(\delta >0\) there exists \(K_1=K_1(\delta ,p,r)>0\) such that for any choice of \(\rho \in \mathbb {R}, \mu >0\) and \(\varepsilon \in (0,1)\), with \(\gamma , N_\varepsilon \) and \(\psi =\psi _{p,\gamma }\) as defined by (5.2), (5.5) as well as (5.3) and (5.4), whenever \(t_0\ge 0\) and \(T\in (t_0,\infty ]\) we have

$$\begin{aligned}&\frac{d}{dt} \int _\Omega |\nabla c_\varepsilon |^{2p} + \frac{2(p-1)}{p} \int _\Omega \Big |\nabla |\nabla c_\varepsilon |^p \Big |^2\nonumber \\&\qquad + \bigg \{ p-K_1 \cdot \Big (M_2^\frac{2r}{r-3}(\varepsilon ,T,r)+M_2^2(\varepsilon ,T,r)\Big )\bigg \} \cdot \int _\Omega |\nabla c_\varepsilon |^{2p} \nonumber \\&\quad \le \delta \int _{\{N_\varepsilon >\gamma \}} N_\varepsilon ^{p-2} |\nabla N_\varepsilon |^2 \nonumber \\&\qquad + K_1 \cdot \bigg \{ \int _\Omega \psi (N_\varepsilon ) + \int _\Omega |\nabla c_\varepsilon |^{2p} \bigg \}^\frac{2p-1}{2p-3} \nonumber \\&\qquad + K_1 \cdot \bigg \{ M_1^\frac{2p(2p-1)}{4p-3} + \gamma ^\frac{2p(2p-1)}{4p-3} + \gamma ^{2p} + \gamma ^\frac{p(2p-1)}{2p-3} \bigg \} \qquad \text{ for } \text{ all } t\in (t_0,T), \end{aligned}$$
(5.7)

where

$$\begin{aligned} M_1(\varepsilon ,T):=\sup _{t\in (t_0,T)} \Vert n_\varepsilon (\cdot ,t)\Vert _{L^1(\Omega )} \end{aligned}$$
(5.8)

and

$$\begin{aligned} M_2(\varepsilon ,T,r):=\sup _{t\in (t_0,T)} \Vert u_\varepsilon (\cdot ,t)\Vert _{L^r(\Omega )}. \end{aligned}$$
(5.9)

Proof

By direct computation using the identity \(\nabla c_\varepsilon \cdot \nabla \Delta c_\varepsilon =\frac{1}{2}\Delta |\nabla c_\varepsilon |^2 - |D^2 c_\varepsilon |^2\), from the second Eq. in (1.7) we obtain that for all \(t>0\) and \(\varepsilon \in (0,1)\),

$$\begin{aligned} \frac{1}{2p} \frac{d}{dt} \int _\Omega |\nabla c_\varepsilon |^{2p}= & {} \int _\Omega |\nabla c_\varepsilon |^{2p-2} \nabla c_\varepsilon \cdot \nabla \Big \{ \Delta c_\varepsilon - c_\varepsilon + n_\varepsilon - u_\varepsilon \cdot \nabla c_\varepsilon \Big \} \nonumber \\= & {} \frac{1}{2} \int _\Omega |\nabla c_\varepsilon |^{2p-2} \Delta |\nabla c_\varepsilon |^2 - \int _\Omega |\nabla c_\varepsilon |^{2p-2} |D^2 c_\varepsilon |^2 - \int _\Omega |\nabla c_\varepsilon |^{2p} \nonumber \\&+ \int _\Omega |\nabla c_\varepsilon |^{2p-2} \nabla c_\varepsilon \cdot \nabla n_\varepsilon - \int _\Omega |\nabla c_\varepsilon |^{2p-2} \nabla c_\varepsilon \cdot \nabla (u_\varepsilon \cdot \nabla c_\varepsilon ),\nonumber \\ \end{aligned}$$
(5.10)

where integrating by parts yields

$$\begin{aligned}&\frac{1}{2} \int _\Omega |\nabla c_\varepsilon |^{2p-2} \Delta |\nabla c_\varepsilon |^2 \nonumber \\&\quad = - \frac{p-1}{2} \int _\Omega |\nabla c_\varepsilon |^{2p-4} \Big |\nabla |\nabla c_\varepsilon |^2 \Big |^2 + \frac{1}{2} \int _{\partial \Omega } |\nabla c_\varepsilon |^{2p-2} \frac{\partial |\nabla c_\varepsilon |^2}{\partial \nu } \nonumber \\&\quad \le - \frac{p-1}{2} \int _\Omega |\nabla c_\varepsilon |^{2p-4} \Big |\nabla |\nabla c_\varepsilon |^2 \Big |^2 \nonumber \\&\quad = - \frac{2(p-1)}{p^2} \int _\Omega \Big |\nabla |\nabla c_\varepsilon |^p \Big |^2 \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1), \end{aligned}$$
(5.11)

because for any such \(\varepsilon \), \(\frac{\partial |\nabla c_\varepsilon |^2}{\partial \nu }\le 0\) on \(\partial \Omega \) due to the convexity of \(\Omega \) and the fact that \(\frac{\partial c_\varepsilon }{\partial \nu }=0\) on \(\partial \Omega \) ([35]).

In the rightmost integral in (5.10), we also integrate by parts and then apply the pointwise inequality \(|\Delta c_\varepsilon | \le \sqrt{3} |D^2c_\varepsilon |\) as well as Young’s inequality to estimate

$$\begin{aligned}&- \int _\Omega |\nabla c_\varepsilon |^{2p-2} \nabla c_\varepsilon \cdot \nabla (u_\varepsilon \cdot \nabla c_\varepsilon ) \nonumber \\&\quad = \int _\Omega |\nabla c_\varepsilon |^{2p-2} \Delta c_\varepsilon (u_\varepsilon \cdot \nabla c_\varepsilon ) \nonumber \\&\qquad + (p-1) \int _\Omega |\nabla c_\varepsilon |^{2p-4} (\nabla |\nabla c_\varepsilon |^2 \cdot \nabla c_\varepsilon ) (u_\varepsilon \cdot \nabla c_\varepsilon ) \nonumber \\&\quad \le \sqrt{3} \int _\Omega |u_\varepsilon | \cdot |\nabla c_\varepsilon |^{2p-1} |D^2c_\varepsilon | \nonumber \\&\qquad + (p-1) \int _\Omega |u_\varepsilon | \cdot |\nabla c_\varepsilon |^{2p-2} \Big |\nabla |\nabla c_\varepsilon |^2 \Big | \nonumber \\&\quad \le \frac{1}{2} \int _\Omega |\nabla c_\varepsilon |^{2p-2} |D^2c_\varepsilon |^2 + \frac{3}{2} \int _\Omega |u_\varepsilon |^2 |\nabla c_\varepsilon |^{2p} \nonumber \\&+ \frac{p-1}{16} \int _\Omega |\nabla c_\varepsilon |^{2p-4} \Big |\nabla |\nabla c_\varepsilon |^2 \Big |^2 + 4(p-1) \int _\Omega |u_\varepsilon |^2 |\nabla c_\varepsilon |^{2p} \nonumber \\&\quad = \frac{1}{2} \int _\Omega |\nabla c_\varepsilon |^{2p-2} |D^2c_\varepsilon |^2 + \frac{p-1}{4p^2} \int _\Omega \Big |\nabla |\nabla c_\varepsilon |^p \Big |^2 \nonumber \\&\qquad + \Big (4p-\frac{5}{2}\Big ) \int _\Omega |u_\varepsilon |^2 |\nabla c_\varepsilon |^{2p} \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1). \end{aligned}$$
(5.12)

Here we use the Hölder inequality and (5.9) to see that whenever \(t_0\ge 0\) and \(T\in (t_0,\infty ]\),

$$\begin{aligned}&\Big (4p-\frac{5}{2}\Big ) \int _\Omega |u_\varepsilon |^2 |\nabla c_\varepsilon |^{2p} \le \Big (4p-\frac{5}{2}\Big ) \bigg \{ \int _\Omega |u_\varepsilon |^r \bigg \}^\frac{2}{r} \bigg \{ \int _\Omega |\nabla c_\varepsilon |^\frac{2pr}{r-2} \bigg \}^\frac{r-2}{r} \\&\quad \le \Big (4p-\frac{5}{2}\Big ) M_2^2(\varepsilon ,T,r) \Big \Vert |\nabla c_\varepsilon |^p \Big \Vert _{L^\frac{2r}{r-2}(\Omega )}^2 \qquad \text{ for } \text{ all } t\in (t_0,T) \text{ and } \varepsilon \in (0,1), \end{aligned}$$

and since \(r>3\) implies that \(\frac{2r}{r-2}<6\), we may invoke the Gagliardo-Nirenberg inequality and Young’s inequality to find \(C_1=C_1(p,r)>0\) and \(C_2=C_2(p,r)>0\) satisfying

$$\begin{aligned}&\Big (4p-\frac{5}{2}\Big ) M_2^2(\varepsilon ,T,r) \Big \Vert |\nabla c_\varepsilon |^p \Big \Vert _{L^\frac{2r}{r-2}(\Omega )}^2 \\&\quad \le C_1 M_2^2(\varepsilon ,T,r) \Big \Vert \nabla |\nabla c_\varepsilon |^p \Big \Vert _{L^2(\Omega )}^\frac{6}{r} \Big \Vert |\nabla c_\varepsilon |^p \Big \Vert _{L^2(\Omega )}^\frac{2(r-3)}{r} + C_1 M_2^2(\varepsilon ,T,r) \Big \Vert |\nabla c_\varepsilon |^p \Big \Vert _{L^2(\Omega )}^2 \\&\quad \le \frac{p-1}{4p^2} \Big \Vert \nabla |\nabla c_\varepsilon |^p \Big \Vert _{L^2(\Omega )}^2 + C_2 M_2^\frac{2r}{r-3}(\varepsilon ,T,r) \Big \Vert |\nabla c_\varepsilon |^p \Big \Vert _{L^2(\Omega )}^2 \\&\qquad + C_1 M_2^2(\varepsilon ,T,r) \Big \Vert |\nabla c_\varepsilon |^p \Big \Vert _{L^2(\Omega )}^2, \qquad \text{ for } \text{ all } t\in (t_0,T) \text{ and } \varepsilon \in (0,1), \end{aligned}$$

so that (5.12) all in all shows that

$$\begin{aligned}&- \int _\Omega |\nabla c_\varepsilon |^{2p-2} \nabla c_\varepsilon \cdot \nabla (u_\varepsilon \cdot \nabla c_\varepsilon ) \nonumber \\&\quad \le \frac{1}{2} \int _\Omega |\nabla c_\varepsilon |^{2p-2} |D^2c_\varepsilon |^2 + \frac{p-1}{2p^2} \int _\Omega \Big | \nabla |\nabla c_\varepsilon |^p \Big |^2 \nonumber \\&\qquad + \Big \{ C_2 M_2^\frac{2r}{r-3}(\varepsilon ,T,r) + C_1 M_2^2(\varepsilon ,T,r) \Big \} \nonumber \\&\qquad \cdot \int _\Omega |\nabla c_\varepsilon |^{2p} \quad \text{ for } \text{ all } t\in (t_0,T) \text{ and } \varepsilon \in (0,1). \end{aligned}$$
(5.13)

We next address the second last integral in (5.10), in which we first partially proceed as above in integrating by parts and applying Young’s inequality to see that

$$\begin{aligned}&\int _\Omega |\nabla c_\varepsilon |^{2p-2} \nabla c_\varepsilon \cdot \nabla n_\varepsilon \nonumber \\&\quad = \int _\Omega |\nabla c_\varepsilon |^{2p-2} \nabla c_\varepsilon \cdot \nabla N_\varepsilon \nonumber \\&\quad = - \int _\Omega N_\varepsilon |\nabla c_\varepsilon |^{2p-2} \Delta c_\varepsilon - (p-1) \int _\Omega N_\varepsilon |\nabla c_\varepsilon |^{2p-4} \nabla c_\varepsilon \cdot \nabla |\nabla c_\varepsilon |^2 \nonumber \\&\quad \le \sqrt{3} \int _\Omega |N_\varepsilon | \cdot |\nabla c_\varepsilon |^{2p-2} |D^2c_\varepsilon | + (p-1) \int _\Omega |N_\varepsilon | \cdot |\nabla c_\varepsilon |^{2p-3} \Big |\nabla |\nabla c_\varepsilon |^2 \Big | \nonumber \\&\quad \le \frac{1}{2} \int _\Omega |\nabla c_\varepsilon |^{2p-2} |D^2c_\varepsilon |^2 + \frac{3}{2} \int _\Omega N_\varepsilon ^2 |\nabla c_\varepsilon |^{2p-2} \nonumber \\&\qquad + \frac{p-1}{8} \int _\Omega |\nabla c_\varepsilon |^{2p-4} \Big |\nabla |\nabla c_\varepsilon |^2 \Big |^2 + 2(p-1) \int _\Omega N_\varepsilon ^2 |\nabla c_\varepsilon |^{2p-2} \nonumber \\&\quad = \frac{1}{4} \int _\Omega |\nabla c_\varepsilon |^{2p-2} |D^2c_\varepsilon |^2 + \frac{p-1}{2p^2} \int _\Omega \Big |\nabla |\nabla c_\varepsilon |^p \Big |^2 \nonumber \\&\qquad + \Big (2p-\frac{1}{2}\Big ) \int _\Omega N_\varepsilon ^2 |\nabla c_\varepsilon |^{2p-2} \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1). \end{aligned}$$
(5.14)

Here in view of the definitions (5.3) and (5.4) of \(\psi \), for \(t>0\) and \(\varepsilon \in (0,1)\) we split the last integral according to

$$\begin{aligned} \Big (2p-\frac{1}{2}\Big )\int _\Omega N_\varepsilon ^2 |\nabla c_\varepsilon |^{2p-2}= & {} \Big (2p-\frac{1}{2}\Big ) \int _{\{N_\varepsilon \le \gamma \}} N_\varepsilon ^2 |\nabla c_\varepsilon |^{2p-2} \nonumber \\&+ \Big (2p-\frac{1}{2}\Big ) \int _{\{N_\varepsilon >\gamma \}} N_\varepsilon ^2 |\nabla c_\varepsilon |^{2p-2}, \end{aligned}$$
(5.15)

where by Young’s inequality

$$\begin{aligned}&\Big (2p-\frac{1}{2}\Big ) \int _{\{N_\varepsilon \le \gamma \}} N_\varepsilon ^2 |\nabla c_\varepsilon |^{2p-2}\nonumber \\&\quad \le \Big (2p-\frac{1}{2}\Big ) \gamma ^2 \int _\Omega |\nabla c_\varepsilon |^{2p-2} \nonumber \\&\quad \le \frac{1}{2} \int _\Omega |\nabla c_\varepsilon |^{2p} + C_4 \gamma ^{2p} \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1)\nonumber \\ \end{aligned}$$
(5.16)

with some \(C_4=C_4(p)>0\), because \(N_\varepsilon \ge -\gamma \) by nonnegativity of \(n_\varepsilon \). Moreover, employing the Hölder, Young and Gagliardo-Nirenberg inequalities we obtain \(C_5=C_5(p)>0\) such that writing

$$\begin{aligned} y_\varepsilon (t):=\int _\Omega \psi (N_\varepsilon (x,t))dx + \int _\Omega |\nabla c_\varepsilon (x,t)|^{2p} dx, \qquad t\ge 0, \ \varepsilon \in (0,1), \end{aligned}$$
(5.17)

we have

$$\begin{aligned}&\Big (2p+\frac{1}{2}\Big ) \int _{\{N_\varepsilon> \gamma \}} N_\varepsilon ^2 |\nabla c_\varepsilon |^{2p-2} \\&\quad \le \Big (2p+\frac{1}{2}\Big ) \bigg \{ \int _{\{N_\varepsilon>\gamma \}} N_\varepsilon ^{2p}\bigg \}^\frac{1}{p} \bigg \{ \int _\Omega |\nabla c_\varepsilon |^{2p} \bigg \}^\frac{p-1}{p} \nonumber \\&\quad \le \Big (2p+\frac{1}{2}\Big ) \bigg \{ \int _{\{N_\varepsilon>\gamma \}} [(N_\varepsilon ^\frac{p}{2}-\gamma ^\frac{p}{2}) + \gamma ^\frac{p}{2}]^4 \bigg \}^\frac{1}{p} \cdot y_\varepsilon ^\frac{p-1}{p}(t) \\&\quad \le \Big (2p+\frac{1}{2}\Big ) \bigg \{ 8\int _{\{N_\varepsilon >\gamma \}} (N_\varepsilon ^\frac{p}{2}-\gamma ^\frac{p}{2})_+^4 + 8|\Omega | \gamma ^{2p} \bigg \}^\frac{1}{p} \cdot y_\varepsilon ^\frac{p-1}{p}(t) \\&\quad = \Big (2p+\frac{1}{2}\Big ) \bigg \{ 8\Big \Vert (N_\varepsilon ^\frac{p}{2}-\gamma ^\frac{p}{2})_+\Big \Vert _{L^4(\Omega )}^4 + 8|\Omega | \gamma ^{2p} \bigg \}^\frac{1}{p} \cdot y_\varepsilon ^\frac{p-1}{p}(t) \\&\quad \le C_5 \bigg \{ \Big \Vert \nabla (N_\varepsilon ^\frac{p}{2}-\gamma ^\frac{p}{2})_+\Big \Vert _{L^2(\Omega )}^\frac{3}{p} \Big \Vert (N_\varepsilon ^\frac{p}{2}-\gamma ^\frac{p}{2})_+\Big \Vert _{L^2(\Omega )}^\frac{1}{p}\\&\qquad + \Big \Vert (N_\varepsilon ^\frac{p}{2}-\gamma ^\frac{p}{2})_+\Big \Vert _{L^\frac{2}{p}(\Omega )}^\frac{4}{p} +\gamma ^2 \bigg \} \cdot y_\varepsilon ^\frac{p-1}{p}(t) \end{aligned}$$

for all \(t>0\) and \(\varepsilon \in (0,1)\). Observing that herein we have

$$\begin{aligned} \Big \Vert \nabla (N_\varepsilon ^\frac{p}{2}-\gamma ^\frac{p}{2})_+\Big \Vert _{L^2(\Omega )}^2 = \frac{p^2}{4} \int _{\{N_\varepsilon>\gamma \}} N_\varepsilon ^{p-2} |\nabla N_\varepsilon |^2 \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1) \end{aligned}$$

as well as

$$\begin{aligned} \Big \Vert (N_\varepsilon ^\frac{p}{2}-\gamma ^\frac{p}{2})_+\Big \Vert _{L^2(\Omega )}^2= & {} \int _{\{N_\varepsilon>\gamma \}} (N_\varepsilon ^\frac{p}{2}-\gamma ^\frac{p}{2})^2 \\\le & {} \int _{\{N_\varepsilon>\gamma \}} N_\varepsilon ^p \\= & {} \int _{\{N_\varepsilon>\gamma \}} \Big \{\psi (N_\varepsilon ) + \frac{2-p}{2} \gamma ^p \Big \} \\\le & {} \int _\Omega \psi (N_\varepsilon ) + C_6 \gamma ^p \\\le & {} y_\varepsilon (t) + C_6 \gamma ^p \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1) \end{aligned}$$

with \(C_6=C_6(p):=\frac{2-p}{2}|\Omega |\), and that given \(t_0\ge 0\) and \(T\in (t_0,\infty ]\), by (5.5) and (5.8) we have

$$\begin{aligned} \Big \Vert (N_\varepsilon ^\frac{p}{2}-\gamma ^\frac{p}{2})_+\Big \Vert _{L^\frac{2}{p}(\Omega )}^\frac{2}{p} = \int _{\{N_\varepsilon>\gamma \}} (N_\varepsilon ^\frac{p}{2}-\gamma ^\frac{p}{2})^\frac{2}{p} \le \int _{\{N_\varepsilon >\gamma \}} N_\varepsilon \le \int _\Omega n_\varepsilon \le M_1(\varepsilon ,T) \end{aligned}$$

for all \(t\in (t_0,T)\) and \(\varepsilon \in (0,1)\), on using Young’s inequality we see that there exists \(C_7=C_7(\delta ,p)>0\) such that for all \(t\in (t_0,T)\) and \(\varepsilon \in (0,1)\),

$$\begin{aligned} \Big (2p+\frac{1}{2}\Big ) \int _{\{N_\varepsilon>\gamma \}} N_\varepsilon ^2 |\nabla c_\varepsilon |^{2p-2}\le & {} \frac{\delta }{2p} \int _{\{N_\varepsilon>\gamma \}} N_\varepsilon ^{p-2} |\nabla N_\varepsilon |^2 \\&+ C_7 \Big \Vert (N_\varepsilon ^\frac{p}{2}-\gamma ^\frac{p}{2})_+\Big \Vert _{L^2(\Omega )}^\frac{2}{2p-3} y^\frac{2p-2}{2p-3}(t) \\&+ C_5 \Big \Vert (N_\varepsilon ^\frac{p}{2}-\gamma ^\frac{p}{2})_+\Big \Vert _{L^\frac{2}{p}(\Omega )}^\frac{4}{p} y_\varepsilon ^\frac{p-1}{p}(t) + C_5 \gamma ^2 y_\varepsilon ^\frac{p-1}{p}(t) \\\le & {} \frac{\delta }{2p} \int _{\{N_\varepsilon >\gamma \}} N_\varepsilon ^{p-2} |\nabla N_\varepsilon |^2 \\&+ C_7 \cdot \Big \{ y_\varepsilon ^{2p-1}(t) + C_6 \gamma ^p y_\varepsilon ^{2p-2}(t) \Big \}^\frac{1}{2p-3} \\&+ C_5 M_1^2(\varepsilon ,T) y_\varepsilon ^\frac{p-1}{p}(t) + C_5 \gamma ^2 y_\varepsilon ^\frac{p-1}{p}(t). \end{aligned}$$

Three more applications of Young’s inequality thereupon provide \(C_8=C_8(p)>0\) and \(C_9=C_9(\delta ,p)>0\) fulfilling

$$\begin{aligned} \Big (2p+\frac{1}{2}\Big ) \int _{\{N_\varepsilon>\gamma \}} N_\varepsilon ^2 |\nabla c_\varepsilon |^{2p-2}\le & {} \frac{\delta }{2p} \int _{\{N_\varepsilon>\gamma \}} N_\varepsilon ^{p-2} |\nabla N_\varepsilon |^2 \\&+ C_7 \cdot \Big \{ 2y_\varepsilon ^{2p-1}(t) + C_8\gamma ^{p(2p-1)} \Big \}^\frac{1}{2p-3} \\&+ C_5 \Big \{ M_1^\frac{2p(2p-1)}{4p-3}(\varepsilon ,T) + y_\varepsilon ^\frac{2p-1}{2p-3}(t) \Big \} \\&+ C_5 \Big \{ \gamma ^\frac{2p(2p-1)}{4p-3} + y_\varepsilon ^\frac{2p-1}{2p-3}(t) \Big \}\\\le & {} \frac{\delta }{2p} \int _{\{N_\varepsilon >\gamma \}} N_\varepsilon ^{p-2} |\nabla N_\varepsilon |^2 \\&+ C_9 y_\varepsilon ^\frac{2p-1}{2p-3}(t) + C_9 \gamma ^\frac{p(2p-1)}{2p-3} \nonumber \\&+ C_5 \gamma ^\frac{2p(2p-1)}{4p-3} + C_5 M_1^\frac{2p(2p-1)}{4p-3}(\varepsilon ,T) \end{aligned}$$

for all \(t\in (t_0,T)\) and \(\varepsilon \in (0,1)\). Combined with (5.15), (5.16) and (5.14), this shows that

$$\begin{aligned}&\int _\Omega |\nabla c_\varepsilon |^{2p-2}\nabla c_\varepsilon \cdot \nabla n_\varepsilon \le \frac{\delta }{2p} \int _{\{N_\varepsilon >\gamma \}} N_\varepsilon ^{p-2} |\nabla N_\varepsilon |^2 \\&\qquad + \frac{1}{2} \int _\Omega |\nabla c_\varepsilon |^{2p-2} |D^2c_\varepsilon |^2 + \frac{p-1}{2p^2} \int _\Omega \Big |\nabla |\nabla c_\varepsilon |^p \Big |^2 + \frac{1}{2} \int _\Omega |\nabla c_\varepsilon |^{2p} \\&\qquad + C_9 y_\varepsilon ^\frac{2p-1}{2p-3}(t) + C_5 M_1^\frac{2p(2p-1)}{4p-3}(\varepsilon ,T) + C_5 \gamma ^\frac{2p(2p-1)}{4p-3}\\&\qquad + C_4 \gamma ^{2p} + C_9 \gamma ^\frac{p(2p-1)}{2p-3} \end{aligned}$$

for all \(t\in (t_0,T)\) and \(\varepsilon \in (0,1)\), which in conjunction with (5.10), (5.11) and (5.13) readily yields (5.7). \(\quad \square \)

In order to prepare an appropriate control of the first summand on the right-hand side of (5.7) in the course of a testing procedure associated with the identity (5.5), let us note the following immediate consequence of the definitions in (5.3) and (5.4).

Lemma 5.2

Let \(p\in (1,2)\), \(\gamma \ge 0\) and \(\psi =\psi _{p,\gamma }\) be defined by (5.3) and (5.4), respectively. Then

$$\begin{aligned} (\psi '(s))^\frac{2-p}{p-1} \psi ''(s) \le p^\frac{1}{p-1} \qquad \text{ for } \text{ all } s\in \mathbb {R}\setminus \{0,\gamma \}. \end{aligned}$$
(5.18)

Proof

If \(s<0\), (5.18) is trivial. If \(\gamma >0\) and \(s\in (0,\gamma )\), then by (5.3) we have \(\psi '(s)=p\gamma ^{p-2} s < p\gamma ^{p-1}\) and \(\psi ''(s)=p\gamma ^{p-2}\), so that since \(p\le 2\) we can estimate

$$\begin{aligned} (\psi '(s))^\frac{2-p}{p-1} \psi ''(s) < (p\gamma ^{p-1})^\frac{2-p}{p-1} \cdot p\gamma ^{p-2} = p^\frac{1}{p-1}. \end{aligned}$$

Finally, in the case \(s>\gamma \) we see from (5.3) and (5.4) that \(\psi '(s)=ps^{p-1}\) and \(\psi ''(s)=p(p-1)s^{p-2}\) and hence

$$\begin{aligned} (\psi '(s))^\frac{2-p}{p-1} \psi ''(s) = (ps^{p-1})^\frac{2-p}{p-1} \cdot p(p-1) s^{p-2} = p^\frac{1}{p-1} (p-1) \le p^\frac{1}{p-1}, \end{aligned}$$

again because \(p\le 2\). \(\quad \square \)

We can thereby establish an ODI that limits the growth of \(\int _\Omega \psi (N_\varepsilon )\) and, yet more importantly, contains the first summand from the right of (5.7) in its dissipative part:

Lemma 5.3

Let \(p\in (\frac{3}{2},2]\). Then for all \(\delta >0\) there exists \(K_2=K_2(\delta ,p)>0\) such that for each \(\rho \in \mathbb {R}, \mu >0\) and \(\varepsilon \in (0,1)\), the quantities \(\gamma ,N_\varepsilon \) and \(\psi =\psi _{p,\gamma }\) introduced in (5.2), (5.5), (5.3) and (5.4) have the property that for any \(t_0\ge 0\) and \(T\in (t_0,\infty ]\),

$$\begin{aligned}&\frac{d}{dt} \int _\Omega \psi (N_\varepsilon ) + \frac{p(p-1)}{2} \int _{\{N_\varepsilon >\gamma \}} N_\varepsilon ^{p-2} |\nabla N_\varepsilon |^2 \nonumber \\&\quad \le \delta \int _\Omega \Big |\nabla |\nabla c_\varepsilon |^p \Big |^2 + \delta \int _\Omega |\nabla c_\varepsilon |^{2p} \nonumber \\&\qquad + K_2 \cdot \bigg \{ \int _\Omega \psi (N_\varepsilon )\bigg \}^\frac{2p-1}{2p-3} \nonumber \\&\qquad + K_2 \cdot \bigg \{ M_1^\frac{p^2}{p-1}(\varepsilon ,T) + \gamma ^\frac{p(2p-1)}{2p-3} + \gamma ^\frac{p^2}{p-1} + \gamma ^\frac{1}{p-1} \bigg \} \end{aligned}$$
(5.19)

for all \(t\in (t_0,T)\) and \(\varepsilon \in (0,1)\), where \(M_1(\cdot ,\cdot )\) is as in (5.8).

Proof

On the basis of (5.6), using that \(\psi \in W^{2,\infty }_{loc}((-\gamma ,\infty ))\), and that \(N_\varepsilon >-\gamma \) in \(\overline{\Omega }\times (0,\infty )\) by positivity of \(n_\varepsilon \), we see that \(0\le t\mapsto \int _\Omega \psi (N_\varepsilon )\) belongs to \(C^0([0,\infty )) \cap C^1((0,\infty ))\) for each \(\varepsilon \in (0,1)\), and that we may integrate by parts in computing

$$\begin{aligned}&\frac{d}{dt} \int _\Omega \psi (N_\varepsilon )\nonumber \\&\quad = \int _\Omega \psi '(N_\varepsilon ) \cdot \bigg \{ \Delta N_\varepsilon - \chi \nabla \cdot \Big ( (N_\varepsilon +\gamma ) \nabla c_\varepsilon \Big ) - |\rho | N_\varepsilon - \mu N_\varepsilon ^2 - u_\varepsilon \cdot \nabla N_\varepsilon \bigg \} \nonumber \\&\quad = - \int _\Omega \psi ''(N_\varepsilon ) |\nabla N_\varepsilon |^2 + \chi \int _\Omega (N_\varepsilon +\gamma ) \psi ''(N_\varepsilon ) \nabla N_\varepsilon \cdot \nabla c_\varepsilon \nonumber \\&\qquad - |\rho | \int _\Omega N_\varepsilon \psi '(N_\varepsilon ) - \mu \int _\Omega N_\varepsilon ^2 \psi '(N_\varepsilon ) \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1), \end{aligned}$$
(5.20)

because \(\nabla \cdot u_\varepsilon \equiv 0\). Since \(\psi '\ge 0\) on \((0,\infty )\) and \(\psi '\equiv 0\) on \((-\infty ,0)\), herein we can estimate

$$\begin{aligned}&-|\rho |\int _\Omega N_\varepsilon \psi '(N_\varepsilon ) \le 0 \quad \text{ and } \quad \nonumber \\&\quad -\mu \int _\Omega N_\varepsilon ^2 \psi '(N_\varepsilon ) \le 0 \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1), \end{aligned}$$
(5.21)

and since \(\psi ''\) is nonnegative, we may invoke Young’s inequality to find that

$$\begin{aligned}&\chi \int _\Omega (N_\varepsilon +\gamma ) \psi ''(N_\varepsilon ) \nabla N_\varepsilon \cdot \nabla c_\varepsilon \nonumber \\&\quad \le \frac{1}{4} \int _\Omega \psi ''(N_\varepsilon ) |\nabla N_\varepsilon |^2 + \chi ^2 \int _\Omega (N_\varepsilon +\gamma )^2 \psi ''(N_\varepsilon ) |\nabla c_\varepsilon |^2 \qquad \nonumber \\&\qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1).\nonumber \\ \end{aligned}$$
(5.22)

On splitting the latter integral and recalling the definition of \(\psi \), we see that in the case \(\gamma >0\),

$$\begin{aligned} \chi ^2 \int _\Omega (N_\varepsilon +\gamma )^2 \psi ''(N_\varepsilon ) |\nabla c_\varepsilon |^2= & {} \chi ^2 \int _{\{0\le N_\varepsilon \le \gamma \}} (N_\varepsilon +\gamma )^2 \psi ''(N_\varepsilon ) |\nabla c_\varepsilon |^2 \\&+ \chi ^2 \int _{\{N_\varepsilon>\gamma \}} (N_\varepsilon +\gamma )^2 \psi ''(N_\varepsilon ) |\nabla c_\varepsilon |^2 \\\le & {} \chi ^2 \int _{\{0\le N_\varepsilon \le \gamma \}} (2\gamma )^2 \cdot p\gamma ^{p-2} |\nabla c_\varepsilon |^2 \\&+ \chi ^2 \int _{\{N_\varepsilon >\gamma \}} (2N_\varepsilon )^2 \cdot p(p-1)N_\varepsilon ^{p-2} |\nabla c_\varepsilon |^2 \end{aligned}$$

for all \(t>0\) and \(\varepsilon \in (0,1)\), which shows that regardless of the sign of \(\rho \) we have

$$\begin{aligned}&\chi ^2 \int _\Omega (N_\varepsilon +\gamma )^2 \psi ''(N_\varepsilon ) |\nabla c_\varepsilon |^2 \nonumber \\&\quad \le 4p\gamma ^p \chi ^2 \int _\Omega |\nabla c_\varepsilon |^2 + 4p(p-1)\chi ^2 \int _{\{N_\varepsilon>\gamma \}} N_\varepsilon ^2 |\nabla c_\varepsilon |^2 \qquad \nonumber \\&\qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1),\nonumber \\ \end{aligned}$$
(5.23)

where by Young’s inequality we obtain \(C_1=C_1(\delta ,p)>0\) such that

$$\begin{aligned} 4p\gamma ^p \chi ^2 \int _\Omega |\nabla c_\varepsilon |^2 \le \frac{\delta }{2} \int _\Omega |\nabla c_\varepsilon |^{2p} + C_1 \gamma ^\frac{p^2}{p-1} \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1).\nonumber \\ \end{aligned}$$
(5.24)

To estimate the second term on the right of (5.23), we first apply the Hölder inequality to obtain

$$\begin{aligned}&\int _{\{N_\varepsilon>\gamma \}} N_\varepsilon ^p |\nabla c_\varepsilon |^2 \le \bigg \{ \int _{\{N_\varepsilon>\gamma \}} N_\varepsilon ^\frac{3p^2}{3p-1}\bigg \}^\frac{3p-1}{3p} \cdot \bigg \{ \int _\Omega |\nabla c_\varepsilon |^{6p} \bigg \}^\frac{1}{3p} \nonumber \\&\quad = \Vert N_\varepsilon ^\frac{p}{2}\Vert _{L^\frac{6p}{3p-1}(\{N_\varepsilon>\gamma \})}^2 \Big \Vert |\nabla c_\varepsilon |^p \Big \Vert _{L^6(\Omega )}^\frac{2}{p} \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1),\nonumber \\ \end{aligned}$$
(5.25)

where

$$\begin{aligned} \Vert N_\varepsilon ^\frac{p}{2}\Vert _{L^\frac{6p}{3p-1}(\{N_\varepsilon>\gamma \})}^2 \le p^{-\frac{p}{p-1}} \Big \Vert \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2} \Big \Vert _{L^\frac{6p}{3p-1}(\Omega )}^2 \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1), \end{aligned}$$

because according to (5.3) and (5.4) we have \(\psi '\ge 0\) on \(\mathbb {R}\) and

$$\begin{aligned} \Big (\psi '(s)\Big )^\frac{p}{2p-2} = (ps^{p-1})^\frac{p}{2p-2} = p^\frac{p}{2p-2} s^\frac{p}{2} \qquad \text{ for } \text{ all } s>\gamma . \end{aligned}$$
(5.26)

Noting that \(\frac{6p}{3p-1}<6\) due to the fact that \(p>\frac{1}{2}\), in (5.25) we may thus employ the Gagliardo-Nirenberg inequality to find \(C_2=C_2(p)>0\) such that

$$\begin{aligned} \Vert N_\varepsilon ^\frac{p}{2}\Vert _{L^\frac{6p}{3p-1}(\{N_\varepsilon >\gamma \})}^2\le & {} C_2 \Big \Vert \nabla \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2} \Big \Vert _{L^2(\Omega )}^\frac{1}{p} \Big \Vert \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2}\Big \Vert _{L^2(\Omega )}^\frac{2p-1}{p}\\&+ C_2 \Big \Vert \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2}\Big \Vert _{L^\frac{2}{p}(\Omega )}^2 \end{aligned}$$

for all \(t>0\) and \(\varepsilon \in (0,1)\). Moreover, thanks to the Sobolev inequality associated with the embedding \(W^{1,2}(\Omega )\hookrightarrow L^6(\Omega )\) we can pick \(C_3>0\) fulfilling

$$\begin{aligned}&\Big \Vert |\nabla c_\varepsilon |^p \Big \Vert _{L^6(\Omega )}^2 \le C_3 \cdot \bigg \{ \Big \Vert \nabla |\nabla c_\varepsilon |^p \Big \Vert _{L^2(\Omega )}^2 +\Big \Vert |\nabla c_\varepsilon |^p \Big \Vert _{L^2(\Omega )}^2 \bigg \} \qquad \nonumber \\&\qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1), \end{aligned}$$

so that by (5.25), using Young’s inequality we can find \(C_4=C_4(\delta ,p)>0\) such that the last summand in (5.23) can be estimated according to

$$\begin{aligned}&4p(p-1)\chi ^2 \int _{\{N_\varepsilon >\gamma \}} N_\varepsilon ^p |\nabla c_\varepsilon |^2 \nonumber \\&\quad \le 4p(p-1)\chi ^2 C_2 C_3^\frac{1}{p} \cdot \bigg \{ \Big \Vert \nabla \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2}\Big \Vert _{L^2(\Omega )}^\frac{1}{p} \Big \Vert \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2}\Big \Vert _{L^2(\Omega )}^\frac{2p-1}{p}\nonumber \\&\qquad + \Big \Vert \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2}\Big \Vert _{L^\frac{2}{p}(\Omega )}^2 \bigg \} \times \nonumber \\&\qquad \times \bigg \{ \Big \Vert \nabla |\nabla c_\varepsilon |^p \Big \Vert _{L^2(\Omega )}^2 + \frac{1}{2} \Big \Vert |\nabla c_\varepsilon |^p \Big \Vert _{L^2(\Omega )}^2 \bigg \}^\frac{1}{p} \nonumber \\&\quad \le \delta \cdot \bigg \{ \Big \Vert \nabla |\nabla c_\varepsilon |^p \Big \Vert _{L^2(\Omega )}^2 + \frac{1}{2} \Big \Vert |\nabla c_\varepsilon |^p \Big \Vert _{L^2(\Omega )}^2 \bigg \} \nonumber \\&\qquad + C_4 \cdot \bigg \{ \Big \Vert \nabla \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2}\Big \Vert _{L^2(\Omega )}^\frac{1}{p} \Big \Vert \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2}\Big \Vert _{L^2(\Omega )}^\frac{2p-1}{p}\nonumber \\&\qquad + \Big \Vert \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2}\Big \Vert _{L^\frac{2}{p}(\Omega )}^2 \bigg \}^\frac{p}{p-1} \nonumber \\&\quad \le \delta \cdot \bigg \{ \Big \Vert \nabla |\nabla c_\varepsilon |^p \Big \Vert _{L^2(\Omega )}^2 + \frac{1}{2} \Big \Vert |\nabla c_\varepsilon |^p \Big \Vert _{L^2(\Omega )}^2 \bigg \} \nonumber \\&\qquad + 2^\frac{1}{p-1} C_4 \Big \Vert \nabla \Big (\psi '(N_\varepsilon )\Big )^\frac{2}{2p-2}\Big \Vert _{L^2(\Omega )}^\frac{1}{p-1} \Big \Vert \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2}\Big \Vert _{L^2(\Omega )}^\frac{2p-1}{p-1}\nonumber \\&\qquad + 2^\frac{1}{p-1} C_4 \Big \Vert \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2}\Big \Vert _{L^\frac{2}{p}(\Omega )}^\frac{2p}{p-1} \end{aligned}$$
(5.27)

for all \(t>0\) and \(\varepsilon \in (0,1)\). Here in order to relate the second last summand to the integral in (5.20) stemming from diffusion, we use Lemma 5.2 to estimate

$$\begin{aligned} \Big \Vert \nabla \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2}\Big \Vert _{L^2(\Omega )}^2= & {} \int _\Omega \Big |\nabla \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2}\Big |^2 \\= & {} \frac{p^2}{4(p-1)^2} \int _\Omega \Big (\psi '(N_\varepsilon )\Big )^\frac{2-p}{p-1} \Big (\psi ''(N_\varepsilon )\Big )^2 |\nabla N_\varepsilon |^2 \\\le & {} C_5 \int _\Omega \psi ''(N_\varepsilon ) |\nabla N_\varepsilon |^2 \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1) \end{aligned}$$

with \(C_5=C_5(p):=\frac{p^2}{4(p-1)^2} \cdot p^\frac{1}{p-1}\).

Since \(\frac{1}{p-1}<2\) thanks to our assumption \(p>\frac{3}{2}\), we may thus again invoke Young’s inequality to find \(C_6=C_6(\delta ,p)>0\) satisfying

$$\begin{aligned}&2^\frac{1}{p-1} C_4 \Big \Vert \nabla \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2}\Big \Vert _{L^2(\Omega )}^\frac{1}{p-1} \Big \Vert \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2}\Big \Vert _{L^2(\Omega )}^\frac{2p-1}{p-1} \nonumber \\&\quad \le 2^\frac{1}{p-1} C_4 C_5^\frac{1}{2p-2} \bigg \{ \int _\Omega \psi ''(N_\varepsilon ) |\nabla N_\varepsilon |^2 \bigg \}^\frac{1}{2p-2} \Big \Vert \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2} \Big \Vert _{L^2(\Omega )}^\frac{2p-1}{p-1} \nonumber \\&\quad \le \frac{1}{4} \int _\Omega \psi ''(N_\varepsilon ) |\nabla N_\varepsilon |^2 + C_6 \Big \Vert \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2} \Big \Vert _{L^2(\Omega )}^\frac{2(2p-1)}{2p-3} \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1),\nonumber \\ \end{aligned}$$
(5.28)

where in the last term we recall (5.26) and (5.3) to estimate

$$\begin{aligned} \Big \Vert \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2}\Big \Vert _{L^2(\Omega )}^2= & {} \int _{\{0\le N_\varepsilon \le \gamma \}} \Big (p\gamma ^{p-2}N_\varepsilon \Big )^\frac{p}{p-1} + \int _{\{N_\varepsilon>\gamma \}} \Big (pN_\varepsilon ^{p-1}\Big )^\frac{p}{p-1} \nonumber \\\le & {} (p\gamma ^{p-1})^\frac{p}{p-1} |\Omega | + p^\frac{p}{p-1} \int _{\{N_\varepsilon>\gamma \}} N_\varepsilon ^p \nonumber \\= & {} (p\gamma ^{p-1})^\frac{p}{p-1} |\Omega | + p^\frac{p}{p-1} \int _{\{N_\varepsilon>\gamma \}} \Big (\psi (N_\varepsilon ) + \frac{2-p}{2}\gamma ^p\Big ) \nonumber \\\le & {} p^\frac{p}{p-1} \int _\Omega \psi (N_\varepsilon ) + \frac{4-p}{2} \cdot p^\frac{p}{p-1} \gamma ^p |\Omega | \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1). \end{aligned}$$

As in the last term in (5.27), given \(t_0\ge 0\) and \(T\in (t_0,\infty ]\) we similarly find that

$$\begin{aligned} \Big \Vert \Big (\psi '(N_\varepsilon )\Big )^\frac{p}{2p-2}\Big \Vert _{L^\frac{2}{p}(\Omega )}^\frac{2}{p}= & {} \int _{\{0\le N_\varepsilon \le \gamma \}} \Big (p\gamma ^{p-2}N_\varepsilon \Big )^\frac{1}{p-1} + \int _{\{N_\varepsilon>\gamma \}} \Big (pN_\varepsilon ^{p-1}\Big )^\frac{1}{p-1} \nonumber \\\le & {} (p\gamma ^{p-1})^\frac{1}{p-1}|\Omega | + p^\frac{1}{p-1} \int _{\{N_\varepsilon >\gamma \}} N_\varepsilon \\\le & {} p^\frac{1}{p-1} |\Omega | \gamma + p^\frac{1}{p-1} \int _\Omega n_\varepsilon \\\le & {} p^\frac{1}{p-1} |\Omega | \gamma + p^\frac{1}{p-1} M_1(\varepsilon ,T) \qquad \text{ for } \text{ all } t\in (t_0,T) \text{ and } \varepsilon \in (0,1) \end{aligned}$$

by definition (5.8) of \(M_1(\varepsilon ,T)\), from (5.27), (5.28) and Young’s inequality we conclude that with some \(C_7=C_7(\delta ,p)>0\) we have

$$\begin{aligned}&4p(p-1)\chi ^2 \int _{\{N_\varepsilon >\gamma \}} N_\varepsilon ^p |\nabla c_\varepsilon |^2 \nonumber \\&\quad \le \delta \cdot \bigg \{ \int _\Omega \Big |\nabla |\nabla c_\varepsilon |^p \Big |^2 + \frac{1}{2} \int _\Omega |\nabla c_\varepsilon |^{2p} \bigg \} \nonumber \\&\qquad + \frac{1}{4} \int _\Omega \psi ''(N_\varepsilon ) |\nabla N_\varepsilon |^2 \nonumber \\&\qquad + C_7 \cdot \bigg \{ \int _\Omega \psi (N_\varepsilon )\bigg \}^\frac{2p-1}{2p-3} + C_7 \gamma ^\frac{p(2p-1)}{2p-3} \nonumber \\&\qquad + C_7 \cdot \Big \{ \gamma ^\frac{p^2}{p-1} + M_1^\frac{p^2}{p-1}(\varepsilon ,T) \Big \} \qquad \text{ for } \text{ all } t\in (t_0,T) \text{ and } \varepsilon \in (0,1). \end{aligned}$$
(5.29)

Finally observing that by (5.3) and (5.4),

$$\begin{aligned} \frac{1}{2} \int _\Omega \psi ''(N_\varepsilon ) |\nabla N_\varepsilon |^2 \ge \frac{p(p-1)}{2} \int _{\{N_\varepsilon>\gamma \}} N_\varepsilon ^{p-2} |\nabla N_\varepsilon |^2 \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1), \end{aligned}$$

we only need to collect (5.21)-(5.24) and (5.29) to infer from (5.20) that indeed (5.19) holds if \(K_2=K_2(\delta ,p)>0\) is appropriately large. \(\quad \square \)

Coupling \(\int _\Omega \psi _{p,\gamma }(N_\varepsilon )\) to \(\int _\Omega |\nabla c_\varepsilon |^{2p}\)

By suitably combining Lemma 5.1 with Lemma 5.3, and by additionally referring to the preliminary bounds obtained in Lemma 2.2, Lemma 3.3 and Lemma 4.2, we can now achieve the main purpose of this section:

Lemma 5.4

Let \(\omega >0\). Then there exists \(\theta _6=\theta _6(\omega )>0\) such that for all \(\delta >0\) one can find \(\eta _6=\eta _6(\omega ,\delta )>0\) and \(\kappa _6=\kappa _6(\omega ,\delta )>0\) with the property that if \(\rho \in \mathbb {R}, \mu >0\) and \(f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)\) are such that (1.10) and (1.11) are valid with some \(\eta <\eta _6\) and \(\kappa <\kappa _6\), then there exists \(t_0=t_0(\omega ,\delta ,\eta ,\kappa ,n_0,c_0,u_0)>0\) such that for any choice of \(\varepsilon \in (0,1)\), we have

$$\begin{aligned} \int _\Omega n_\varepsilon ^p(x,t)dx + \int _\Omega |\nabla c_\varepsilon (x,t)|^{2p}dx < \delta \qquad \text{ for } \text{ all } t\ge t_0, \end{aligned}$$
(5.30)

where \(p=p(\omega ):=\frac{3}{2}+\theta _6\).

Proof

For fixed \(\omega >0\), we let \(\theta _1:=\theta _1(\omega )\in (0,\frac{1}{2})\), \(\theta _3:=\theta _3(\omega )>0\) and \(\theta _5:=\theta _5(\omega )>0\) be as given by Lemma 2.2, Lemma 3.3 and Lemma 4.2, respectively, and set \(\theta _6=\theta _6(\omega ):=\min \{\theta _1,\frac{\theta _5}{2}\}\), \(p=p(\omega ):=\frac{3}{2}+\theta _6\) and \(r=r(\omega ):=3+\theta _3\) We then choose positive numbers \(\delta _1=\delta _1(\omega )\) and \(\delta _2=\delta _2(\omega )\) satisfying

$$\begin{aligned} \delta _1 \le \frac{p(p-1)}{2} \end{aligned}$$
(5.31)

as well as

$$\begin{aligned} \delta _2 \le \frac{2(p-1)}{p} \qquad \text{ and } \qquad \delta _2 \le \frac{p}{2}, \end{aligned}$$
(5.32)

and let \(K_1:=K_1(\delta _1,p,r)\) and \(K_2:=K_2(\delta _2,p)\) denote the constants provided by Lemma 5.1 and Lemma 5.3, respectively.

Now given \(\delta >0\), we fix \(\delta _3=\delta _3(\omega ,\delta )>0\) small enough such that writing \(\alpha =\alpha (\omega ):=\frac{2p-1}{2p-3}>1\) we have

$$\begin{aligned} (K_1+K_2)\delta _3^{\alpha -1} \le \frac{1}{8}, \end{aligned}$$
(5.33)

and such that

$$\begin{aligned} 2^p \delta _3 < \frac{\delta }{2}, \end{aligned}$$
(5.34)

and thereafter we pick \(\gamma _0=\gamma _0(\omega ,\delta )>0\) suitably small fulfilling

$$\begin{aligned} (4-p) \cdot 2^{p-1} |\Omega | \cdot \gamma _0^p < \frac{\delta }{2} \end{aligned}$$
(5.35)

and

$$\begin{aligned} K_1 \cdot \bigg \{ \Big (2|\Omega |\gamma _0\Big )^\frac{2p(2p-1)}{4p-3} + \gamma _0^\frac{2p(2p-1)}{4p-3} + \gamma _0^{2p} + \gamma _0^\frac{2(2p-1)}{2p-3} \bigg \} < \frac{\delta _3}{16} \end{aligned}$$
(5.36)

as well as

$$\begin{aligned} K_2 \cdot \bigg \{ \Big (2|\Omega |\gamma _0\Big )^\frac{p^2}{p-1} + \gamma _0^\frac{p(2p-1)}{2p-3} + \gamma _0^\frac{p^2}{p-1} + \gamma _0^\frac{1}{p-1} \bigg \} < \frac{\delta _3}{16}. \end{aligned}$$
(5.37)

We now pick \(\delta _4=\delta _4(\omega ,\delta )>0\) and \(\delta _5=\delta _5(\omega ,\delta )>0\) such that

$$\begin{aligned} 2\delta _4 \le \Big (\frac{\delta _3}{4}\Big )^\frac{2}{p} \end{aligned}$$
(5.38)

and

$$\begin{aligned} K_1 \cdot \Big \{ \delta _5^\frac{2r}{r-3} + \delta _5^2 \Big \} \le \frac{p}{2}, \end{aligned}$$
(5.39)

and let \(\eta _1:=\eta _1(\omega ,\delta _4)\), \(\eta _3:=\eta _3(\omega ,\delta _5)\), \(\kappa _3:=\kappa _3(\omega ,\delta _5)\), \(\eta _5:=\eta _5(\omega ,\delta _4)\) and \(\kappa _5:=\kappa _5(\omega ,\delta _4)\) be as correspondingly provided by Lemma 2.2, Lemma 3.3 and Lemma 4.2, respectively.

We thereupon claim that the statement of the lemma holds if we pick \(\eta _6=\eta _6(\omega ,\delta )>0\) in such a way that

$$\begin{aligned} \eta _6 \le \min \{\eta _1,\eta _3,\eta _5\} \qquad \text{ and } \qquad \eta _6 < \gamma _0, \end{aligned}$$
(5.40)

and define

$$\begin{aligned} \kappa _6=\kappa _6(\omega ,\delta ):=\min \{\kappa _3,\kappa _5\}. \end{aligned}$$
(5.41)

To see this, we suppose that \(\rho \in \mathbb {R},\mu >0\) and \(f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)\) are such that (1.10) and (1.11) hold with some \(\eta <\eta _6\) and \(\kappa <\kappa _6\). On applying Lemma 2.2, we then obtain that since \(\eta <\eta _1\) and \(p=\frac{3}{2}+\theta _6 \le \frac{3}{2}+\theta _1\), there exists \(t_1=t_1(\eta )>0\) such that for all \(\varepsilon \in (0,1)\) we have

$$\begin{aligned} \int _{t-1}^t \Vert n_\varepsilon (\cdot ,s)\Vert _{L^p(\Omega )}^2 ds < \delta _4 \qquad \text{ for } \text{ all } t\ge t_1. \end{aligned}$$
(5.42)

Likewise, Lemma 3.3 asserts the existence of \(t_2=t_2(\omega ,\eta ,\kappa ,n_0,u_0)>t_1\) such that for each \(\varepsilon \in (0,1)\),

$$\begin{aligned} \Vert u_\varepsilon (\cdot ,t)\Vert _{L^r(\Omega )} < \delta _5 \qquad \text{ for } \text{ all } t\ge t_2, \end{aligned}$$
(5.43)

because \(\eta<\eta _3, \kappa <\kappa _3\) and \(r\le 3+\theta _3\), and using that \(\eta<\eta _5, \kappa <\kappa _5\) and \(2p\le 3+\theta _5\) we infer from Lemma 4.2 that with some \(t_3=t_3(\omega ,\delta ,\eta ,\kappa ,n_0,c_0,u_0)>t_2\) we have

$$\begin{aligned} \int _{t-1}^t \Vert \nabla c_\varepsilon (\cdot ,s)\Vert _{L^{2p}(\Omega )}^2 ds < \delta _4 \qquad \text{ for } \text{ all } t\ge t_3 \text{ and } \varepsilon \in (0,1). \end{aligned}$$
(5.44)

Finally, noting that \(\gamma :=\frac{\rho _+}{\mu }\) satisfies

$$\begin{aligned} \gamma <\gamma _0 \end{aligned}$$
(5.45)

by (5.40), and that hence \(\rho _0:=\mu \gamma _0\) is positive with \(\rho _0>\rho \), we may invoke Lemma 2.1 to find \(t_0=t_0(\omega ,\delta ,\eta ,\kappa ,n_0,c_0,u_0)>t_3\) such that

$$\begin{aligned} \int _\Omega n_\varepsilon (x,t)dx < 2|\Omega |\gamma _0 \qquad \text{ for } \text{ all } t\ge t_0 \text{ and } \varepsilon \in (0,1). \end{aligned}$$
(5.46)

In order to prove that this choice ensures that (5.30) holds for each fixed \(\varepsilon \in (0,1)\), given \(t\ge t_0\) we first apply the Hölder inequality to obtain from (5.42) and (5.44) that

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

whence we can find \(t_\varepsilon \in (t-1,t)\) fulfilling

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

Since \((\xi _1+\xi _2)^\frac{1}{p} \le \xi _1^\frac{1}{p}+\xi _2^\frac{1}{p}\) for all nonnegative \(\xi _1\) and \(\xi _2\), in view of (5.38) this entails that

$$\begin{aligned} \Vert n_\varepsilon (\cdot ,t_\varepsilon )\Vert _{L^p(\Omega )}^p + \Vert \nabla c_\varepsilon (\cdot ,t_\varepsilon )\Vert _{L^{2p}(\Omega )}^{2p} < (2\delta _4)^p \le \frac{\delta _3}{4}, \end{aligned}$$

and that therefore the function \(y\in C^1([t_\varepsilon ,\infty ))\) defined by

$$\begin{aligned} y(s):=\int _\Omega \psi (N_\varepsilon (x,s))dx + \int _\Omega |\nabla c_\varepsilon (x,s)|^{2p}dx, \qquad s\ge t_\varepsilon , \end{aligned}$$

where \(\psi =\psi _{p,\gamma }\) is as in (5.3) and (5.4), satisfies

$$\begin{aligned} y(t_\varepsilon ) <\frac{\delta _3}{4}, \end{aligned}$$
(5.47)

because in both cases \(\gamma >0\) and \(\gamma =0\) we have \(\psi (s)\le s^p\) for all \(s>0\).

In particular, (5.47) implies that

$$\begin{aligned} S:=\Big \{ T_0\in [t_\varepsilon ,t] \ \Big | \ y(s) < \delta _3 \ \text{ for } \text{ all } s\in [t_\varepsilon ,T_0] \Big \} \end{aligned}$$

is not empty and hence \(T:=\sup S\) well-defined with \(T>t_\varepsilon \) by continuity of y. To see that actually \(T=t\), we estimate y by analyzing an ODI which can be derived for this function from Lemma 5.1 and Lemma 5.3. To this end, we let

$$\begin{aligned} \overline{M}_1:=\sup _{s>t_\varepsilon } \Vert n_\varepsilon (\cdot ,s)\Vert _{L^1(\Omega )} \qquad \text{ and } \qquad \overline{M}_2:=\sup _{s>t_\varepsilon } \Vert u_\varepsilon (\cdot ,s)\Vert _{L^r(\Omega )} \end{aligned}$$

and then obtain from (5.46) that

$$\begin{aligned} \overline{M}_1 \le 2|\Omega |\gamma _0, \end{aligned}$$
(5.48)

whereas (5.43) combined with (5.39) entails that

$$\begin{aligned} K_1 \cdot \Big \{ \overline{M}_2^\frac{2r}{r-3} + \overline{M}_2^2 \Big \} \le \frac{p}{2}. \end{aligned}$$
(5.49)

In conjunction with the outcome of Lemma 5.1, (5.31), (5.45) and (5.36), the latter two inequalities say that

$$\begin{aligned} \frac{d}{dt} \int _\Omega |\nabla c_\varepsilon |^{2p}+ & {} \frac{2(p-1)}{p} \int _\Omega \Big |\nabla |\nabla c_\varepsilon |^p \Big |^2 + \frac{p}{2} \int _\Omega |\nabla c_\varepsilon |^{2p} \nonumber \\\le & {} \frac{p(p-1)}{2} \int _{\{N_\varepsilon>\gamma \}} N_\varepsilon ^{p-2} |\nabla N_\varepsilon |^2 \nonumber \\&+ K_1 \cdot \bigg \{ \int _\Omega \psi (N_\varepsilon ) + \int _\Omega |\nabla c_\varepsilon |^{2p} \bigg \}^\alpha \nonumber \\&+ K_1 \cdot \bigg \{ \Big (2|\Omega | \gamma _0\Big )^\frac{2p(2p-1)}{4p-3} + \gamma _0^\frac{2p(2p-1)}{4p-3} + \gamma _0^{2p} + \gamma _0^\frac{2(2p-1)}{2p-3} \bigg \} \nonumber \\\le & {} \frac{p(p-1)}{2} \int _{\{N_\varepsilon >\gamma \}} N_\varepsilon ^{p-2} |\nabla N_\varepsilon |^2 + K_1 y^\alpha + \frac{\delta _3}{16} \qquad \text{ on } (t_\varepsilon ,\infty ),\nonumber \\ \end{aligned}$$
(5.50)

while using (5.48) along with Lemma 5.3, (5.32), (5.45) and (5.37) shows that

$$\begin{aligned}&\frac{d}{dt} \int _\Omega \psi (N_\varepsilon ) + \frac{p(p-1)}{2} \int _{\{N_\varepsilon >\gamma \}} N_\varepsilon ^{p-2} |\nabla N_\varepsilon |^2 \nonumber \\&\quad \le \frac{2(p-1)}{p} \int _\Omega \Big |\nabla |\nabla c_\varepsilon |^p \Big |^2 + \frac{p}{2} \int _\Omega |\nabla c_\varepsilon |^{2p} \nonumber \\&\qquad + K_2 \cdot \bigg \{ \int _\Omega \psi (N_\varepsilon )\Big \}^\alpha \nonumber \\&\qquad + K_2\cdot \bigg \{ \Big (2|\Omega |\gamma _0\Big )^\frac{p^2}{p-1} + \gamma _0^\frac{p(2p-1)}{2p-3} + \gamma _0^\frac{p^2}{p-1} + \gamma _0^\frac{1}{p-1} \bigg \} \nonumber \\&\quad \le \frac{2(p-1)}{p} \int _\Omega \Big |\nabla |\nabla c_\varepsilon |^p \Big |^2 + \frac{p}{2} \int _\Omega |\nabla c_\varepsilon |^{2p} + K_2y^\alpha + \frac{\delta _3}{16} \qquad \text{ on } (t_\varepsilon ,\infty ).\nonumber \\ \end{aligned}$$
(5.51)

Adding (5.50) to (5.51) yields the inequality

$$\begin{aligned} y'(s) \le (K_1+K_2) y^\alpha (s) + \frac{\delta _3}{16} + \frac{\delta _3}{16} = (K_1+K_2) y^\alpha (s) + \frac{\delta _3}{8} \qquad \text{ for } \text{ all } s>t_\varepsilon ,\nonumber \\ \end{aligned}$$
(5.52)

so that by definition of T and (5.33) we obtain that

$$\begin{aligned} y'(s)\le & {} (K_1+K_2) \delta _3^\alpha + \frac{\delta _3}{8} \\= & {} \delta _3 \cdot \Big \{ (K_1+K_2) \delta _3^{\alpha -1} + \frac{1}{8} \Big \} \\\le & {} \delta _3 \cdot \Big \{ \frac{1}{8}+\frac{1}{8}\Big \} = \frac{\delta _3}{4} \qquad \text{ for } \text{ all } s\in (t_\varepsilon ,T). \end{aligned}$$

According to (5.47) and the fact that \(T \le t<t_\varepsilon +1\), on integration this implies that

$$\begin{aligned} y(s) \le y(t_\varepsilon ) + \frac{\delta _3}{4} \cdot (t-t_\varepsilon ) \le y(t_\varepsilon ) + \frac{\delta _3}{4} < \frac{\delta _3}{2} \qquad \text{ for } \text{ all } s\in (t_\varepsilon ,T). \end{aligned}$$

Again by continuity of y, this rules out the possibility that \(T<t\), meaning that indeed

$$\begin{aligned} y(s) < \delta _3 \qquad \text{ for } \text{ all } s\in [t_\varepsilon ,t) \end{aligned}$$

and that hence

$$\begin{aligned} y(t) \le \delta _3 \qquad \text{ for } \text{ all } t\ge t_0. \end{aligned}$$
(5.53)

Now since by (5.3) and (5.4), in both cases \(\gamma >0\) and \(\gamma =0\) we have

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

and

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

it follows that

$$\begin{aligned} \int _\Omega n_\varepsilon ^p\le & {} 2^p \cdot \bigg \{ \int _\Omega \psi (N_\varepsilon ) + |\Omega |\gamma ^p + \frac{2-p}{2}|\Omega |\gamma ^p \bigg \} \\= & {} 2^p \int _\Omega \psi (N_\varepsilon ) + (4-p) \cdot 2^{p-1} |\Omega |\gamma ^p \qquad \text{ for } \text{ all } t>0. \end{aligned}$$

In view of (5.45), (5.34) and (5.35), the inequality (5.53) therefore implies that

$$\begin{aligned} \int _\Omega n_\varepsilon ^p(x,t)dx + \int _\Omega |\nabla c_\varepsilon (x,t)|^{2p}dx\le & {} 2^p y(t) + (4-p) \cdot 2^{p-1} |\Omega |\gamma ^p \\\le & {} 2^p \delta _3 + (4-p)\cdot 2^{p-1} |\Omega | \gamma _0^p \\< & {} \frac{\delta }{2}+\frac{\delta }{2}=\delta \qquad \text{ for } \text{ all } t\ge t_0, \end{aligned}$$

as claimed. \(\quad \square \)

Boundedness of \(n_\varepsilon \)

Now thanks to the fact that in (5.30) we have \(2p>3\), we may rely on known smoothing properties of the heat semigroup to assert eventual \(L^\infty \) bounds for \(n_\varepsilon \) in the following sense.

Lemma 6.1

Let \(\omega >0\). Then there exist \(\eta _7=\eta _7(\omega )>0\) and \(\kappa _7=\kappa _7(\omega )>0\) such that whenever \(\rho \in \mathbb {R}, \mu >0\) and \(f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)\) are such that (1.10) and (1.11) hold with some \(\eta <\eta _7\) and \(\kappa <\kappa _7\), one can find \(t_0=t_0(\omega ,\eta ,\kappa ,n_0,c_0,u_0)>0\) and \(C=C(\omega )>0\) such that

$$\begin{aligned} \Vert n_\varepsilon (\cdot ,t)\Vert _{L^\infty (\Omega )} \le C \qquad \text{ for } \text{ all } t\ge t_0 \end{aligned}$$
(6.1)

and each \(\varepsilon \in (0,1)\).

Proof

With \(\theta _3:=\theta _3(\omega ), \eta _3:=\eta _3(\omega ,1)\) and \(\kappa _3:=\kappa _3(\omega ,1)\) taken from Lemma 3.3 and \(\theta _6(\omega )>0, \eta _6(\omega ,1)\) and \(\kappa _6(\omega ,1)\) as provided by Lemma 5.4, we let \(\eta _7\equiv \eta _7(\omega ):=\min \{\eta _3,\eta _6\}\) and \(\kappa _7\equiv \kappa _7(\omega ):=\min \{\kappa _3,\kappa _6\}\) and assume that (1.10) and (1.11) hold with some \(\eta <\eta _7\) and \(\kappa <\kappa _7\). Then applying Lemma 3.3 and Lemma 5.4, we see that writing \(p=p(\omega ):=3+\min \{\theta _3,2\theta _6\}\), we can fix \(t_1=t_1(\omega ,\eta ,\kappa ,n_0,c_0,u_0)>0\) and \(C_1=C_1(\omega )>0\) such that for all \(\varepsilon \in (0,1)\) we have

$$\begin{aligned} \Vert u_\varepsilon (\cdot ,t)\Vert _{L^p(\Omega )} \le C_1 \quad \text{ and } \quad \Vert \nabla c_\varepsilon (\cdot ,t)\Vert _{L^p(\Omega )} \le C_1 \qquad \text{ for } \text{ all } t\ge t_1 \text{ and } \varepsilon \in (0,1),\nonumber \\ \end{aligned}$$
(6.2)

where in view of Lemma 2.1 we may also assume on enlarging \(t_1\) and \(C_1\) if necessary that

$$\begin{aligned} \Vert n_\varepsilon (\cdot ,t)\Vert _{L^1(\Omega )} \le C_1 \qquad \text{ for } \text{ all } t\ge t_1 \text{ and } \varepsilon \in (0,1). \end{aligned}$$
(6.3)

In order to see that (6.1) holds for \(t_0=t_0(\omega ,\eta ,\kappa ,n_0,c_0,u_0):=t_1+1\), we estimate the finite numbers

$$\begin{aligned} M_\varepsilon (T):=\sup _{t\in (t_0,T)} \Vert n_\varepsilon (\cdot ,t)\Vert _{L^\infty (\Omega )}, \qquad T>t_0, \ \varepsilon \in (0,1), \end{aligned}$$
(6.4)

by decomposing

$$\begin{aligned} n_\varepsilon (\cdot ,t)= & {} e^{\Delta } n_\varepsilon (\cdot ,t-1) - \int _{t-1}^t e^{(t-s)\Delta } \nabla \cdot \Big \{ h_{1\varepsilon }(\cdot ,s) n_\varepsilon (\cdot ,s)\Big \} ds \nonumber \\&+ \int _{t-1}^t e^{(t-s)\Delta } h_{2\varepsilon }(\cdot ,s) ds \nonumber \\=: & {} n_{1\varepsilon }(\cdot ,t) + n_{2\varepsilon }(\cdot ,t)+ n_{3\varepsilon }(\cdot ,t), \qquad t\ge t_0, \ \varepsilon \in (0,1), \end{aligned}$$
(6.5)

again with \((e^{\tau \Delta })_{\tau \ge 0}\) denoting the Neumann heat semigroup in \(\Omega \) and

$$\begin{aligned}&h_{1\varepsilon }(x,t):=\chi \nabla c_\varepsilon (x,t) + u_\varepsilon (x,t) \quad \text{ as } \text{ well } \text{ as } \\&\quad h_{2\varepsilon }(x,t):=\rho n_\varepsilon (x,t)-\mu n_\varepsilon ^2(x,t), \qquad t\ge t_0, \ \varepsilon \in (0,1). \end{aligned}$$

Here thanks to a standard \(L^1\)-\(L^\infty \) smoothing property of \((e^{\tau \Delta })_{\tau \ge 0}\), there exists \(C_2>0\) such that due to (6.3) we have

$$\begin{aligned} \Vert n_{1\varepsilon }(\cdot ,t)\Vert _{L^\infty (\Omega )} \le C_2\Vert n_\varepsilon (\cdot ,t-1)\Vert _{L^1(\Omega )} \le C_1 C_2 \qquad \text{ for } \text{ all } t\ge t_0 \text{ and } \varepsilon \in (0,1).\nonumber \\ \end{aligned}$$
(6.6)

and since \(\rho \xi - \mu \xi ^2 \le \frac{\rho _+^2}{4\mu }\) for all \(\xi \ge 0\), we may invoke the maximum principle to obtain the one-sided estimate

$$\begin{aligned} n_{3\varepsilon }(\cdot ,t) \le \frac{\rho _+^2}{4\mu } \quad \text{ in } \Omega \qquad \text{ for } \text{ all } t\ge t_0 \text{ and } \varepsilon \in (0,1). \end{aligned}$$
(6.7)

Finally, fixing any \(p_0=p_0(\omega )\in (3,p)\) we can make use of a further known smoothing property of the Neumann heat semigroup (cf. [13, Lemma 3.3] for a version precisely covering the present situation) to find \(C_3=C_3(\omega )>0\) fulfilling

$$\begin{aligned}&\Vert n_{2\varepsilon }(\cdot ,t)\Vert _{L^\infty (\Omega )} \nonumber \\&\quad \le C_3 \int _{t-1}^t (t-s)^{-\frac{1}{2}-\frac{3}{2p_0}} \Big \Vert h_{1\varepsilon }(\cdot ,s) n_\varepsilon (\cdot ,s)\Big \Vert _{L^{p_0}(\Omega )} ds \qquad \nonumber \\&\qquad \text{ for } \text{ all } t\ge t_0 \text{ and } \varepsilon \in (0,1), \nonumber \\ \end{aligned}$$
(6.8)

where by the Hölder inequality, (6.2), (6.4) and (6.3), given \(T>t_0\) we have

$$\begin{aligned}&\Big \Vert h_{1\varepsilon }(\cdot ,s) n_\varepsilon (\cdot ,s)\Big \Vert _{L^{p_0}(\Omega )}\nonumber \\&\quad \le \Vert h_{1\varepsilon }(\cdot ,s)\Vert _{L^p(\Omega )} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^\frac{pp_0}{p-p_0}(\Omega )} \nonumber \\&\quad \le \Vert h_{1\varepsilon }(\cdot ,s)\Vert _{L^p(\Omega )} \Vert n_\varepsilon (\cdot ,s)\Vert _{L^\infty (\Omega )}^a \Vert n_\varepsilon (\cdot ,s)\Vert _{L^1(\Omega )}^{1-a} \nonumber \\&\quad \le (\chi +1) C_1 \cdot M_\varepsilon ^a(T) \cdot C_1^{1-a} \qquad \text{ for } \text{ all } s\in (t_0,T) \text{ and } \varepsilon \in (0,1)\nonumber \\ \end{aligned}$$
(6.9)

with \(a=a(\omega ):=1-\frac{p-p_0}{pp_0} \in (0,1)\). Since \(p_0>3\) entails that \(\int _0^1 \sigma ^{-\frac{1}{2}-\frac{3}{2p_0}} d\sigma \) is finite, combining (6.5)-(6.9) we conclude that there exists \(C_4=C_4(\omega )>0\) such that whenever \(T>t_0\) and \(\varepsilon \in (0,1)\),

$$\begin{aligned} \sup _{x\in \Omega } n_\varepsilon (x,t) \le C_4 + C_4 M_\varepsilon ^a(T) \qquad \text{ for } \text{ all } t\in (t_0,T). \end{aligned}$$

This clearly implies that

$$\begin{aligned} M_\varepsilon (T) \le \max \Big \{ 1 \, , \, (2C_4)^\frac{1}{1-a} \Big \} \qquad \text{ for } \text{ all } T>t_0 \text{ and } \text{ each } \varepsilon \in (0,1), \end{aligned}$$

and thereby establishes (6.1) on taking \(T\rightarrow \infty \). \(\quad \square \)

A Hölder Bound for \(u_\varepsilon \)

Together with the hypothesis (1.12), which is now explicitly referred to for the first time, the outcome of Lemma 6.1 next provides regularity information on the forcing terms in the Navier-Stokes part of (1.7) that is sufficient to infer the following consequence concerning eventual Hölder regularity of the velocity field.

Lemma 7.1

Let \(\omega >0\). Then there exist \(\eta _8=\eta _8(\omega )>0\) and \(\kappa _8=\kappa _8(\omega )>0\) with the following property: If \(\rho \in \mathbb {R}, \mu >0\) and \(f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)\) are such that (1.10) and (1.11) hold with some \(\eta <\eta _8\) and \(\kappa <\kappa _8\), and if furthermore there exist \(\mathbf{p}>\frac{3}{2}\) and \(\mathbf{q}>\frac{2\mathbf{p}}{2\mathbf{p}-3}\) such that (1.12) is valid, then one can find \(\alpha =\alpha (\omega ,f)\in (0,1)\), \(t_0=t_0(\omega ,\eta ,f,n_0,c_0,u_0)>0\) and \(C=C(\omega ,f)>0\) such that for all \(\varepsilon \in (0,1)\),

$$\begin{aligned} \Vert u_\varepsilon \Vert _{C^{\alpha ,\frac{\alpha }{2}}(\bar{\Omega }\times [t,t+1])} \le C \qquad \text{ for } \text{ all } t\ge t_0. \end{aligned}$$
(7.1)

Proof

With \(\theta _3:=\theta _3(\omega )\), \(\eta _3:=\eta _3(\omega ,1)\) and \(\kappa _3:=\kappa _3(1)\) as provided by Lemma 3.3 and \(\eta _7:=\eta _7(\omega )\) and \(\kappa _7:=\kappa _7(\omega )\) as in Lemma 6.1, we let \(\eta _8=\eta _8(\omega ):=\min \{\eta _3,\eta _7\}\) and \(\kappa _8=\kappa _8(\omega ):=\min \{\kappa _3,\kappa _7\}\) and suppose that \(\rho ,\mu \) and f have the assumed properties. We then pick \(\beta _0=\beta _0(\omega )>\frac{1}{4}\) such that \(\beta _0\le \frac{1}{4}+\theta _3\), so that in particular \(2\beta _0-\frac{3}{2}>-1\) and hence there exists \(p_1=p_1(\omega )>3\) fulfilling

$$\begin{aligned} 2\beta _0-\frac{3}{2}>-\frac{3}{p_1}. \end{aligned}$$
(7.2)

We next fix any \(p_2=p_2(\omega )>3\) such that \(p_2<p_1\), choose \(p_3=p_3(\omega )>3\) such that

$$\begin{aligned} p_3 \le 3+\theta _3 \qquad \text{ and } \qquad p_3<\frac{p_1 p_2}{p_1-p_2} \end{aligned}$$
(7.3)

and write \(\lambda =\lambda (\omega ,f):=\max \{\mathbf{p},p_2\}\). Then according to the inequalities \(p_2>3\) and \(\mathbf{q}>\frac{2\mathbf{p}}{2\mathbf{p}-3}\) it is possible to fix \(\beta =\beta (\omega )\in (0,1)\) in such a way that

$$\begin{aligned} \beta >\frac{3}{2\lambda } \end{aligned}$$
(7.4)

and

$$\begin{aligned} \beta <1-\frac{1}{\mathbf{q}} - \frac{3}{2\mathbf{p}} + \frac{3}{2\lambda } \end{aligned}$$
(7.5)

as well as

$$\begin{aligned} \beta <\frac{1}{2}-\frac{3}{2p_2}+\frac{3}{2\lambda }, \end{aligned}$$
(7.6)

where (7.4) guarantees that for some \(\alpha _1=\alpha _1(\omega ,f)>0\) we have \(D(A_\lambda ^\beta ) \hookrightarrow C^{\alpha _1}(\bar{\Omega };\mathbb {R}^3)\) ([15, 21]), implying that there exists \(C_1=C_1(\omega )>0\) such that

$$\begin{aligned} \Vert \varphi \Vert _{L^\infty (\Omega )} \le \Vert \varphi \Vert _{C^{\alpha _1}(\bar{\Omega })} \le C_1\Vert A^\beta \varphi \Vert _{L^\lambda (\Omega )} \qquad \text{ for } \text{ all } \varphi \in D(A_\lambda ^\beta ). \end{aligned}$$
(7.7)

Moreover, since \(\beta _0\le \frac{1}{4}+\theta _3\) and \(p_3\le 3+\theta _3\), Lemma 3.3 and Lemma 6.1 yield positive constants \(t_1=t_1(\omega ,\eta ,\kappa ,n_0,c_0,u_0)\) and \(C_2=C_2(\omega ,f)\) such that for all \(\varepsilon \in (0,1)\) we have

$$\begin{aligned}&\Vert A^{\beta _0}u_\varepsilon (\cdot ,t)\Vert _{L^2(\Omega )} \le 1, \quad \Vert u_\varepsilon (\cdot ,t)\Vert _{L^{p_3}(\Omega )} \le 1 \nonumber \\&\quad \text{ and } \quad \Vert n_\varepsilon (\cdot ,t)\Vert _{L^\lambda (\Omega )} \le C_2 \qquad \text{ for } \text{ all } t\ge t_1, \end{aligned}$$
(7.8)

and we first claim that these inequalities imply a uniform bound, independent of \(\varepsilon \in (0,1)\) and \(T > t_0\) with \(t_0 = t_0(\omega ,\eta ,f,n_0,c_0,u_0) := t_1+1\), for the numbers

$$\begin{aligned} M_\varepsilon (T):=\sup _{t\in (t_0,T)} \Vert A^\beta u_\varepsilon (\cdot ,t)\Vert _{L^\lambda (\Omega )}, \qquad T>t_0, \ \varepsilon \in (0,1). \end{aligned}$$
(7.9)

To achieve this, we recall that if for vectors \(v=(v_1,v_2,v_3)\in \mathbb {R}^3\) and \(w=(w_1,w_2,w_3)\in \mathbb {R}^3\) we define \(v\otimes w:=(a_{ij})_{i,j\in \{1,2,3\}}\) by letting \(a_{ij}:=v_i w_j\) for \(i,j\in \{1,2,3\}\), then since \(Y_\varepsilon u_\varepsilon \) is solenoidal, we have \((Y_\varepsilon u_\varepsilon \cdot \nabla )u_\varepsilon \equiv \nabla \cdot (Y_\varepsilon u_\varepsilon \otimes u_\varepsilon )\) ([46, cf. p. 265]). We thereby obtain the representation

$$\begin{aligned} u_\varepsilon (\cdot ,t)= & {} e^{-A} u_\varepsilon (\cdot ,t-1) - \int _{t-1}^t e^{-(t-s)A} \mathcal{P}\Big [ \nabla \cdot \Big (Y_\varepsilon u_\varepsilon (\cdot ,s) \otimes u_\varepsilon (\cdot ,s) \Big ) \Big ] ds \nonumber \\&+ \int _{t-1}^t e^{-(t-s)A} \mathcal{P}\Big [ n_\varepsilon (\cdot ,s)\nabla \phi \Big ] ds + \int _{t-1}^t e^{-(t-s)A} \mathcal{P}f(\cdot ,s) ds \nonumber \\=: & {} u_{1\varepsilon }(\cdot ,t)+\cdots +u_{4\varepsilon }(\cdot ,t), \qquad t\ge t_0, \end{aligned}$$
(7.10)

to which we apply the fractional power \(A^\beta \). By (7.8) and standard smoothing properties of the Stokes semigroup ([46, 16, p. 201]), we see that with some \(C_3=C_3(\omega ,f)>0\),

$$\begin{aligned} \Vert A^\beta u_{1\varepsilon }(\cdot ,t)\Vert _{L^\lambda (\Omega )} \le C_3 \Vert u_\varepsilon (\cdot ,t-1)\Vert _{L^{p_3}(\Omega )} \le C_3 \qquad \text{ for } \text{ all } t\ge t_0. \end{aligned}$$
(7.11)

Likewise, the third inequality in (7.8) along with the boundedness of \(\mathcal{P}\) in \(L^\lambda (\Omega ;\mathbb {R}^3)\) ([14]) and of \(\nabla \phi \) in \(L^\infty (\Omega )\) entails the existence of positive constants \(C_4(\omega ,f)\) and \(C_5(\omega ,f)\) such that

$$\begin{aligned} \Vert A^\beta u_{3\varepsilon }(\cdot ,t)\Vert _{L^\lambda (\Omega )}\le & {} C_4 \int _{t-1}^t (t-s)^{-\beta } \Big \Vert \mathcal{P}\Big [ n_\varepsilon (\cdot ,s)\nabla \phi \Big ] \Big \Vert _{L^\lambda (\Omega )} ds \nonumber \\\le & {} C_5 \int _{t-1}^t (t-s)^{-\beta } \Vert n_\varepsilon (\cdot ,s)\Vert _{L^\lambda (\Omega )} ds \nonumber \\\le & {} C_2 C_5 \int _{t-1}^t (t-s)^{-\beta } ds = \frac{C_2 C_5}{1-\beta } \qquad \text{ for } \text{ all } t\ge t_0. \end{aligned}$$
(7.12)

In quite a similar manner, recalling (1.12) and, again, standard \(L^p\)-\(L^q\) estimates for the Stokes semigroup, we can find \(C_6=C_6(\omega ,f)>0\) and \(C_7=C_7(\omega ,f)>0\) such that

$$\begin{aligned}&\Vert A^\beta u_{\varepsilon 4}(\cdot ,t)\Vert _{L^\lambda (\Omega )} \nonumber \\&\quad \le C_6 \int _{t-1}^t (t-s)^{-\beta -\frac{3}{2}(\frac{1}{\mathbf{p}}-\frac{1}{\lambda })} \Vert f(\cdot ,s)\Vert _{L^\mathbf{p}(\Omega )} ds \nonumber \\&\quad \le C_6 \bigg \{ \int _{t-1}^t (t-s)^{-[\beta +\frac{3}{2\mathbf{p}}-\frac{3}{2\lambda }]\cdot \frac{\mathbf{q}}{\mathbf{q}-1}} ds \bigg \}^\frac{\mathbf{q}-1}{\mathbf{q}} \cdot \bigg \{ \int _{t-1}^t \Vert f(\cdot ,s)\Vert _{L^\mathbf{p}(\Omega )}^\mathbf{q}ds \bigg \}^\frac{1}{\mathbf{q}} \nonumber \\&\quad \le C_7 \qquad \text{ for } \text{ all } t\ge t_0, \end{aligned}$$
(7.13)

where we also have used the Hölder inequality and the fact that

$$\begin{aligned} \Big [ \beta +\frac{3}{2\mathbf{p}}-\frac{3}{2\lambda }\Big ] \cdot \frac{\mathbf{q}}{\mathbf{q}-1} < \Big (1-\frac{1}{\mathbf{q}}\Big ) \cdot \frac{\mathbf{q}}{\mathbf{q}-1} { =1, } \end{aligned}$$

as asserted by (7.5).

In order to estimate \(A^\beta u_{2\varepsilon }\) appropriately, we once more invoke \(L^p\)-\(L^q\) estimates for the Stokes evolution operator to obtain \(C_8=C_8(\omega ,f)>0\) such that

$$\begin{aligned}&\Vert A^\beta u_{2\varepsilon }(\cdot ,t)\Vert _{L^\lambda (\Omega )} \nonumber \\&\quad \le C_8 \int _{t-1}^t (t-s)^{-\beta -\frac{1}{2}-\frac{3}{2}(\frac{1}{p_2}-\frac{1}{\lambda })} \Big \Vert Y_\varepsilon u_\varepsilon (\cdot ,s) \otimes u_\varepsilon (\cdot ,s)\Big \Vert _{L^{p_2}(\Omega )} ds \qquad \text{ for } \text{ all } t>0.\nonumber \\ \end{aligned}$$
(7.14)

Here by the Hölder inequality and the fact that \(D(A_2^{\beta _0})\hookrightarrow L^{p_1}(\Omega ;\mathbb {R}^3)\) by (7.2), we can find positive constants \(C_9=C_9(\omega ,f)\) and \(C_{10}=C_{10}(\omega ,f)\) such that due to (7.8) we have

$$\begin{aligned}&\Big \Vert Y_\varepsilon u_\varepsilon (\cdot ,s) \otimes u_\varepsilon (\cdot ,s)\Big \Vert _{L^{p_2}(\Omega )} \nonumber \\&\quad \le C_9 \Vert Y_\varepsilon u_\varepsilon (\cdot ,s)\Vert _{L^{p_1}(\Omega )} \Vert u_\varepsilon (\cdot ,s)\Vert _{L^\frac{p_1p_2}{p_1-p_2}(\Omega )} \nonumber \\&\quad \le C_{10} \Vert A^{\beta _0} Y_\varepsilon u_\varepsilon (\cdot ,s)\Vert _{L^2(\Omega )} \Vert u_\varepsilon (\cdot ,s)\Vert _{L^\infty (\Omega )}^a \Vert u_\varepsilon (\cdot ,s)\Vert _{L^{p_3}(\Omega )}^{1-a} \nonumber \\&\quad \le C_{10} \Vert A^{\beta _0} u_\varepsilon (\cdot ,s)\Vert _{L^2(\Omega )} \Vert u_\varepsilon (\cdot ,s)\Vert _{L^\infty (\Omega )}^a \Vert u_\varepsilon (\cdot ,s)\Vert _{L^{p_3}(\Omega )}^{1-a} \nonumber \\&\quad \le C_{10} \Vert u_\varepsilon (\cdot ,s)\Vert _{L^\infty (\Omega )}^a \qquad \text{ for } \text{ all } s>0 \end{aligned}$$
(7.15)

with \(a:=1-\frac{(p_1-p_2)p_3}{p_1 p_2} \in (0,1)\), where we have once more made use of the fact that \(Y_\varepsilon \) and \(A^{\beta _0}\) commute on \(D(A_2^{\beta _0})\), and that \(Y_\varepsilon \) is nonexpansive on \(L^2_\sigma (\Omega )\). In light of (7.7) and the definition (7.9) of \(M_\varepsilon (T)\), thanks to the fact that

$$\begin{aligned} \beta +\frac{1}{2}+\frac{3}{2}\Big (\frac{1}{p_2}-\frac{1}{\lambda }\Big )<1 \end{aligned}$$

guaranteed by (7.6), combining (7.14) with (7.15) yields \(C_{11}=C_{11}(\omega ,f)>0\) such that

$$\begin{aligned} \Vert A^\beta u_{2\varepsilon }(\cdot ,t)\Vert _{L^\lambda (\Omega )} \le C_{11} M_\varepsilon ^a(T) \qquad \text{ for } \text{ all } t\in (t_0,T). \end{aligned}$$

Together with (7.10)-(7.13), this shows that with some \(C_{12}=C_{12}(\omega ,f)>0\) we have

$$\begin{aligned} \Vert A^\beta u_\varepsilon (\cdot ,t)\Vert _{L^\lambda (\Omega )} \le C_{12} + C_{12} M_\varepsilon ^a(T) \qquad \text{ for } \text{ all } t\in (t_0,T), \end{aligned}$$

and that hence

$$\begin{aligned} M_\varepsilon (T) \le C_{13}=C_{13}(\omega ,f):=\max \Big \{ 1 \, , \, (2C_{12})^\frac{1}{1-a} \Big \} \qquad \text{ for } \text{ all } T>t_0, \end{aligned}$$

in view of (7.7) implying that

$$\begin{aligned} \Vert u_\varepsilon (\cdot ,t)\Vert _{C^{\alpha _1}(\bar{\Omega })} \le C_1 C_{13} \qquad \text{ for } \text{ all } t\ge t_0. \end{aligned}$$
(7.16)

Now by means of a standard adaptation of the above reasoning ([11]), involving estimates quite similar to those used before, it is next possible to find \(C_{14}=C_{14}(\omega ,f)>0\) and \(\alpha _2=\alpha _2(\omega ,f)>0\) such that for all \(\varepsilon \in (0,1)\),

$$\begin{aligned} \Vert A^\beta u_\varepsilon (\cdot ,t) - A^\beta u_\varepsilon (\cdot ,\tau )\Vert _{L^\lambda (\Omega )} \le C_{14} (t-\tau )^{\alpha _2} \qquad \text{ for } \text{ all } \tau \ge t_0 \text{ and } t\in [\tau ,\tau +1]. \end{aligned}$$

In conjunction with (7.16), this readily yields the claim for suitably small \(\alpha =\alpha (\omega ,f)\in (0,1)\). \(\quad \square \)

Eventual Regularity. Proof of Theorem 1.2

Straightforward applications of results on parabolic Hölder regularity, and of Schauder theories for the linear inhomogeneous heat and the Stokes evolution equations, finally turn the above into the following higher order estimates.

Lemma 8.1

Let \(\omega >0\). Then there exist \(\eta =\eta (\omega )>0\) and \(\kappa =\kappa (\omega )>0\) such that if \(\rho \in \mathbb {R}, \mu >0\) and \(f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)\) are such that (1.10), (1.11) and (1.12) are valid with some \(\mathbf{p}>\frac{3}{2}\) and \(\mathbf{q}>\frac{2\mathbf{p}}{2\mathbf{p}-3}\), then there exist \(\alpha =\alpha (\omega ,f)\in (0,1)\) and \(t_0=t_0(\omega ,\eta ,f,n_0,c_0,u_0)>0\) such that for each \(T>t_0\) and \(\varepsilon \in (0,1)\),

$$\begin{aligned} \Vert n_\varepsilon \Vert _{C^{2+\alpha ,1+\frac{\alpha }{2}}(\bar{\Omega }\times [t_0,T])} + \Vert c_\varepsilon \Vert _{C^{2+\alpha ,1+\frac{\alpha }{2}}(\bar{\Omega }\times [t_0,T])} + \Vert u_\varepsilon \Vert _{C^{2+\alpha ,1+\frac{\alpha }{2}}(\bar{\Omega }\times [t_0,T])} \le C \end{aligned}$$
(8.1)

holds with some \(C=C(\omega ,f,T)>0\).

Proof

For \(i\in \{3,6\}\) we let \(\theta _i:=\theta _i(\omega ), \eta _i:=\eta _i(\omega ,1)\) and \(\kappa _i:=\kappa _i(\omega ,1)\) be as in Lemma 3.3 and Lemma 5.4, and for \(i\in \{7,8\}\) we take \(\eta _i:=\eta _i(\omega )\) and \(\kappa _i:=\kappa _i(\omega )\) from Lemma 6.1 and Lemma 7.1. Then assuming (1.10), (1.11) and (1.12) to be satisfied with some \(\eta <\min \{\eta _3,\eta _6,\eta _7,\eta _8\}\), \(\kappa <\min \{\kappa _3,\kappa _6, \kappa _7,\kappa _8\}\), \(\mathbf{p}>\frac{3}{2}\) and \(\mathbf{q}>\frac{2\mathbf{p}}{2\mathbf{p}-3}\), we once more write the first Eq. in (1.7) in the form

$$\begin{aligned} n_{\varepsilon t} = \Delta n_\varepsilon - \nabla \cdot \Big (n_\varepsilon h_{1\varepsilon }(x,t)\Big ) + h_{2\varepsilon }(x,t), \qquad x\in \Omega , \ t>0, \ \varepsilon \in (0,1), \end{aligned}$$
(8.2)

with

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

Here we see from Lemma 3.3, Lemma 5.4 and Lemma 6.1 that there exist \(t_1=t_1(\omega ,\eta ,f,n_0,c_0,u_0)>0\) and \(C_1=C_1(\omega ,f)>0\) such that with \(p_1=p_1(\omega ):=3+\min \{2\theta _3,2\theta _6\}\), for all \(\varepsilon \in (0,1)\) we have

$$\begin{aligned}&\Vert n_\varepsilon (\cdot ,t)\Vert _{L^\infty (\Omega )} \le C_1, \quad \Vert h_{1\varepsilon }(\cdot ,t)\Vert _{L^{p_1}(\Omega )} \le C_1 \quad \text{ and }\\&\qquad { \Vert h_{2\varepsilon }(\cdot ,t)\Vert _{L^\infty (\Omega )} \le C_1 } \qquad \text{ for } \text{ all } t\ge t_1. \end{aligned}$$

Since \(p_1>3\), we can therefore apply standard parabolic Hölder regularity theory ([44, Theorem 1.3]) to see that there exists \(\alpha _1=\alpha _1(\omega )>0\) with the property that for each \(T>t_2=t_2(\omega ,\eta ,f,n_0,c_0,u_0):=t_1+1\) we can find \(C_2=C_2(\omega ,f,T)>0\) fulfilling

$$\begin{aligned} \Vert n_\varepsilon \Vert _{C^{\alpha _1,\frac{\alpha _1}{2}}(\bar{\Omega }\times [t_2,T])} \le C_2 \qquad \text{ for } \text{ all } \varepsilon \in (0,1). \end{aligned}$$
(8.3)

Along with the outcome of Lemma 7.1, this shows that the coefficient functions \(n_\varepsilon \) and \(u_\varepsilon \) appearing in the second Eq. in (1.7) both satisfy \(\varepsilon \)-independent estimates in \(C^{\alpha _2,\frac{\alpha _2}{2}}(\bar{\Omega }\times [t_3,T])\) for some \(\alpha _2=\alpha _2(\omega ,f)\in (0,1)\) and \(t_3>t_2\) and each \(T>t_3\), which by parabolic Schauder theory ([30]) implies the existence of \(\alpha _3=\alpha _3(\omega ,f)\in (0,1)\) such that for all \(T>t_4=t_4(\omega ,\eta ,f,n_0,c_0,u_0):=t_3+1\) we have

$$\begin{aligned} \Vert c_\varepsilon \Vert _{C^{2+\alpha _3,1+\frac{\alpha _3}{2}}(\bar{\Omega }\times [t_4,T])} \le C_3 \qquad \text{ for } \text{ all } \varepsilon \in (0,1) \end{aligned}$$
(8.4)

with some \(C_3=C_3(\omega ,f,T)>0\). This in turn suggests to re-interpret (8.2) as

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

with \(h_{1\varepsilon }\) as above and

$$\begin{aligned} h_{3\varepsilon }(x,t):=\chi n_\varepsilon (x,t)\Delta c_\varepsilon (x,t) + \rho n_\varepsilon (x,t)-\mu n_\varepsilon ^2(x,t), \qquad x\in \Omega , t>0, \varepsilon \in (0,1). \end{aligned}$$

Now, namely, we know that for some \(\alpha _4=\alpha _4(\omega ,f)\in (0,1)\), all \(T>t_4\) and any \(\varepsilon \in (0,1)\) we have

$$\begin{aligned} \Vert h_{i\varepsilon }\Vert _{C^{\alpha _4,\frac{\alpha _4}{2}}(\bar{\Omega }\times [t_4,T])} \le C_4 \qquad \text{ for } i\in \{1,3\} \end{aligned}$$

with some \(C_4=C_4(\omega ,f,T)>0\), so that again by parabolic Schauder theory we obtain \(\alpha _5=\alpha _5(\omega ,f)\in (0,1)\) with the property that for all \(T>t_5=t_5(\omega ,\eta ,f,n_0,c_0,u_0):=t_4+1\) we can find \(C_5=C_5(\omega ,f,T)>0\) such that

$$\begin{aligned} \Vert n_\varepsilon \Vert _{C^{2+\alpha _5,1+\frac{\alpha _5}{2}}(\bar{\Omega }\times [t_5,T])} \le C_5 \qquad \text{ for } \text{ all } \varepsilon \in (0,1). \end{aligned}$$
(8.5)

Finally, combining (8.3) with Lemma 7.1 and our overall assumption \(f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)\) allows us to follow a standard reasoning involving Schauder estimates for the linear inhomogeneous Stokes evolution Eq. ([47]) to see that

$$\begin{aligned} \Vert u_\varepsilon \Vert _{C^{2+\alpha _6,1+\frac{\alpha _6}{2}}(\bar{\Omega }\times [t_6,T])} \le C_6 \end{aligned}$$

is valid for some \(\alpha _6=\alpha _6(\omega ,f)\in (0,1)\) and \(t_6=t_6(\omega ,\eta ,f,n_0,c_0,u_0)>t_5\), each \(T>t_6\) and \(\varepsilon \in (0,1)\) and some appropriately large \(C_6=C_6(\omega ,f,T)>0\). Together with (8.4) and (8.5), this proves (8.1). \(\quad \square \)

In view of the Arzelà-Ascoli theorem, the following consequence of the latter is immediate:

Lemma 8.2

Let \(\omega >0\). Then there exist \(\eta =\eta (\omega )>0\) and \(\kappa =\kappa (\omega )>0\) such that if \(\rho \in \mathbb {R}, \mu >0\) and \(f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)\) are such that (1.10), (1.11) and (1.12) hold with some \(\mathbf{p}>\frac{3}{2}\) and \(\mathbf{q}>\frac{2\mathbf{p}}{2\mathbf{p}-3}\), then there exists \(t_0=t_0(\omega ,\eta ,f,n_0,c_0,u_0)>0\) with the property that given any \((\varepsilon _j)_{j\in \mathbb {N}} \subset (0,1)\) satisfying \(\varepsilon _j\rightarrow 0\) as \(j\rightarrow \infty \) one can find \((n,c,u,P) \in (C^{2,1}(\bar{\Omega }\times [t_0,\infty )))^2 \times C^{2,1}(\bar{\Omega }\times [t_0,\infty );\mathbb {R}^3) \times C^{1,0}(\bar{\Omega }\times [t_0,\infty ))\) and a subsequence \((\varepsilon _{j_k})_{k\in \mathbb {N}}\) such that

$$\begin{aligned} \left\{ \begin{array}{l} n_\varepsilon \rightarrow n \qquad \text{ in } C^{2,1}_{loc}(\bar{\Omega }\times [t_0,\infty )), \\ c_\varepsilon \rightarrow c \qquad \text{ in } C^{2,1}_{loc}(\bar{\Omega }\times [t_0,\infty )) \qquad \text{ and } \\ u_\varepsilon \rightarrow u \qquad \text{ in } C^{2,1}_{loc}(\bar{\Omega }\times [t_0,\infty )), \end{array} \right. \end{aligned}$$
(8.6)

and such that (ncuP) is a classical solution of (1.1), (1.5) in \(\bar{\Omega }\times [t_0,\infty )\).

Proof

All statements directly result from Lemma 8.1 upon an application of the Arzelà-Ascoli theorem and thereafter taking \(\varepsilon =\varepsilon _{j_k}\rightarrow 0\) in each of the expressions in the PDE system in (1.7) separately, finally constructing the associated pressure P by means of a standard procedure ([46, 51]). \(\quad \square \)

The derivation of our main results hence reduces to suitably collecting the essence of the above:

.

Proof of Theorem 1.2

Taking \(\eta =\eta (\omega )\) and \(\kappa =\kappa (\omega )\) as in Lemma 8.2, from a combination of the latter lemma with Proposition 1.1 we immediately obtain that for some \(t_0=t_0(\omega ,\eta ,f,n_0,c_0,u_0)\ge 0\), the global generalized solution (ncu) from said proposition indeed enjoys the eventual smoothness properties listed in (1.13), and that with some \(P\in C^{1,0}(\bar{\Omega }\times [t_0,\infty ))\), (ncuP) actually solves (1.1), (1.5) classically in \(\bar{\Omega }\times [t_0,\infty )\). Finally, the boundedness properties claimed in (1.14) are implied by Lemma 6.1, Lemma 4.1, Lemma 5.4 and Lemma 7.1 upon noting that \(W^{1,2p}(\Omega ) \hookrightarrow L^\infty (\Omega )\) for each \(p>\frac{3}{2}\). \(\quad \square \)

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Acknowledgements

The author is grateful to the anonymous reviewers for several useful comments which lead to significant improvements, and to Tobias Black for numerous helpful remarks concerning the arrangement of the arguments. He moreover acknowledges support of the Deutsche Forschungsgemeinschaft in the context of the project Emergence of structures and advantages in cross-diffusion systems (Project No. 411007140, GZ: WI 3707/5-1).

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Appendix: The Underlying Solution Concept

Appendix: The Underlying Solution Concept

For completeness, we finally import from [59] the framework of generalized solvability that underlies the existence statement from Proposition 1.1. Here with regard to the requirements imposed on candidates for solutions, the main relaxation consists in considering the first sub-problem in (1.1)-(1.4)-(1.5) fulfilled if merely certain transformed versions of n play roles of weak sub- and supersolutions thereof in a sense specified as follows:

Definition 9.1

Let \(\Phi \in C^2([0,\infty ))\) be nonnegative with \(\Phi '>0\) on \([0,\infty )\), and let \(n_0: \Omega \rightarrow [0,\infty )\), \(c_0: \Omega \rightarrow \mathbb {R}\) and \(u_0:\Omega \rightarrow \mathbb {R}^3\) are such that \(\Phi (n_0)\in L^1(\Omega )\). Suppose that \(n:\Omega \times (0,\infty )\rightarrow [0,\infty )\), \(c:\Omega \times (0,\infty )\rightarrow \mathbb {R}\) and \(u:\Omega \times (0,\infty )\rightarrow \mathbb {R}^3\) are such that \(\nabla n, \nabla c\) and \(\Delta c\) are measurable, that

$$\begin{aligned} \Big \{ \Phi (n), \Phi ''(n) |\nabla n|^2, \Phi (n)\Delta c, n\Phi '(n) \Delta c, n\Phi '(n), n^2\Phi '(n) \Big \} \subset L^1_{loc}(\bar{\Omega }\times [0,\infty )) \end{aligned}$$

and

$$\begin{aligned} \Big \{ \Phi '(n)\nabla n, \Phi (n)\nabla c, \Phi (n)u \Big \} \subset L^1_{loc}(\bar{\Omega }\times [0,\infty );\mathbb {R}^3) \end{aligned}$$

and that \(\nabla \cdot u\equiv 0\) in \(\mathcal{D}'(\Omega \times (0,\infty ))\). Then we say that (ncu) is a global weak \(\Phi \)-subsolution (resp., a global weak \(\Phi \)-supersolution) of the first Eqs. in (1.1)-(1.4)-(1.5) if

$$\begin{aligned}&- \int _0^\infty \int _\Omega \Phi (n)\varphi _t - \int _\Omega \Phi (n_0) \varphi (\cdot ,0) {\mathop {\le }\limits ^{(\ge )}} - \int _0^\infty \int _\Omega \Phi ''(n)|\nabla n|^2 \varphi \\&\qquad - \int _0^\infty \int _\Omega \Phi '(n) \nabla n \cdot \nabla \varphi \nonumber \\&\qquad + \chi \int _0^\infty \int _\Omega \Phi (n) \nabla c \cdot \nabla \varphi + \chi \int _0^\infty \int _\Omega \Phi (n) \Delta c \varphi - \chi \int _0^\infty \int _\Omega n\Phi '(n) \Delta c \varphi \nonumber \\&\quad + \rho \int _0^\infty \int _\Omega n\Phi '(n) \varphi - \mu \int _0^\infty \int _\Omega n^2\Phi '(n) \varphi + \int _0^\infty \int _\Omega \Phi (n) u\cdot \nabla \varphi \end{aligned}$$

holds for all nonnegative \(\varphi \in C_0^\infty (\bar{\Omega }\times [0,\infty ))\).

The solution concept under consideration, partially resembling that also used in related contexts (cf. e.g. [2]) can now be specified as follows, again letting \(v\otimes w:=(a_{ij})_{i,j\in \{1,2,3\}}\), with \(a_{ij}:=v_i w_j\), \(i,j\in \{1,2,3\}\), for \(v=(v_1,v_2,v_3)\in \mathbb {R}^3\) and \(w=(w_1,w_2,w_3)\in \mathbb {R}^3\).

Definition 9.2

Let \(n_0\in L^1(\Omega ), c_0\in L^1(\Omega )\) and \(u_0\in L^1(\Omega ;\mathbb {R}^3)\), and assume that

$$\begin{aligned} \left\{ \begin{array}{l} n \in L^1_{loc}(\bar{\Omega }\times [0,\infty )), \\ c \in L^1_{loc}([0,\infty );W^{1,1}(\Omega )) \qquad \text{ and } \\ u \in L^1_{loc}([0,\infty );W_0^{1,1}(\Omega ;\mathbb {R}^3)) \end{array} \right. \end{aligned}$$

are such that

$$\begin{aligned} cu \in L^1_{loc}(\bar{\Omega }\times [0,\infty );\mathbb {R}^3) \qquad \text{ and } \qquad u\otimes u\in L^1_{loc}(\Omega \times [0,\infty );\mathbb {R}^{3\times 3}), \end{aligned}$$

and that \(n\ge 0\) a.e. in \(\Omega \times (0,\infty )\). Then (ncu) is called a global generalized solution of (1.1)-(1.4)-(1.5) if \(\nabla \cdot u=0\) in \(\mathcal{D}'(\Omega \times (0,\infty ))\), if

$$\begin{aligned}&- \int _0^\infty \int _\Omega c\varphi _t - \int _\Omega c_0\varphi (\cdot ,0) \\&\quad = - \int _0^\infty \int _\Omega \nabla c \cdot \nabla \varphi - \int _0^\infty \int _\Omega c\varphi + \int _0^\infty \int _\Omega n\varphi + \int _0^\infty \int _\Omega cu\cdot \nabla \varphi \end{aligned}$$

for all \(\varphi \in C_0^\infty (\bar{\Omega }\times [0,\infty ))\) and

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

for all \(\varphi \in C_0^\infty (\Omega \times [0,\infty );\mathbb {R}^3)\) fulfilling \(\nabla \cdot \varphi \equiv 0\), and if there exist \(\Phi _1,\Phi _2\in C^2([0,\infty ))\) such that \(\Phi _1'>0\) and \(\Phi _2'>0\) on \([0,\infty )\), and such that (ncu) is a weak \(\Phi _1\)-subsolution and a weak \(\Phi _2\)-supersolution of the first Eqs. in (1.1)-(1.4)-(1.5) in the sense of Definition 9.1.

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Winkler, M. Reaction-Driven Relaxation in Three-Dimensional Keller–Segel–Navier–Stokes Interaction. Commun. Math. Phys. 389, 439–489 (2022). https://doi.org/10.1007/s00220-021-04272-y

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