Keller–Segel–Navier–Stokes System Involving Tensor-Valued Sensitivity

Abstract

Chemotaxis, the biased movement of cells in response to chemical gradients, plays an important role in coordinating cell migration in many biological phenomena (see Hillen and Painter 2009). For example, the fruit fly Drosophila melanogaster navigates up gradients of attractive odors during food location, and male moths follow pheromone gradients released by the female during mate location.

2.1 Introduction

Chemotaxis, the biased movement of cells in response to chemical gradients, plays an important role in coordinating cell migration in many biological phenomena (see Hillen and Painter 2009). For example, the fruit fly Drosophila melanogaster navigates up gradients of attractive odors during food location, and male moths follow pheromone gradients released by the female during mate location. In 1970, Keller and Segel (1971b) proposed a mathematical model describing chemotactic aggregation of cellular slime molds

$$\begin{aligned} \left\{ \begin{aligned}&n_t=\varDelta n-\nabla \cdot (n\nabla c),\quad&x\in \varOmega ,~ t>0, \\&c_t=\varDelta c- c +n\quad&x\in \varOmega , ~t>0, \end{aligned} \right. \end{aligned}$$

(2.1.1)

where $\varOmega \subset \mathbb {R}^N$, and n and c denote the density of the cell population and the concentration of the attracting chemical substance, respectively. One of the most characteristic mathematical features of system (2.1.1) is the possibility of blow-up of solutions in a finite or infinite time. It is well known that solutions of system (2.1.1) may blow up when $N=2$ with large total mass of cells and $N\ge 3$ with arbitrarily small prescribed total mass of cells (see Bellomo et al. 2015; Horstmann 2003; Nagai et al. 1997; Winkler 2013). In order to describe the nonlinear dependence on the cell density in cell movement, the following variant has also been widely studied:

$$\begin{aligned} \left\{ \begin{aligned}&n_t=\varDelta n^m-\nabla \cdot ( n\nabla c), \quad&x\in \varOmega ,~ t>0, \\&c_t=\varDelta c- c +n,\quad&x\in \varOmega , ~t>0, \end{aligned} \right. \end{aligned}$$

(2.1.2)

where $m>0$. Recent results indicates that $m=2-\frac{2}{N}$ is the critical blow-up exponent of (2.1.2) in some sense. Indeed, all solutions are global and uniformly bounded if $m>2-\frac{2}{N}$ (see Tao and Winkler 2012a; Winkler 2010b); whereas if $m<2-\frac{2}{N}$, (2.1.2) has some solutions which blow up in a finite time (see Cieślak and Stinner 2012; Winkler 2010b).

Recent analytical findings show that already cell transport through a given fluid can substantially influence the solution behavior in certain Keller–Segel-type chemotaxis systems (Kiselev and Ryzhik 2012a, b); in fact, even complete suppression of blow-up may occur (Kiselev and Xu 2016). To the best of our knowledge, however, the full mutual coupling of equations from fluid dynamics to chemotaxis systems, including buoyancy-driven feedback on the fluid motion, has been considered in the analytical literature in cases when the signal substance is produced by the cells, which seems rather thin compared with that in cases for the oxygen-consumed though, such as the Keller–Segel–Navier–Stokes of the form

$$\begin{aligned} \left\{ \begin{aligned}&n_t+u\cdot \nabla n=\varDelta n-\nabla \cdot (nS(x, n, c)\nabla c),\quad x\in \varOmega , t>0,\\&c_t+u\cdot \nabla c=\varDelta c-c+n,\quad x\in \varOmega , t>0,\\&u_t+\kappa (u\cdot \nabla )u+\nabla P=\varDelta u+n\nabla \phi ,\quad x\in \varOmega , t>0,\\&\nabla \cdot u=0,\quad x\in \varOmega , t>0,\\&\displaystyle {(\nabla n-nS(x, n, c))\cdot \nu =\nabla c\cdot \nu =0,\;u=0,}\quad x\in \partial \varOmega , t>0,\\&\displaystyle {n(x,0)=n_0(x),c(x,0)=c_0(x),\;u(x,0)=u_0(x),}\quad x\in \varOmega ,\\ \end{aligned}\right. \end{aligned}$$

(2.1.3)

where n and c are defined as before and $\varOmega \subset \mathbb {R}^3$ is a bounded domain with a smooth boundary. Here, u, $P,\phi$ and $\kappa \in \mathbb {R}$ denote, respectively, the velocity field, the associated pressure of the fluid, the potential of the gravitational field and the strength of nonlinear fluid convection. S(x, n, c) is a chemotactic sensitivity tensor satisfying

$$\begin{aligned} S\in C^2(\bar{\varOmega }\times [0,\infty )^2;\mathbb {R}^{3\times 3}) \end{aligned}$$

(2.1.4)

and

$$\begin{aligned} |S(x, n, c)|\le C_S(1 + n)^{-\alpha } ~~~~\text{ for } \text{ all }~~ (x, n, c)\in \varOmega \times [0,\infty )^2 \end{aligned}$$

(2.1.5)

with some $C_S > 0$ and $\alpha > 0$. Problem (2.1.3) is proposed to describe the chemotaxis–fluid interaction in cases when the evolution of the chemoattractant is essentially dominated by production through cells (see Winkler et al. 2015 and Hillen and Painter 2009). For example, in two dimensions, if $S=S(x, n, c)$ is a tensor-valued sensitivity fulfilling (2.1.4) and (2.1.5), Wang and Xiang (2015) proved that the Stokes version ($\kappa =0$ in the first equation of (2.1.3)) of system (2.1.3) admits a unique global classical solution that is bounded. Recently, Wang, Winkler and Xiang (2018) extended the above result Wang and Xiang (2015) to the Navier–Stokes version ($\kappa \ne 0$ in the first equation of (2.1.3)). In both papers Wang et al. (2018) and Wang and Xiang (2015), the condition $\alpha >0$ is optimal for the existence of the solution. Furthermore, similar results are also valid for the three-dimensional Stokes version ($\kappa =0$ in the first equation of (2.1.3)) of system (2.1.3) with $\alpha >\frac{1}{2}$ (see Wang and Xiang 2016). In the three-dimensional case, Wang and Liu (2017) showed that the Keller–Segel–Navier–Stokes ($\kappa \ne 0$ in the first equation of (2.1.3)) system (2.1.3) admits a global weak solution for tensor-valued sensitivity S(x, n, c) satisfying (2.1.4) and (2.1.5) with $\alpha > \frac{3}{7}$. Recently, due to the lack of enough regularity and compactness properties for the first equation, by using the idea proposed by Winkler (2015a), Wang (2017) presented the existence of global very weak solutions for the system (2.1.3) under the assumption that S satisfies (2.1.4) and (2.1.5) with $\alpha > \frac{1}{3}$, which, in light of the known results for the fluid-free system mentioned above, is an optimal restriction on $\alpha$. However, the existence of global (stronger than the result of Wang 2017) weak solutions is still open.

When taking the nonlinear diffusion of the cells into account, the system above may be reformed as

$$\begin{aligned} \left\{ \begin{aligned}&n_t+u\cdot \nabla n=\varDelta n^m-\nabla \cdot (nS(x, n, c)\nabla c),\quad&x\in \varOmega , t>0, \\&c_t+u\cdot \nabla c=\varDelta c-c+n,\quad&x\in \varOmega , t>0, \\&u_t+\kappa (u\cdot \nabla )u+\nabla P=\varDelta u+n\nabla \phi ,\quad&x\in \varOmega , t>0, \\&\nabla \cdot u=0,\quad&x\in \varOmega , t>0, \end{aligned}\right. \end{aligned}$$

(2.1.6)

with S fulfilling

$$ |S(x,n,c)|\le C_S $$

for some positive constant $C_S$. When $\kappa =0$, Li et al. (2016) and Zheng (2019) considered the chemotaxis–Stokes system (2.1.6) for $N=2$ and $N=3$, respectively. They concluded that when $m>2-\frac{2}{N}$, the weak solutions of the simplified system (2.1.6) ($\kappa =0$) are global existent and bounded. But till now, as far as we know, it is still not clear that in the case that $\kappa \not =0$, whether the solution of the chemotaxis–Navier–Stokes system (2.1.6) is bounded or not.

The emergence of degenerate diffusion, full Navier–Stokes fluid ($\kappa \ne 0$) and rotational flux (tensor-valued sensitivity S) makes the system (2.1.6) contain a more complex cross-diffusion mechanism, which brings more mathematical difficulties to the problem. In fact, if $\kappa =0$, by utilizing the $L^1$ estimate on n, one can invoke Lemma 2.4 in Wang and Xiang (2015) and the Sobolev embedding theorem (Theorem 5.6.6 in Evans 2010) to obtain the regularity of u in arbitrary $L^p$ spaces (see Lemma 2.4 in Li et al. 2016). Then one can also obtain $L^p$ estimate on c, by using the variation-of-constants representation for c (see the proof of Lemma 2.6 in Wang and Xiang 2015 and Lemma 2.6 in Li et al. 2016). By using the estimates on c and u, one can finally derive the entropy-like estimate involving the functional $\int _{\varOmega }n^{p} +\int _{\varOmega } |\nabla c|^{2q}$ (see Lemma 2.9 in Li et al. 2016 or Lemma 2.10 in Wang and Xiang 2015). Once the crucial step has been accomplished, the main results can be easily obtained by using the standard Alikakos–Moser iteration. However, when $\kappa \ne 0$, one cannot acquire the regularity of u in arbitrary $L^p$ spaces directly. Here, we develop some $L^p$-estimate techniques to raise the a priori estimates of solutions from $L^1(\varOmega )\rightarrow L^{m-1}(\varOmega )\rightarrow L^m(\varOmega )\rightarrow L^p(\varOmega )$ (for any $p>2)$, which even seems a new method in the case of fluid-free system.

The first part of this chapter is concerned with system (2.1.6) along with the initial data

$$\begin{aligned} \displaystyle {n(x,0)=n_0(x),\quad c(x,0)=c_0(x),\quad u(x,0)=u_0(x),}\qquad x\in \varOmega , \end{aligned}$$

(2.1.7)

and under the boundary conditions

$$\begin{aligned} \displaystyle {\left( \nabla n^m-nS(x, n, c)\nabla c\right) \cdot \nu =\nabla c\cdot \nu =0,\quad u=0,}\qquad x\in \partial \varOmega , t>0 \end{aligned}$$

(2.1.8)

in a bounded domain $\varOmega \subset \mathbb {R}^2$ with smooth boundary, where the chemotactic sensitivity tensor S(x, n, c) satisfies

$$\begin{aligned} S\in C^2(\bar{\varOmega }\times [0,\infty )^2;\mathbb {R}^{2\times 2}) \end{aligned}$$

(2.1.9)

and

$$\begin{aligned} |S(x, n, c)|\le C_S ~~~~\text{ for } \text{ all }~~ (x, n, c)\in \varOmega \times [0,\infty )^2 \end{aligned}$$

(2.1.10)

with some $C_S > 0$. Throughout this part $\phi \in W^{2,\infty }(\varOmega )$ and the initial data $(n_0, c_0, u_0)$ fulfills

$$\begin{aligned} \left\{ \begin{aligned}&\displaystyle {n_0\in C^\kappa (\bar{\varOmega })~~\text{ for } \text{ certain }~~ \kappa > 0~~ \text{ with }~~ n_0\ge 0 ~~\text{ in }~~\varOmega }, \\&\displaystyle {c_0\in W^{2,\infty }(\varOmega )~~\text{ with }~~c_0\ge 0~~\text{ in }~~\bar{\varOmega },} \\&\displaystyle {u_0\in D(A),} \end{aligned} \right. \end{aligned}$$

(2.1.11)

where A denotes the Stokes operator with domain $D(A) := W^{2,{2}}(\varOmega )\cap W^{1,{2}}_0(\varOmega ) \cap L^{2}_{\sigma }(\varOmega )$, and $L^{2}_{\sigma }(\varOmega ) := \{\varphi \in L^{2}(\varOmega )|\nabla \cdot \varphi = 0\}$ (see Sohr 2001).

Within the above frameworks, the main result on global existence and boundedness of solutions to (2.1.6)–(2.1.8) is stated as follows (Zheng and Ke 2020).

Theorem 2.1

Let $m>1$, $\varOmega \subset \mathbb {R}^2$ be a bounded domain with smooth boundary, and assume (2.1.9)–(2.1.11) hold. Then the problem (2.1.6)–(2.1.8) admits a global-in-time weak solution (n, c, u, P), which is uniformly bounded in the sense that

$$\begin{aligned} \Vert n(\cdot , t)\Vert _{L^\infty (\varOmega )}+\Vert c(\cdot , t)\Vert _{W^{1,\infty }(\varOmega )}+\Vert u(\cdot , t)\Vert _{L^{\infty }(\varOmega )}\le C~~ \text{ for } \text{ all }~~ t>0 \end{aligned}$$

(2.1.12)

with some positive constant C.

Remark 2.1

(i) If $u\equiv 0$, Theorem 2.1 coincides with Theorem 5.1 in Winkler (2010b), which seems to be optimal according to the two-dimensional fluid-free system.

(ii) Theorem 2.1 extends the results of Li et al. (2016), in which the authors discussed the chemotaxis–Stokes system ($\kappa =0$) in a two-dimensional convex domain. As mentioned earlier, we not only extend the results to the chemotaxis–Navier–Stokes system ($\kappa \not =0$), but also remove the convexity assumption of the domain. In Li et al. (2016), in order to get the regularity of $\nabla c$, the assumption that the domain should be convex is required. Applying the boundedness of $\Vert \nabla c\Vert _{L^2(\varOmega )}$ (see Lemma 2.8) and the fractional Gagliardo–Nirenberg inequality (see Lemma 2.5 in Ishida et al. 2014) to gain the regularity of $\nabla c$ in arbitrary $L^p$ spaces, the hypothesis of convexity for $\varOmega$ is removed herein.

The second part of this chapter considers the globally defined weak solution (see Definition 2.1) to system (2.1.3) with the initial data $(n_0, c_0, u_0)$ fulfilling

$$\begin{aligned} \left\{ \begin{aligned}&\displaystyle {n_0\in C^\kappa (\bar{\varOmega })~~\text{ for } \text{ certain }~~ \kappa > 0~~ \text{ with }~~ n_0\ge 0 ~~\text{ in }~~\varOmega },\\&\displaystyle {c_0\in W^{1,\infty }(\varOmega )~~\text{ with }~~c_0\ge 0~~\text{ in }~~\bar{\varOmega },}\\&\displaystyle {u_0\in D(A^\gamma _{r})~~\text{ for } \text{ some }~~\gamma \in ( {3}/{4}, 1)~~\text{ and } \text{ any }~~ {r}\in (1,\infty ),}\\ \end{aligned} \right. \end{aligned}$$

(2.1.13)

where $A_{r}$ denotes the Stokes operator with domain $D(A_{r}) := W^{2,{r}}(\varOmega )\cap W^{1,{r}}_0(\varOmega ) \cap L^{r}_{\sigma }(\varOmega )$ and $L^{r}_{\sigma }(\varOmega ) := \{\varphi \in L^{r}(\varOmega )|\nabla \cdot \varphi = 0\}$ for ${r}\in (1,\infty )$ (similar to that in Sohr 2001).

Theorem 2.2

Let $\varOmega \subset \mathbb {R}^3$ be a bounded domain with a smooth boundary, and (2.1.13) hold. Suppose that S satisfies (2.1.4) and (2.1.5) with some $\alpha >\frac{1}{3}$. Then problem (2.1.3) possesses at least one global weak solution (n, c, u, P) in the sense of Definition 2.1.

2.2 Preliminaries

In order to construct the weak solutions to (2.1.6)–(2.1.8) by an approximation procedure, we consider the approximate variant of (2.1.3) given by

$$\begin{aligned} \left\{ \begin{aligned}&n_{\varepsilon t}+u_{\varepsilon }\cdot \nabla n_{\varepsilon } =\varDelta (n_{\varepsilon }+\varepsilon )^m-\nabla \cdot (n_{\varepsilon }S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_{\varepsilon }),\quad x\in \varOmega ,\; t>0, \\&c_{\varepsilon t}+u_{\varepsilon }\cdot \nabla c_{\varepsilon }=\varDelta c_{\varepsilon }-c_{\varepsilon }+n_{\varepsilon },\quad x\in \varOmega ,\; t>0, \\&u_{\varepsilon t}+\nabla P_{\varepsilon }=\varDelta u_{\varepsilon }-\kappa (u_{\varepsilon } \cdot \nabla )u_{\varepsilon }+n_{\varepsilon }\nabla \phi ,\quad x\in \varOmega ,\; t>0, \\&\nabla \cdot u_{\varepsilon }=0,\quad x\in \varOmega ,\; t>0, \\&\displaystyle {\nabla n_{\varepsilon }\cdot \nu =\nabla c_{\varepsilon }\cdot \nu =0,u_{\varepsilon }=0},\quad x\in \partial \varOmega ,\; t>0, \\&\displaystyle {n_{\varepsilon }(x,0)=n_0(x),c_{\varepsilon }(x,0)=c_0(x),\;u_{\varepsilon }(x,0)=u_0(x)},\quad x\in \varOmega , \end{aligned}\right. \end{aligned}$$

(2.2.1)

where $S_\varepsilon (x,n,c):= \rho _\varepsilon (x)\chi _\varepsilon (n)S(x, n, c)$, $n\ge 0,~c\ge 0$, $\rho _\varepsilon \in C^\infty _0 (\varOmega )~~\text{ such } \text{ that } 0\le \rho _\varepsilon \le 1~~\text{ in }~~\varOmega ~~\text{ and }~~\rho _\varepsilon \nearrow 1~~\text{ in }~~\varOmega ~~\text{ as }~~\varepsilon \searrow 0,$ $\chi _\varepsilon \in C^\infty _0 ([0,\infty ))~~\text{ such } \text{ that }~~0\le \chi _\varepsilon \le 1~~\text{ in }~~[0,\infty )~~\text{ and }~~\chi _\varepsilon \nearrow 1~~\text{ in }~~[0,\infty )~~ \text{ as }~~\varepsilon \searrow 0.$

By the well-established fixed-point arguments (see Lemma 2.1 in Winkler 2016v, Winkler 2015b and Lemma 2.1 in Painter and Hillen 2002), we could show the local solvability of system (2.2.1).

Lemma 2.1

Let $\varOmega \subset \mathbb {R}^2$ be a bounded domain with smooth boundary, and assume (2.1.9)–(2.1.11) hold. For any $\varepsilon \in (0,1)$, there exist $T_{max,\varepsilon }\in (0,\infty ]$ and a classical solution $(n_\varepsilon , c_\varepsilon , u_\varepsilon , P_\varepsilon )$ of system (2.2.1) in $\varOmega \times [0,T_{max,\varepsilon })$. Here,

$$\begin{aligned} \left\{ \begin{aligned}&n_\varepsilon \in C^0(\bar{\varOmega }\times [0,T_{max,\varepsilon }))\cap C^{2,1}(\bar{\varOmega }\times (0,T_{max,\varepsilon })), \\&c_\varepsilon \in C^0(\bar{\varOmega }\times [0,T_{max,\varepsilon }))\cap C^{2,1}(\bar{\varOmega }\times (0,T_{max,\varepsilon }))\cap \bigcap _{p>1} L^\infty ([0,T_{max,\varepsilon }); W^{1,p}(\varOmega )), \\&u_\varepsilon \in C^0(\bar{\varOmega }\times [0,T_{max,\varepsilon }))\cap C^{2,1}(\bar{\varOmega }\times (0,T_{max,\varepsilon }))\cap \bigcap _{\gamma \in (0,1)}C^0([0,T_{max,\varepsilon }); D(A^\gamma )), \\&P_\varepsilon \in C^{1,0}(\bar{\varOmega }\times (0,T_{max,\varepsilon })). \end{aligned}\right. \end{aligned}$$

(2.2.2)

Moreover, $n_\varepsilon$ and $c_\varepsilon$ are nonnegative in $\varOmega \times (0, T_{max,\varepsilon })$, and if $T_{max,\varepsilon }<+\infty$, then

$$ \limsup _{t\nearrow T_{max,\varepsilon }}[\Vert n_\varepsilon (\cdot , t)\Vert _{L^\infty (\varOmega )}+\Vert c_\varepsilon (\cdot , t)\Vert _{W^{1,\infty }(\varOmega )}+\Vert A^\gamma u_\varepsilon (\cdot , t)\Vert _{L^{2}(\varOmega )}]=\infty $$

for all $p > 2$ and $\gamma \in (\frac{1}{2}, 1)$.

Lemma 2.2

(Tao and Winkler 2015b) Let $T\in (0,\infty ]$, $\sigma \in (0,T)$, $A>0$ and $B>0$, and suppose that $y:[0,T)\rightarrow [0,\infty )$ is absolutely continuous such that $y'(t)+Ay(t)\le h(t) ~~\text{ for } \text{ a.e. }~~t\in (0,T)$ with some nonnegative function $h\in L^1_{loc}([0, T))$ satisfying $\int _{t}^{t+\sigma }h(s)ds\le B~~\text{ for } \text{ all }~~t\in (0,T-\sigma ).$ Then $y(t)\le \max \{y_0+B,\frac{B}{A\tau }+2B\}~~\text{ for } \text{ all }~~t\in (0,T).$

In light of the strong nonlinear term $(u \cdot \nabla )u$, problem (2.1.3) has no classical solutions in general, thus we consider its weak solutions.

Definition 2.1

Let $T > 0$ and assume that $(n_0, c_0, u_0)$ fulfills (2.1.13). Then a triple of functions (n, c, u) is called a weak solution of (2.1.3) if the following conditions are satisfied:

$$\begin{aligned} \left\{ \begin{aligned} n\in L_{loc}^1(\bar{\varOmega }\times [0,T)), \\ c \in L_{loc}^1([0,T); W^{1,1}(\varOmega )),\\ u \in L_{loc}^1([0,T); W^{1,1}(\varOmega );\mathbb {R}^{3}), \\ \end{aligned}\right. \end{aligned}$$

(2.2.3)

where $n\ge 0$ and $c\ge 0$ in $\varOmega \times (0, T)$ as well as $\nabla \cdot u = 0$ in the distributional sense in $\varOmega \times (0, T)$. Moreover,

$$\begin{aligned} \begin{aligned}&u\otimes u \in L^1_{loc}(\bar{\varOmega }\times [0, \infty );\mathbb {R}^{3\times 3})~\text{ and }~ n~\text{ belongs } \text{ to }~ L^1_{loc}(\bar{\varOmega }\times [0, \infty )),\\&cu,~ nu, ~\text{ and }~nS(x,n,c)\nabla c~ \text{ belong } \text{ to }~ L^1_{loc}(\bar{\varOmega }\times [0, \infty );\mathbb {R}^{3}) \end{aligned} \end{aligned}$$

(2.2.4)

and

$$\begin{aligned} \begin{aligned}&\displaystyle {-\int _0^{T}\int _{\varOmega }n\varphi _t-\int _{\varOmega }n_0\varphi (\cdot ,0) } \\ =&\displaystyle {- \int _0^T\int _{\varOmega }\nabla n\cdot \nabla \varphi +\int _0^T\int _{\varOmega }n S(x,n,c)\nabla c\cdot \nabla \varphi } +\displaystyle {\int _0^T\int _{\varOmega }nu\cdot \nabla \varphi } \end{aligned} \end{aligned}$$

(2.2.5)

for any $\varphi \in C_0^{\infty } (\bar{\varOmega }\times [0, T))$ satisfying $\frac{\partial \varphi }{\partial \nu }= 0$ on $\partial \varOmega \times (0, T)$, as well as

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

(2.2.6)

for any $\varphi \in C_0^{\infty } (\bar{\varOmega }\times [0, T))$ and

$$\begin{aligned} \begin{aligned}&\displaystyle {-\int _0^{T}\int _{\varOmega }u\varphi _t-\int _{\varOmega }u_0\varphi (\cdot ,0) -\kappa \int _0^T\int _{\varOmega } u\otimes u\cdot \nabla \varphi } \\ =&\displaystyle {- \int _0^T\int _{\varOmega }\nabla u\cdot \nabla \varphi - \int _0^T\int _{\varOmega }n\nabla \phi \cdot \varphi } \end{aligned} \end{aligned}$$

(2.2.7)

for any $\varphi \in C_0^{\infty } (\bar{\varOmega }\times [0, T);\mathbb {R}^3)$ fulfilling $\nabla \varphi \equiv 0$ in $\varOmega \times (0, T)$.

If (n, c, u) :  $\varOmega \times (0,\infty )\longrightarrow \mathbb {R}^5$ is a weak solution of (2.1.3) in $\varOmega \times (0, T)$ for all $T > 0$, then (n, c, u) is called a global weak solution of (2.1.3).

To obtain the solution of system (2.1.3), we first consider the following approximate system of (2.1.3):

$$\begin{aligned} \left\{ \begin{aligned}&n_{\varepsilon t}+u_{\varepsilon }\cdot \nabla n_{\varepsilon }=\varDelta n_{\varepsilon }-\nabla \cdot (n_{\varepsilon }F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_{\varepsilon }),\quad x\in \varOmega ,\; t>0,\\&c_{\varepsilon t}+u_{\varepsilon }\cdot \nabla c_{\varepsilon }=\varDelta c_{\varepsilon }-c_{\varepsilon }+F_{\varepsilon }(n_{\varepsilon }),\quad x\in \varOmega ,\; t>0,\\&u_{\varepsilon t}+\nabla P_{\varepsilon }=\varDelta u_{\varepsilon }-\kappa (Y_{\varepsilon }u_{\varepsilon } \cdot \nabla )u_{\varepsilon }+n_{\varepsilon }\nabla \phi ,\quad x\in \varOmega ,\; t>0,\\&\nabla \cdot u_{\varepsilon }=0,\quad x\in \varOmega ,\; t>0,\\&\displaystyle {\nabla n_{\varepsilon }\cdot \nu =\nabla c_{\varepsilon }\cdot \nu =0,u_{\varepsilon }=0,\quad x\in \partial \varOmega ,\; t>0,}\\&\displaystyle {n_{\varepsilon }(x,0)=n_0(x),c_{\varepsilon }(x,0)=c_0(x),\;u_{\varepsilon }(x,0)=u_0(x)},\quad x\in \varOmega ,\\ \end{aligned}\right. \end{aligned}$$

(2.2.8)

where

$$\begin{aligned} F_{\varepsilon }(s):=\frac{1}{\varepsilon }\ln (1+\varepsilon s)~\quad ~\text{ for } \text{ all }~~s \ge 0~~\text{ and }~~\varepsilon > 0, \end{aligned}$$

(2.2.9)

as well as

$$\begin{aligned} \begin{aligned} S_\varepsilon (x, n, c) := \rho _\varepsilon (x)S(x, n, c),~~ x\in \bar{\varOmega },~~n\ge 0,~~c\ge 0 \end{aligned} \end{aligned}$$

(2.2.10)

and

$$ Y_{\varepsilon }w := (1 + \varepsilon A)^{-1}w \quad ~\text{ for } \text{ all }~ w\in L^2_{\sigma }(\varOmega ) $$

is a standard Yosida approximation and A is the realization of the Stokes operator (see Sohr 2001). Here, $(\rho _\varepsilon )_{\varepsilon \in (0,1)} \in C^\infty _0 (\varOmega )$ is a family of standard cutoff functions satisfying $0\le \rho _\varepsilon \le 1$ in $\varOmega$ and $\rho _\varepsilon \nearrow 1$ in $\varOmega$ as $\varepsilon \searrow 0$.

The local solvability of (2.2.8) can be derived by a suitable extensibility criterion and a slight modification of the well-established fixed-point arguments in Lemma 2.1 of Winkler (2016v) (see also Winkler 2015b and Lemma 2.1 of Painter and Hillen 2002), so here we omit the proof.

Lemma 2.3

For each $\varepsilon \in (0,1)$, there exist $T_{max,\varepsilon }\in (0,\infty ]$ and a classical solution $(n_\varepsilon , c_\varepsilon , u_\varepsilon , P_\varepsilon )$ of (2.2.8) in $\varOmega \times (0, T_{max,\varepsilon })$ such that

$$ \left\{ \begin{aligned}&n_\varepsilon \in C^0(\bar{\varOmega }\times [0,T_{max,\varepsilon }))\cap C^{2,1}(\bar{\varOmega }\times (0,T_{max,\varepsilon })),\\&c_\varepsilon \in C^0(\bar{\varOmega }\times [0,T_{max,\varepsilon }))\cap C^{2,1}(\bar{\varOmega }\times (0,T_{max,\varepsilon })),\\&u_\varepsilon \in C^0(\bar{\varOmega }\times [0,T_{max,\varepsilon }); \mathbb {R}^3)\cap C^{2,1}(\bar{\varOmega }\times (0,T_{max,\varepsilon }); \mathbb {R}^3),\\&P_\varepsilon \in C^{1,0}(\bar{\varOmega }\times (0,T_{max,\varepsilon })), \end{aligned}\right. $$

classically solving (2.2.8) in $\varOmega \times [0,T_{max,\varepsilon })$. Moreover, $n_\varepsilon$ and $c_\varepsilon$ are nonnegative in $\varOmega \times (0, T_{max,\varepsilon })$, and

$$ \Vert n_\varepsilon (\cdot , t)\Vert _{L^\infty (\varOmega )}+\Vert c_\varepsilon (\cdot , t)\Vert _{W^{1,\infty }(\varOmega )}+\Vert A^\gamma u_\varepsilon (\cdot , t)\Vert _{L^{2}(\varOmega )}\rightarrow \infty ~~ \text{ as }~~ t\rightarrow T_{max,\varepsilon }, $$

where $\gamma$ is given by (2.1.13).

Lemma 2.4

(Winkler 2010; Zheng 2017c) Let $(e^{\tau \varDelta })_{\tau \ge 0}$ be the Neumann heat semigroup in $\varOmega$ and $p>3$. Then there exist positive constants $k_1:=k_1(\varOmega ),$ $k_2:=k_2(\varOmega )$ and $k_3:=k_3(\varOmega )$ such that for all $\tau >0$ and any $\varphi \in W^{1,p}(\varOmega )$,

$$ \Vert \nabla e^{\tau \varDelta }\varphi \Vert _{L^p(\varOmega )} \le k_1\Vert \nabla \varphi \Vert _{L^p(\varOmega )}, $$

and for all $\tau > 0$ and each $\varphi \in L^\infty (\varOmega )$

$$ \Vert \nabla e^{\tau \varDelta }\varphi \Vert _{L^p(\varOmega )} \le k_2(1+\tau ^{-\frac{1}{2}})\Vert \varphi \Vert _{L^\infty (\varOmega )}, $$

as well as for all $\tau > 0$ and all $\varphi \in C^{1}(\bar{\varOmega }; \mathbb {R}^3)$ fulfilling $\varphi \cdot \nu =0~\text{ on }~\partial \varOmega$

$$ \Vert e^{\tau \varDelta }\nabla \cdot \varphi \Vert _{L^\infty (\varOmega )}\le k_3(1+\tau ^{-\frac{1}{2}-\frac{3}{2p}})\Vert \varphi \Vert _{L^p(\varOmega )}. $$

2.3 Blow-Up Prevention by Nonlinear Diffusion to a Two-Dimensional Keller–Segel–Navier–Stokes System

2.3.1 Some Basic a Priori Estimates

In order to establish the global solvability of system (2.2.1), this section is to derive some necessary estimates for the approximate system (2.2.1). Let us first state two basic estimates on $n_{\varepsilon }$ and $c_{\varepsilon }$.

Lemma 2.5

(Ke and Zheng 2019) The solution of (2.2.1) satisfies

$$\begin{aligned} \int _{\varOmega }{n_{\varepsilon }}= \int _{\varOmega }{n_{0}}~~\text{ for } \text{ all }~~ t\in (0, T_{max,\varepsilon }) \end{aligned}$$

(2.3.1)

as well as

$$ \int _{\varOmega }{c_{\varepsilon }}\le \max \{\int _{\varOmega }{n_{0}},\int _{\varOmega }{c_{0}}\}~~\text{ for } \text{ all }~~ t\in (0, T_{max,\varepsilon }). $$

According to Lemma 2.5, we will derive some information on $(n_{\varepsilon }+\varepsilon )^{m-1}, \nabla (n_{\varepsilon }+\varepsilon )^{m-1}, c_{\varepsilon }^2$ and $|\nabla {c_{\varepsilon }}|^2$. This approach has been undertaken previously in, e.g., Wang et al. (2018), Ke and Zheng (2019) and Liu and Wang (2016).

Lemma 2.6

Let $m>1$. Then there exists $C>0$ independent of $\varepsilon$ such that the solution of (2.2.1) satisfies

$$\begin{aligned} \displaystyle {\int _{\varOmega } (n_{\varepsilon }+\varepsilon )^{m-1}+\int _{\varOmega } c_{\varepsilon }^2 +\int _{\varOmega } | {u_{\varepsilon }}|^2\le C~~~\text{ for } \text{ all }~~ t\in (0, T_{max,\varepsilon })} \end{aligned}$$

(2.3.2)

as well as

$$\begin{aligned} \displaystyle {\int _{t}^{t+\tau }\int _{\varOmega } \left[ (n_{\varepsilon }+\varepsilon )^{2m-4} |\nabla {n_{\varepsilon }}|^2 +|\nabla {c_{\varepsilon }}|^2+ |\nabla {u_{\varepsilon }}|^2\right] \le C} \end{aligned}$$

(2.3.3)

for all $t\in (0, T_{max,\varepsilon }-\tau )$ with $\tau =\min \{1,\frac{1}{6}T_{max,\varepsilon }\}.$

Utilizing the latter spatio-temporal bound for $\nabla (n_{\varepsilon }+\varepsilon )^{m-1}$, we can establish the regularity of $c_\varepsilon$ beyond Lemma 2.6.

Lemma 2.7

Let $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon , P_\varepsilon )$ be the solution of (2.2.1). Then for any $q>2$, there exists $C: = C(q)$ independent of $\varepsilon$ such that

$$\begin{aligned} \Vert c_{\varepsilon }(\cdot , t)\Vert _{L^q(\varOmega )}\le C~~ \text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon }). \end{aligned}$$

(2.3.4)

Proof

Multiplying the second equation in (2.2.1) by ${c^{p-1}_{\varepsilon }}$ with $p>3+4(m-1)$, using the fact $\nabla \cdot u_{\varepsilon }=0$, we have

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{p}\frac{d}{dt}\int _{\varOmega }c^{{{p}}}_{\varepsilon }+({{p}-1}) \int _{\varOmega }c^{{{p}-2}}_{\varepsilon }|\nabla c_{\varepsilon }|^2+\int _{\varOmega }c^{{{p}}}_{\varepsilon }} \\ \le&\displaystyle {\int _\varOmega c^{p-1}_{\varepsilon }(n_{\varepsilon }+\varepsilon )} \\ \le&\displaystyle {\Vert n_{\varepsilon }+\varepsilon \Vert _{L^\frac{p-2(m-1)}{p-4(m-1)}(\varOmega )} \left( \int _\varOmega c^{\frac{(p-1)[p-2(m-1)]}{{m-1}}}_{\varepsilon }\right) ^{\frac{{m-1}}{p-2(m-1)}} ~~}\\ \le&\displaystyle {C_{1}\Vert n_{\varepsilon }+\varepsilon \Vert _{L^\frac{p-2(m-1)}{p-4(m-1)}(\varOmega )}} \displaystyle {(\Vert \nabla c^{\frac{p}{2}}_{\varepsilon }\Vert _{L^2(\varOmega )}^{\frac{p[p-2(m-1)-1]}{[p-1][p-2(m-1)]}}\Vert c^{\frac{p}{2}}_{\varepsilon }\Vert _{L^{\frac{2}{p}}(\varOmega )}^{\frac{{2(m-1)}}{(p-1)[p-2(m-1)]}} +\Vert c^{\frac{p}{2}}_{\varepsilon }\Vert _{L^\frac{2}{p}(\varOmega )})^{\frac{2(p-1)}{p}}}\\ \le&\displaystyle {C_{2} \Vert n_{\varepsilon }+\varepsilon \Vert _{L^\frac{p-2(m-1)}{p-4(m-1)}(\varOmega )}} \displaystyle {(\Vert \nabla c^{\frac{p}{2}}_{\varepsilon }\Vert _{L^2(\varOmega )}^{\frac{2[p-2(m-1)-1]}{p-2(m-1)}}+1)}\\ \le&\displaystyle {\frac{({{p}-1})}{2}\int _{\varOmega }c^{{{p}-2}}_{\varepsilon }|\nabla c_{\varepsilon }|^2 +C_3\Vert n_{\varepsilon }+\varepsilon \Vert _{L^\frac{p-2(m-1)}{p-4(m-1)}(\varOmega )}^{p-2(m-1)}+C_3} \end{aligned} \end{aligned}$$

(2.3.5)

for some positive constants $C_i,$ $(i=1,2,3)$, due to the Gagliardo–Nirenberg inequality.

By appropriate reformulation, if follows from (2.3.5) that for all $t\in (0,T_{max,\varepsilon })$,

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{p}\frac{d}{dt}\int _{\varOmega }c^{{{p}}}_{\varepsilon }+\frac{({{p}-1})}{2}\int _{\varOmega }c^{{{p}-2}}_{\varepsilon }|\nabla c_{\varepsilon }|^2+\int _{\varOmega }c^{{{p}}}_{\varepsilon }} \\ \le&\displaystyle {C_3\Vert n_{\varepsilon }+\varepsilon \Vert _{L^\frac{p-2(m-1)}{p-4(m-1)}(\varOmega )}^{p-2(m-1)}+C_3~\text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon }).}\\ \end{aligned} \end{aligned}$$

(2.3.6)

In the following, we will estimate the integrals on the right-hand side of (2.3.6). In view of the Gagliardo–Nirenberg inequality, for some $C_4>0$ and $C_5> 0$, we may derive from (2.3.3) that

$$ \begin{aligned}&\Vert n_{\varepsilon }+\varepsilon \Vert _{L^\frac{p-2(m-1)}{p-4(m-1)}(\varOmega )}^{p-2(m-1)}\\ =&\Vert (n_{\varepsilon }+\varepsilon )^{m-1}\Vert ^{\frac{p-2(m-1)}{m-1}}_{L^{\frac{p-2(m-1)}{[p-4(m-1)](m-1)}}(\varOmega )} \\ \le&\Vert \nabla { (n_{\varepsilon }+\varepsilon )^{m-1}}\Vert ^{2}_{L^{2}(\varOmega )}\Vert { (n_{\varepsilon }+\varepsilon )^{m-1}}\Vert ^{{\frac{p}{m-1}}}_{L^{\frac{1}{m-1}}(\varOmega )}+ \Vert { (n_{\varepsilon }+\varepsilon )^{m-1}}\Vert ^{\frac{p-2(m-1)}{m-1}}_{L^{\frac{1}{m-1}}(\varOmega )} \\ \le&\Vert \nabla { (n_{\varepsilon }+\varepsilon )^{m-1}}\Vert ^{2}_{L^{2}(\varOmega )} \end{aligned} $$

which along with (2.3.6) and Lemma 2.2 leads to (2.3.4).

Based on the information from Lemma 2.7, we can derive the more regularity property of solutions than that in Lemma 2.6 asserted in the following lemma.

Lemma 2.8

Let $m>1$. Then the solution of (2.2.1) satisfies

$$\begin{aligned} \int _{\varOmega }(n_\varepsilon +\varepsilon )^{m}+\int _{\varOmega }|\nabla c_{\varepsilon }|^{2} \le C ~~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon }) \end{aligned}$$

(2.3.7)

as well as

$$\begin{aligned} \int _{t}^{t+\tau }\int _{\varOmega } |\nabla (n_\varepsilon +\varepsilon )^{\frac{2m-1}{2}}|^2 \le C~~ \text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon }-\tau ), \end{aligned}$$

(2.3.8)

where $\tau =\min \{1,\frac{1}{6}T_{max,\varepsilon }\}.$

Proof

Multiplying the first equation of (2.2.1) by ${(n_{\varepsilon }+\varepsilon )^{m-1}}$ and noticing $\nabla \cdot u_\varepsilon =0$, one obtains

$$ \begin{aligned}&\displaystyle {\frac{1}{{m}}\frac{d}{dt}\Vert n_\varepsilon +\varepsilon \Vert ^{{m}}_{L^{{m}}(\varOmega )} +({m-1})\int _{\varOmega } (n_\varepsilon +\varepsilon )^{{{2{m-3}}}}|\nabla n_\varepsilon |^2} \\ =&\displaystyle {-\int _\varOmega (n_{\varepsilon }+\varepsilon )^{m-1}\nabla \cdot (n_\varepsilon S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon }) \nabla c_\varepsilon ) } \\ \le&\displaystyle { C_S({m-1}) \int _\varOmega (n_{\varepsilon }+\varepsilon ) ^{m-1} |\nabla n_\varepsilon ||\nabla c_\varepsilon |~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon })} \end{aligned} $$

by using (2.1.10). From the Young inequality, it follows that for any $\eta >0$, there exists $C_1(\eta )>0$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{m}}\frac{d}{dt}\Vert n_\varepsilon +\varepsilon \Vert ^{{{m}}}_{L^{{m}}(\varOmega )} +({m-1})\int _{\varOmega } (n_{\varepsilon }+\varepsilon )^{{{{2m-3}}}}|\nabla n_\varepsilon |^2} \\ \le&\displaystyle {\frac{{m-1}}{2}\int _{\varOmega } (n_{\varepsilon }+\varepsilon )^{{{{2m-3}}}}|\nabla n_\varepsilon |^2 +\frac{(m-1)C_S^2}{2}\int _\varOmega (n_{\varepsilon }+\varepsilon ) |\nabla c_\varepsilon |^2}\\ \le&\displaystyle {\frac{{m-1}}{2}\int _{\varOmega } (n_{\varepsilon }+\varepsilon )^{{{{2m-3}}}}|\nabla n_\varepsilon |^2} + \displaystyle {\eta \int _\varOmega (n_\varepsilon +\varepsilon )^{2m}+C_1(\eta )\int _\varOmega |\nabla c_\varepsilon |^{\frac{4m}{2m-1}} .} \end{aligned} \end{aligned}$$

(2.3.9)

On the other hand, in view of Lemma 2.5 and the Gagliardo–Nirenberg inequality, we infer that for some $C_{2}> 0$,

$$\begin{aligned} \int _\varOmega (n_\varepsilon +\varepsilon )^{2m} = \Vert (n_\varepsilon +\varepsilon ) ^{\frac{2{m}-1}{2}}\Vert _{L^{\frac{4m}{2m-1}}(\varOmega )}^{\frac{4m}{2m-1}} \le C_2\Vert \nabla (n_\varepsilon +\varepsilon ) ^{\frac{2{m}-1}{2}}\Vert _{L^{2}(\varOmega )}^{2}+C_2. \end{aligned}$$

(2.3.10)

Inserting (2.3.10) into (2.3.9) and choosing $\eta$ appropriately small, we then get

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{m}}\frac{d}{dt}\Vert n_\varepsilon +\varepsilon \Vert ^{{{m}}}_{L^{{m}}(\varOmega )} +\frac{m-1}{4}\int _{\varOmega } (n_{\varepsilon }+\varepsilon )^{{{{2m-3}}}}|\nabla n_\varepsilon |^2} \\ \le&C_3\displaystyle {\int _\varOmega |\nabla c_\varepsilon |^{\frac{4m}{2m-1}}. } \end{aligned} \end{aligned}$$

In light of (2.3.4), there exist positive constants $l_0>\frac{1}{m-1}$ and $C_2$, such that

$$\begin{aligned} \Vert c_\varepsilon (\cdot , t)\Vert _{L^{l_0}(\varOmega )}\le C_2~~ \text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon }). \end{aligned}$$

(2.3.11)

Next, with the help of the Gagliardo–Nirenberg inequality and (2.3.11), we derive that

$$ \begin{aligned} \displaystyle {\int _\varOmega |\nabla c_\varepsilon |^{\frac{4m}{2m-1}}} \le&\displaystyle {C_4\Vert \varDelta c_\varepsilon \Vert _{L^{2}(\varOmega )}^{a\frac{4m}{2m-1}}\Vert c_\varepsilon \Vert _{L^{l_0}(\varOmega )}^{(1-a)\frac{4m}{2m-1}} +C_4\Vert c_\varepsilon \Vert _{L^{l_0}(\varOmega )}^{\frac{4m}{2m-1}}} \\ \le&\displaystyle {C_{5}\Vert \varDelta c_\varepsilon \Vert _{L^{2}(\varOmega )}^{a\frac{4m}{2m-1}}+C_{5}} \end{aligned} $$

with some positive constants $C_3$ and $C_{4}$, and

$$a=\frac{\frac{1}{2}+\frac{1}{l_0}-\frac{2{m}-1}{4m}}{\frac{1}{2}+\frac{1}{l_0}}\in (0,1),$$

which together with the fact that $\frac{4am}{2m-1}<2$ (due to $l_0>\frac{1}{m-1}$), yields

$$\begin{aligned} \displaystyle {C_3\int _\varOmega |\nabla c_\varepsilon |^{\frac{4m}{2m-1}}\le \frac{1}{8}\Vert \varDelta c_\varepsilon \Vert _{L^{2}(\varOmega )}^{2}+C_{6}.} \end{aligned}$$

(2.3.12)

On the other hand, taking $-\varDelta {c_{\varepsilon }}$ as the test function for the second equation of (2.2.1), and using the Young inequality, it yields that for all $t\in (0,T_{max,\varepsilon })$

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{1}{{2}}\displaystyle \frac{d}{dt}\Vert \nabla {c_{\varepsilon }}\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}+ \int _{\varOmega } |\varDelta c_{\varepsilon }|^2+ \int _{\varOmega } | \nabla c_{\varepsilon }|^2 \\ =&\displaystyle {-\int _{\varOmega } n_{\varepsilon }\varDelta c_{\varepsilon }-\int _{\varOmega }\nabla c_{\varepsilon }\nabla (u_{\varepsilon }\cdot \nabla c_{\varepsilon })} \\ =&\displaystyle {-\int _{\varOmega } n_{\varepsilon }\varDelta c_{\varepsilon }-\int _{\varOmega }\nabla c_{\varepsilon }(\nabla u_{\varepsilon }\cdot \nabla c_{\varepsilon })}\\ \le&\displaystyle {\frac{1}{2}} \int _{\varOmega }|\varDelta c_{\varepsilon }|^2+\displaystyle {\frac{1}{2}} \int _{\varOmega }n_{\varepsilon }^2 + \displaystyle {\Vert \nabla u_{\varepsilon }\Vert _{L^{2}(\varOmega )}\Vert \nabla c_{\varepsilon }\Vert _{L^{4}(\varOmega )}^2}\\ \le&\displaystyle {\frac{1}{2}} \int _{\varOmega }|\varDelta c_{\varepsilon }|^2+\displaystyle {\frac{1}{2}} \int _{\varOmega }n_{\varepsilon }^2 + \displaystyle {C_{7}\Vert \nabla u_{\varepsilon }\Vert _{L^{2}(\varOmega )}\Vert \varDelta c_{\varepsilon }\Vert _{L^{2}(\varOmega )}\Vert \nabla c_{\varepsilon }\Vert _{L^{2}(\varOmega )}}\\ \le&\displaystyle {\frac{3}{4}} \int _{\varOmega }|\varDelta c_{\varepsilon }|^2+\displaystyle {\frac{1}{2}} \int _{\varOmega }n_{\varepsilon }^2 + \displaystyle {C_{8}\Vert \nabla u_{\varepsilon }\Vert ^2_{L^{2}(\varOmega )}\Vert \nabla c_{\varepsilon }\Vert ^2_{L^{2}(\varOmega )}} \end{aligned} \end{aligned}$$

(2.3.13)

where we have used the fact that

$$ \displaystyle {\int _{\varOmega }\nabla c_{\varepsilon }\cdot (D^2 c_{\varepsilon }\cdot u_{\varepsilon }) =\frac{1}{2}\int _{\varOmega } u_{\varepsilon }\cdot \nabla |\nabla c_{\varepsilon }|^2=0 ~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon })} $$

as well as

$$ \displaystyle \Vert \nabla c_{\varepsilon }\Vert _{L^{4}(\varOmega )}^2\le \displaystyle {C_{7}\Vert \varDelta c_{\varepsilon }\Vert _{L^{2}(\varOmega )}\Vert \nabla c_{\varepsilon }\Vert _{L^{2}(\varOmega )} ~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon })} $$

for some $C_{7}> 0$ by the elliptic regularity (Gilbarg and Trudinger 2001).

Hence by appropriate reformulation, (2.3.9), (2.3.12) and (2.3.13), we derive that for all $t\in (0,T_{max,\varepsilon })$,

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{d}{dt}(\Vert n_\varepsilon +\varepsilon \Vert ^{{{m}}}_{L^{{m}}(\varOmega )}+ \displaystyle \Vert \nabla {c_{\varepsilon }}\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}) +{C_8}\int _{\varOmega } | \nabla (n_\varepsilon +\varepsilon ) ^{\frac{2{m}-1}{2}}|^2} \\&+\displaystyle {C_8\int _{\varOmega } (|\varDelta c_{\varepsilon }|^2+ | \nabla c_{\varepsilon }|^2)}\\ \le&\displaystyle {C_9\int _\varOmega (n_\varepsilon +\varepsilon ) ^{2}+ C_9\Vert \nabla u_{\varepsilon }\Vert _{L^{2}(\varOmega )}^2\Vert \nabla c_{\varepsilon }\Vert _{L^{2}(\varOmega )}^2+C_{9}~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon })} \end{aligned} \end{aligned}$$

(2.3.14)

with some $C_8>0$ and $C_9>0$. So by (2.3.10) and $m>1$, we can see that

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{d}{dt}(\Vert n_\varepsilon +\varepsilon \Vert ^{{{m}}}_{L^{{m}}(\varOmega )}+ \displaystyle \Vert \nabla {c_{\varepsilon }}\Vert ^{{{2}}}_{L^{{2}}(\varOmega )})+ \frac{C_8}{2}\int _{\varOmega } | \nabla (n_\varepsilon +\varepsilon ) ^{\frac{2{m}-1}{2}}|^2} \\&+\displaystyle {C_8\int _{\varOmega } (|\varDelta c_{\varepsilon }|^2+ | \nabla c_{\varepsilon }|^2)}\\ \le&\displaystyle { C_9\Vert \nabla u_{\varepsilon }\Vert _{L^{2}(\varOmega )}^2\Vert \nabla c_{\varepsilon }\Vert _{L^{2}(\varOmega )}^2+C_{10}~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon })} \end{aligned} \end{aligned}$$

(2.3.15)

for some constant $C_{10}>0$.

Note that from the Gagliardo–Nirenberg inequality and (2.3.3), it follows that there exist constants $C_{i}> 0$, $(i=11,12,13)$, such that

$$\begin{aligned} \begin{aligned}&\displaystyle \int _{t}^{t+\tau }\int _\varOmega (n_\varepsilon +\varepsilon ) ^{{m}} \\ =&\displaystyle \int _{t}^{t+\tau }\Vert (n_\varepsilon +\varepsilon ) ^{m-1}\Vert _{L^{\frac{m}{m-1}}(\varOmega )}^{\frac{m}{m-1}} \\ \le&C_{11}\left( \displaystyle \int _{t}^{t+\tau }\Vert \nabla (n_\varepsilon +\varepsilon ) ^{m-1}\Vert _{L^{2}(\varOmega )}^ {\frac{{m-1}}{m}} \Vert (n_\varepsilon +\varepsilon ) ^{m-1}\Vert _{L^{\frac{1}{m-1}}(\varOmega )}^{\frac{1}{m}}\right. \\&+\left. \displaystyle \int _{t}^{t+\tau }\Vert (n_\varepsilon +\varepsilon \right) ^{m-1}\Vert _{L^{\frac{1}{m-1}}(\varOmega )})^{\frac{{m}}{m-1}}\\ \le&C_{12}\displaystyle \int _{t}^{t+\tau }\Vert \nabla (n_\varepsilon +\varepsilon ) ^{m-1}\Vert _{L^{2}(\varOmega )}^{2}+C_{12} \\ \le&C_{13}. \end{aligned} \end{aligned}$$

(2.3.16)

Therefore, if we write $y(t) :=\Vert n_\varepsilon (\cdot , t)+\varepsilon \Vert ^{{{m}}}_{L^{{m}}(\varOmega )}+ \Vert \nabla {c_{\varepsilon }}(\cdot , t)\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}+1$ and $\rho (t) =C_{9}\int _{\varOmega }|\nabla u_{\varepsilon }(\cdot , t)|^2+C_{10},$ (2.3.15) implies that

$$\begin{aligned} y'(t)+h(t) \le \displaystyle { \rho (t)y(t)~~~\text{ for } \text{ all }~t\in (0,T_{max,\varepsilon }),} \end{aligned}$$

(2.3.17)

with

$$ h(t)=\frac{C_8}{2} \int _{\varOmega } |\nabla (n_\varepsilon +\varepsilon )^{\frac{2m-1}{2}}|^2 +C_8\int _{\varOmega } |\varDelta c_{\varepsilon }|^2. $$

Next, by estimates (2.3.3) and (2.3.16), one obtains

$$\begin{aligned} \int _{t}^{t+\tau }\rho (s)ds\le \displaystyle { C_{14}} \end{aligned}$$

(2.3.18)

as well as

$$\begin{aligned} \int _{t}^{t+\tau }y(s)ds\le \displaystyle { C_{14}}, \end{aligned}$$

(2.3.19)

for all $t\in (0,T_{max,\varepsilon }-\tau )$ and some $C_{14}>0$. For given $t\in (0, T_{max,\varepsilon })$, by estimate (2.3.19), one can see that there exists $t_0 \ge 0$ such that $t_0\in [t-\tau , t]$, $y(\cdot ,t_0)\le \frac{ C_{14}}{\tau }$ and hence

$$\begin{aligned} y(t)\le \displaystyle {y(t_0)e^{\int _{t_0}^t\rho (s)ds}} \le \displaystyle {\frac{C_{14}}{\tau } e^{C_{14}}} \end{aligned}$$

(2.3.20)

due to (2.3.18) and the Gronwall lemma. Moreover, combining (2.3.17), (2.3.18) and (2.3.20), we immediately get the desired inequality (2.3.8).

In order to obtain the global boundedness of $n_{\varepsilon }$, a further regularity of $\nabla n_{\varepsilon }$ beyond (2.3.7) seems to be required. Indeed, drawing on (2.3.3) and (2.3.8), we first have the following.

Lemma 2.9

Let $m>1.$ There exists a positive constant C independent of $\varepsilon$, such that

$$\begin{aligned} \int _{\varOmega }{|\nabla u_{\varepsilon }(\cdot ,t)|^2}\le C~~\text{ for } \text{ all }~~ t\in (0, T_{max,\varepsilon }). \end{aligned}$$

(2.3.21)

Proof

Applying the Helmholtz projection to the third equation in (2.2.1), and also multiplying the result identified by $Au_{\varepsilon }$, we then find that

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{2}}\frac{d}{dt}\Vert A^{\frac{1}{2}}u_{\varepsilon }\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}+ \int _{\varOmega }|Au_{\varepsilon }|^2 } \\ =&\displaystyle { \int _{\varOmega }Au_{\varepsilon }\mathscr {P}(-\kappa (Y_{\varepsilon }u_{\varepsilon } \cdot \nabla )u_{\varepsilon })+ \int _{\varOmega }\mathscr {P}(n_{\varepsilon }\nabla \phi ) Au_{\varepsilon }} \\ \le&\displaystyle { \frac{1}{2}\int _{\varOmega }|Au_{\varepsilon }|^2+\kappa ^2\int _{\varOmega } |(Y_{\varepsilon }u_{\varepsilon } \cdot \nabla )u_{\varepsilon }|^2} \\&+ \Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}\int _{\varOmega }n_{\varepsilon }^2~~\text{ for } \text{ all }~~t\in (0,T_{max,\varepsilon }). \end{aligned} \end{aligned}$$

(2.3.22)

Noticing that since $\Vert Y_{\varepsilon }u_{\varepsilon }\Vert _{L^2(\varOmega )}\le \Vert u_{\varepsilon }\Vert _{L^2(\varOmega )},$ it follows from the Gagliardo–Nirenberg inequality and the Cauchy–Schwarz inequality that with some $C_1 >0$ and $C_2 > 0$

$$\begin{aligned} \begin{aligned}&\kappa ^2\displaystyle \int _{\varOmega } |(Y_{\varepsilon }u_{\varepsilon } \cdot \nabla )u_{\varepsilon }|^2\\ \le&\displaystyle { \kappa ^2\Vert Y_{\varepsilon }u_{\varepsilon }\Vert ^2_{L^4(\varOmega )}\Vert \nabla u_{\varepsilon }\Vert ^2_{L^4(\varOmega )}} \\ \le&\displaystyle { \kappa ^2C_1[\Vert \nabla Y_{\varepsilon }u_{\varepsilon }\Vert _{L^2(\varOmega )}\Vert Y_{\varepsilon }u_{\varepsilon }\Vert _{L^2(\varOmega )}] [\Vert A u_{\varepsilon }\Vert _{L^2(\varOmega )}\Vert \nabla u_{\varepsilon }\Vert _{L^2(\varOmega )}] }\\ \le&\displaystyle { \kappa ^2C_{2}\Vert \nabla Y_{\varepsilon }u_{\varepsilon }\Vert _{L^2(\varOmega )} \Vert A u_{\varepsilon }\Vert _{L^2(\varOmega )}\Vert \nabla u_{\varepsilon }\Vert _{L^2(\varOmega )} ~~\text{ for } \text{ all }~~t\in (0,T_{max,\varepsilon })}. \end{aligned} \end{aligned}$$

(2.3.23)

Now, from the fact that $D( A^{\frac{1}{2}}) :=W^{1,2}_0(\varOmega ;\mathbb {R}^2) \cap L_{\sigma }^{2}(\varOmega )$ and (2.3.2), it follows that

$$\begin{aligned} \Vert \nabla Y_{\varepsilon }u_{\varepsilon }\Vert _{L^2(\varOmega )}=\Vert A^{\frac{1}{2}} Y_{\varepsilon }u_{\varepsilon }\Vert _{L^2(\varOmega )} =\Vert Y_{\varepsilon } A^{\frac{1}{2}} u_{\varepsilon }\Vert _{L^2(\varOmega )}\le \Vert A^{\frac{1}{2}} u_{\varepsilon }\Vert _{L^2(\varOmega )}\le \Vert \nabla u_{\varepsilon }\Vert _{L^2(\varOmega )}. \end{aligned}$$

This, together with (2.3.23), yields

$$ \begin{aligned}&\kappa ^2\displaystyle \int _{\varOmega }|(Y_{\varepsilon }u_{\varepsilon } \cdot \nabla )u_{\varepsilon }|^2 \\ \le&\displaystyle { C_{3}\Vert A u_{\varepsilon }\Vert _{L^2(\varOmega )}\Vert \nabla u_{\varepsilon }\Vert _{L^2(\varOmega )}^2} \\ \le&\displaystyle { \frac{1}{4}\Vert A u_{\varepsilon }\Vert ^2_{L^2(\varOmega )}+C_{3}^2\Vert \nabla u_{\varepsilon }\Vert _{L^2(\varOmega )}^4 ~~\text{ for } \text{ all }~~t\in (0,T_{max,\varepsilon }),} \end{aligned} $$

which combining with (2.3.22) implies that

$$ \displaystyle \frac{1}{{2}}\frac{d}{dt}\Vert A^{\frac{1}{2}}u_{\varepsilon }\Vert ^{{{2}}}_{L^{{2}}(\varOmega )} \le \displaystyle {C_{3}^2\Vert \nabla u_{\varepsilon }\Vert _{L^2(\varOmega )}^4+ \Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}\int _{\varOmega }n_{\varepsilon }^2~~\text{ for } \text{ all }~t\in (0,T_{max,\varepsilon }).} $$

By the fact that $\Vert A^{\frac{1}{2}}u_{\varepsilon }\Vert ^{{{2}}}_{L^{{2}}(\varOmega )} = \Vert \nabla u_{\varepsilon }\Vert ^{{{2}}}_{L^{{2}}(\varOmega )},$ we conclude that

$$\begin{aligned} z'(t)\le \rho (t)z(t)+ h(t)\displaystyle {~~\text{ for } \text{ all }~~t\in (0,T_{max,\varepsilon })}, \end{aligned}$$

(2.3.24)

where $z(t) :=\int _{\varOmega }|\nabla u_{\varepsilon }(\cdot , t)|^2,$ as well as $\rho (t) =2C_{3}^2\int _{\varOmega }|\nabla u_{\varepsilon }(\cdot , t)|^2$ and

$$ h(t)=2 \Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}\int _{\varOmega }n_{\varepsilon }^2(\cdot ,t). $$

Note that (2.3.3), (2.3.8) and (2.3.10) warrant that for some positive constant $C_4$,

$$ \displaystyle \int _{t}^{t+\tau } (\rho (s)+ h(s))ds \le C_4. $$

Therefore, by an argument similar to the proof of (2.3.8), we can arrive at (2.3.21) and thus complete the proof of this lemma.

Lemma 2.10

Let $m>1$. Then there exists a positive constant C independent of $\varepsilon$ such that the solution of (2.2.1) satisfies

$$\begin{aligned} \displaystyle {\Vert \nabla c_\varepsilon (\cdot , t)\Vert _{L^{2m}(\varOmega )}\le C~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon }).} \end{aligned}$$

(2.3.25)

Proof

Considering the fact that $\nabla c_{\varepsilon }\cdot \nabla \varDelta c_{\varepsilon } = \frac{1}{2}\varDelta |\nabla c_{\varepsilon }|^2-|D^2c_{\varepsilon }|^2$, the straightforward computation implies that

$$\begin{aligned}&\displaystyle {\frac{1}{{2m}}\frac{d}{dt} \Vert \nabla c_{\varepsilon }\Vert ^{{{2m}}}_{L^{{2m}}(\varOmega )}} \nonumber \\ =&\displaystyle {\int _{\varOmega } |\nabla c_{\varepsilon }|^{2m-2}\nabla c_{\varepsilon }\cdot \nabla (\varDelta c_{\varepsilon } -c_{\varepsilon }+n_{\varepsilon }-u_{\varepsilon }\cdot \nabla c_{\varepsilon })} \nonumber \\ =&\displaystyle {\frac{1}{{2}}\int _{\varOmega } |\nabla c_{\varepsilon }|^{2m-2}\varDelta |\nabla c_{\varepsilon }|^2 -\int _{\varOmega } |\nabla c_{\varepsilon }|^{2m-2}|D^2 c_{\varepsilon }|^2-\int _{\varOmega } |\nabla c_{\varepsilon }|^{2m}} \nonumber \\&-\displaystyle {\int _\varOmega n_{\varepsilon }\nabla \cdot ( |\nabla c_{\varepsilon }|^{2m-2}\nabla c_{\varepsilon }) +\int _\varOmega (u_{\varepsilon }\cdot \nabla c_{\varepsilon })\nabla \cdot ( |\nabla c_{\varepsilon }|^{2m-2}\nabla c_{\varepsilon })} \\ =&\displaystyle {-\frac{m-1}{{2}}\int _{\varOmega } |\nabla c_{\varepsilon }|^{2m-4}\left| \nabla |\nabla c_{\varepsilon }|^{2}\right| ^2 +\frac{1}{{2}}\int _{\partial \varOmega } |\nabla c_{\varepsilon }|^{2m-2}\frac{\partial |\nabla c_{\varepsilon }|^{2}}{\partial \nu } -\int _{\varOmega } |\nabla c_{\varepsilon }|^{2m}} \nonumber \\&-\displaystyle {\int _{\varOmega } |\nabla c_{\varepsilon }|^{2m-2}|D^2 c_{\varepsilon }|^2 -\int _\varOmega n_{\varepsilon } |\nabla c_{\varepsilon }|^{2m-2}\varDelta c_{\varepsilon }-\int _\varOmega n_{\varepsilon }\nabla c_{\varepsilon }\cdot \nabla ( |\nabla c_{\varepsilon }|^{2m-2})} \nonumber \\&+\displaystyle {\int _\varOmega (u_{\varepsilon }\cdot \nabla c_{\varepsilon }) |\nabla c_{\varepsilon }|^{2m-2}\varDelta c_{\varepsilon } +\int _\varOmega (u_{\varepsilon }\cdot \nabla c_{\varepsilon })\nabla c_{\varepsilon }\cdot \nabla ( |\nabla c_{\varepsilon }|^{2m-2})} \nonumber \\ =&\displaystyle {-\frac{2({m}-1)}{{{m}^2}}\int _{\varOmega }\left| \nabla |\nabla c_{\varepsilon }|^{m}\right| ^2 +\frac{1}{{2}}\int _{\partial \varOmega } |\nabla c_{\varepsilon }|^{2m-2}\frac{\partial |\nabla c_{\varepsilon }|^{2}}{\partial \nu } -\int _{\varOmega } |\nabla c_{\varepsilon }|^{2m-2}|D^2 c_{\varepsilon }|^2} \nonumber \\&-\displaystyle {\int _\varOmega n_{\varepsilon } |\nabla c_{\varepsilon }|^{2m-2}\varDelta c_{\varepsilon } -\int _\varOmega n_{\varepsilon }\nabla c_{\varepsilon }\cdot \nabla ( |\nabla c_{\varepsilon }|^{2m-2})-\int _{\varOmega } |\nabla c_{\varepsilon }|^{2m}} \nonumber \\&+\displaystyle {\int _\varOmega (u_{\varepsilon }\cdot \nabla c_{\varepsilon }) |\nabla c_{\varepsilon }|^{2m-2}\varDelta c_{\varepsilon } +\int _\varOmega (u_{\varepsilon }\cdot \nabla c_{\varepsilon })\nabla c_{\varepsilon }\cdot \nabla ( |\nabla c_{\varepsilon }|^{2m-2})} \nonumber \end{aligned}$$

(2.3.26)

for all $t\in (0,T_{max,\varepsilon })$. Since $|\varDelta c_{\varepsilon }| \le \sqrt{2}|D^2c_{\varepsilon }|$, we can estimate

$$\begin{aligned} \begin{aligned} \displaystyle \int _\varOmega n_{\varepsilon } |\nabla c_{\varepsilon }|^{2m-2}\varDelta c_{\varepsilon } \le&\displaystyle {\sqrt{2}\int _\varOmega n_{\varepsilon } |\nabla c_{\varepsilon }|^{2m-2}|D^2c_{\varepsilon }|} \\ \le&\displaystyle {\frac{1}{4}\int _\varOmega |\nabla c_{\varepsilon }|^{2m-2}|D^2c_{\varepsilon }|^2+{2}\int _\varOmega n^2_{\varepsilon } |\nabla c_{\varepsilon }|^{2m-2}} \\ \le&\displaystyle {\frac{1}{4}\int _\varOmega |\nabla c_{\varepsilon }|^{2m-2}|D^2c_{\varepsilon }|^2+{2}\int _\varOmega (n_{\varepsilon }+\varepsilon )^2 |\nabla c_{\varepsilon }|^{2m-2}} \end{aligned} \end{aligned}$$

(2.3.27)

and

$$\begin{aligned} \begin{aligned} \displaystyle \int _\varOmega (u_{\varepsilon }\cdot \nabla c_{\varepsilon }) |\nabla c_{\varepsilon }|^{2{m}-2}\varDelta c_{\varepsilon } \le&\displaystyle {\sqrt{2}\int _\varOmega |u_{\varepsilon }\cdot \nabla c_{\varepsilon }| |\nabla c_{\varepsilon }|^{2{m}-2}|D^2c_{\varepsilon }|} \\ \le&\displaystyle {\frac{1}{4}\int _\varOmega |\nabla c_{\varepsilon }|^{2{m}-2}|D^2c_{\varepsilon }|^2 +2\int _\varOmega |u_{\varepsilon }\cdot \nabla c_{\varepsilon }|^2 |\nabla c_{\varepsilon }|^{2{m}-2}} \\ \le&\displaystyle {\frac{1}{4}\int _\varOmega |\nabla c_{\varepsilon }|^{2{m}-2}|D^2c_{\varepsilon }|^2 +2\int _\varOmega |u_{\varepsilon }|^2 |\nabla c_{\varepsilon }|^{2{m}}} \\ \le&\displaystyle {\frac{1}{4}\int _\varOmega |\nabla c_{\varepsilon }|^{2{m}-2}|D^2c_{\varepsilon }|^2 +2\int _\varOmega |u_{\varepsilon }|^2 |\nabla c_{\varepsilon }|^{2{m}}} \end{aligned} \end{aligned}$$

(2.3.28)

for all $t\in (0,T_{max,\varepsilon })$. Again, from the Young inequality, we have

$$\begin{aligned} \begin{aligned}&-\displaystyle \int _\varOmega n_{\varepsilon }\nabla c_{\varepsilon }\cdot \nabla ( |\nabla c_{\varepsilon }|^{2{m}-2}) \\ =&\displaystyle {-({m}-1)\int _\varOmega n_{\varepsilon } |\nabla c_{\varepsilon }|^{2({m}-2)}\nabla c_{\varepsilon }\cdot \nabla |\nabla c_{\varepsilon }|^{2}} \\ \le&\displaystyle {\frac{{m}-1}{8}\int _{\varOmega } |\nabla c_{\varepsilon }|^{2{m}-4}\left| \nabla |\nabla c_{\varepsilon }|^{2}\right| ^2+2({m}-1) \int _\varOmega |n_{\varepsilon }|^2 |\nabla c_{\varepsilon }|^{2{m}-2}} \\ \le&\displaystyle {\frac{({m}-1)}{2{{m}^2}}\int _{\varOmega }\left| \nabla |\nabla c_{\varepsilon }|^{m}\right| ^2+2({m}-1) \int _\varOmega |n_{\varepsilon }|^2 |\nabla c_{\varepsilon }|^{2{m}-2}} \end{aligned} \end{aligned}$$

(2.3.29)

and

$$\begin{aligned}&\displaystyle {\int _\varOmega (u_{\varepsilon }\cdot \nabla c_{\varepsilon })\nabla c_{\varepsilon }\cdot \nabla ( |\nabla c_{\varepsilon }|^{2{m}-2})} \nonumber \\ =&\displaystyle {({m}-1)\int _\varOmega (u_{\varepsilon }\cdot \nabla c_{\varepsilon }) |\nabla c_{\varepsilon }|^{2(m-2)}\nabla c_{\varepsilon }\cdot \nabla |\nabla c_{\varepsilon }|^{2}} \nonumber \\ \le&\displaystyle {\frac{{m}-1}{8}\int _{\varOmega } |\nabla c_{\varepsilon }|^{2{m}-4}\left| \nabla |\nabla c_{\varepsilon }|^{2}\right| ^2} \\&+\displaystyle {2({m}-1)\int _\varOmega |u_{\varepsilon }\cdot \nabla c_{\varepsilon }|^2 |\nabla c_{\varepsilon }|^{2{m}-2}} \nonumber \\ \le&\displaystyle {\frac{({m}-1)}{2{{m}^2}}\int _{\varOmega }\left| \nabla |\nabla c_{\varepsilon }|^{m}\right| ^2 +2({m}-1)\int _\varOmega |u_{\varepsilon }|^2 |\nabla c_{\varepsilon }|^{2{m}}.} \nonumber \end{aligned}$$

(2.3.30)

Observe that

$$\begin{aligned} \begin{aligned} \displaystyle {\int _{\partial \varOmega }\frac{\partial |\nabla c_{\varepsilon }|^2}{\partial \nu } |\nabla c_{\varepsilon }|^{2{m}-2} } \le&\displaystyle {C_\varOmega \int _{\partial \varOmega } |\nabla c_{\varepsilon }|^{2{m}} }\\ =&\displaystyle {C_\varOmega | |\nabla c_{\varepsilon }|^{m}\Vert ^2_{L^2(\partial \varOmega )}.}\\ \end{aligned} \end{aligned}$$

(2.3.31)

Due to Proposition 4.22 (ii) in Haroske and Triebel (2008), $W^{r+\frac{1}{2},2}(\varOmega )\hookrightarrow L^2(\partial \varOmega )$ with $r\in (0,\frac{1}{2})$ is compact and thus

$$\begin{aligned} \begin{aligned}&\displaystyle {\Vert |\nabla c_{\varepsilon }|^{m}\Vert ^2_{L^2{(\partial \varOmega })}\le C_1\Vert |\nabla c_{\varepsilon }|^{m}\Vert ^2_{W^{r+\frac{1}{2},2}(\varOmega )}.}\\ \end{aligned} \end{aligned}$$

(2.3.32)

Therefore, from the fractional Gagliardo–Nirenberg inequality and Lemma 2.8, it follows that for some positive constants $C_2$ and $C_3$,

$$\begin{aligned} \begin{aligned}&\displaystyle {\Vert |\nabla c_{\varepsilon }|^{m}\Vert ^2_{W^{r+\frac{1}{2},2}(\varOmega )}} \\ \le&\displaystyle {C_2\Vert \nabla |\nabla c_{\varepsilon }|^{m}\Vert ^{2a}_{L^2(\varOmega )}\Vert |\nabla c_{\varepsilon }|^m\Vert ^{2-2a}_{L^{\frac{2}{m}}(\varOmega )} +\delta _1\Vert |\nabla c_2|^m\Vert ^2_{L^{\frac{2}{m}}(\varOmega )}} \\ \le&\displaystyle {C_3\Vert \nabla |\nabla c_{\varepsilon }|^{m}\Vert ^a_{L^2(\varOmega )}+C_3} \end{aligned} \end{aligned}$$

(2.3.33)

with $a=\frac{2{m}+2r-1}{2{m}}$. Note that $r\in (0,\frac{1}{2})$ and $m>1$, $0< a<1$. Hence, combining (2.3.31)–(2.3.33) and using the fact that $a\in (0,1)$, we can see that

$$\begin{aligned} \begin{aligned} \displaystyle \int _{\partial \varOmega }\frac{\partial |\nabla c_{\varepsilon }|^2}{\partial \nu } |\nabla c_{\varepsilon }|^{2{m}-2} \le \displaystyle {\frac{({m}-1)}{2{{m}^2}}\int _{\varOmega }\left| \nabla |\nabla c_{\varepsilon }|^{m}\right| ^2+C_4}. \end{aligned} \end{aligned}$$

(2.3.34)

Now from (2.3.26)–(2.3.30) and (2.3.34), we obtain that for some positive constant $C_5$,

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{2{m}}}\frac{d}{dt}\Vert \nabla c_{\varepsilon }\Vert ^{{{2{m}}}}_{L^{{2{m}}}(\varOmega )} +\frac{{m}-1}{2{{m}^2}}\int _{\varOmega }\left| \nabla |\nabla c_{\varepsilon }|^{{m}}\right| ^2 +\frac{1}{2}\int _\varOmega |\nabla c_{\varepsilon }|^{2{m}-2}|D^2c_{\varepsilon }|^2+\int _{\varOmega } |\nabla c_{\varepsilon }|^{2{m}}} \\ \le&\displaystyle {2{m}\int _\varOmega n^2_{\varepsilon } |\nabla c_{\varepsilon }|^{2{m}-2} +2{m}\int _\varOmega |u_{\varepsilon }|^2 |\nabla c_{\varepsilon }|^{2{m}}+C_5~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon }).} \end{aligned} \end{aligned}$$

(2.3.35)

Next we turn to estimate terms on the right of (2.3.35). By the Young inequality, we have

$$\begin{aligned} \begin{aligned} \displaystyle {2{m}\int _\varOmega n^2_{\varepsilon } |\nabla c_{\varepsilon }|^{2{m}-2}} \le&\displaystyle {2{m}\int _\varOmega (n_{\varepsilon }+\varepsilon )^2 |\nabla c_{\varepsilon }|^{2{m}-2}} \\ \le&\displaystyle {\frac{1}{2}\int _\varOmega |\nabla c_{\varepsilon }|^{2{m}}+C_5\int _\varOmega (n_{\varepsilon }+\varepsilon )^{2m}~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon })} \end{aligned} \end{aligned}$$

(2.3.36)

and

$$\begin{aligned} 2{m}\displaystyle \int _\varOmega |u_{\varepsilon }|^2 |\nabla c_{\varepsilon }|^{2{m}} \le \displaystyle {\int _\varOmega |\nabla c_{\varepsilon }|^{2{m}+1}+C_6\int _\varOmega u_{\varepsilon }^{4{m}+2}~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon }),} \end{aligned}$$

(2.3.37)

where $C_5=\frac{m}{m-1}\left( \frac{1}{2}m\right) ^{-\frac{1}{m-1}}(2{m})^{m}$ and $C_6=(2{m})^{2{m}+1}$. On the other hand due to (2.3.7), we derive from the Gagliardo–Nirenberg inequality that for some positive constants $C_i$, $(i=7,8,9)$,

$$ \begin{aligned} \displaystyle {\int _\varOmega |\nabla c_{\varepsilon }|^{2{m}+1}} =&\displaystyle {\Vert |\nabla c_{\varepsilon }|^{m}\Vert _{L^{\frac{2{m}+1}{m}}(\varOmega )}^{\frac{2{m}+1}{m}}} \\ \le&\displaystyle {C_7(\Vert \nabla |\nabla c_{\varepsilon }|^m\Vert _{L^2(\varOmega )}^{\frac{2m-1}{2m+1}}\Vert |\nabla c_{\varepsilon }|^m\Vert _{L^\frac{2}{m}(\varOmega )}^{\frac{2}{2m+1}} +\Vert |\nabla c_{\varepsilon }|^m\Vert _{L^\frac{2}{m} (\varOmega )})^{\frac{2{m}+1}{m}}} \\ \le&\displaystyle {C_8(\Vert \nabla |\nabla c_{\varepsilon }|^m\Vert _{L^2(\varOmega )}^{\frac{2m-1}{m}}+1),}\\ \le&\displaystyle {\frac{{m}-1}{2{{m}^2}}\int _{\varOmega } \left| \nabla |\nabla c_{\varepsilon }|^{{m}}\right| ^2+C_9} \end{aligned} $$

which along with (2.3.37) implies that

$$\begin{aligned} \begin{aligned}&2{m}\displaystyle \int _\varOmega |u_{\varepsilon }|^2 |\nabla c_{\varepsilon }|^{2{m}} \\ \le&\displaystyle {\frac{{m}-1}{2{{m}^2}}\int _{\varOmega } \left| \nabla |\nabla c_{\varepsilon }|^{{m}}\right| ^2+C_6\int _\varOmega u_{\varepsilon }^{4{m}+2}+C_9~~~\text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon }).} \end{aligned} \end{aligned}$$

(2.3.38)

Substituting (2.3.36) and (2.3.38) into (2.3.35), we have

$$ \begin{aligned}&\displaystyle \frac{1}{{2{m}}}\frac{d}{dt}\Vert \nabla c_{\varepsilon }\Vert ^{{{2{m}}}}_{L^{{2{m}}}(\varOmega )}+\frac{1}{{2{}}}\int _{\varOmega } |\nabla c_{\varepsilon }|^{2{m}} \\ \le&\displaystyle {C_5\int _\varOmega (n_{\varepsilon }+\varepsilon )^{2m}+C_6\int _\varOmega u_{\varepsilon }^{4{m}+2}+C_{10}~~~\text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon }).} \end{aligned} $$

Hence, due to $W^{1,2}(\varOmega )\hookrightarrow L^p(\varOmega )$ for any $p>1,$ the boundedness of $\Vert \nabla u_{\varepsilon }(\cdot , t)\Vert _{L^2(\varOmega )}$ (see Lemma 2.9) implies that there exists a positive constant $C_{11}$ such that $\Vert u_\varepsilon (\cdot , t)\Vert _{L^{4m+2}(\varOmega )}\le C_{11}~~ \text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon }),$ which together with (2.3.8), (2.3.10) leads to (2.3.25) by Lemma 2.2. This completes the proof of Lemma 2.10.

Lemma 2.11

Let $m>1$. Then for all $p>1,$ there exists a positive constant C independent of $\varepsilon$, such that the solution of (2.2.1) from Lemma 2.2 satisfies

$$\begin{aligned} \begin{aligned}&\displaystyle {\Vert n_\varepsilon (\cdot , t)\Vert _{L^{p}(\varOmega )}\le C~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon }).}\\ \end{aligned} \end{aligned}$$

(2.3.39)

Proof

Taking ${(n_{\varepsilon }+\varepsilon )^{p-1}}$ with $p>\max \{1,m-1\}$ as the test function for the first equation of (2.2.1), combining with the second equation, and using $\nabla \cdot u_\varepsilon =0$, we obtain that for all $t\in (0,T_{max,\varepsilon })$,

$$ \begin{aligned}&\displaystyle {\frac{1}{{p}}\frac{d}{dt}\Vert n_{\varepsilon }+\varepsilon \Vert ^{{{p}}}_{L^{{p}}(\varOmega )}+ m(p-1)\int _{\varOmega } (n_{\varepsilon }+\varepsilon )^{m+p-3}|{\nabla } {n}_{\varepsilon }|^2 } \\ \le&\displaystyle {(p-1)\int _\varOmega (n_{\varepsilon }+\varepsilon )^{p-2}n_{\varepsilon }{|\nabla } {n}_{\varepsilon }||S_{\varepsilon }(x,n_{\varepsilon },c_{\varepsilon }) ||\nabla c_{\varepsilon }| } \\ \le&\displaystyle {(p-1)C_S\int _\varOmega (n_{\varepsilon }+\varepsilon )^{p-1}{|\nabla } {n}_{\varepsilon }||\nabla c_{\varepsilon }| } \\ \le&\displaystyle {\frac{m(p-1)}{2}\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+p-3} |\nabla n_{\varepsilon }|^2 +\frac{(p-1)C_S^2}{2m}\int _\varOmega {(n_{\varepsilon }+\varepsilon )^{p+1-m}}|\nabla c_{\varepsilon }|^2}, \end{aligned} $$

and hence

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{1}{{p}}\displaystyle \frac{d}{dt}\Vert n_{\varepsilon }+\varepsilon \Vert ^{{{p}}}_{L^{{p}}(\varOmega )}+ \displaystyle \frac{m(p-1)}{2}\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+p-3} |\nabla n_{\varepsilon }|^2 \\ \le&\displaystyle {\frac{(p-1)C_S^2}{2m}\int _\varOmega (n_{\varepsilon }+\varepsilon )^{p+1-m}|\nabla c_{\varepsilon }|^2} \end{aligned} \end{aligned}$$

(2.3.40)

for all $t\in (0,T_{max,\varepsilon })$. In the following, we will estimate the right-hand side of (2.3.40). In fact, due to $m>1$, we conclude from (2.3.25) that

$$\begin{aligned} \begin{aligned}&\displaystyle { \int _\varOmega (n_{\varepsilon }+\varepsilon )^{p+1-m} |\nabla c_{\varepsilon }|^2} \\ \le&\displaystyle { \left( \int _\varOmega (n_{\varepsilon } +\varepsilon )^{\frac{m(p+1-m)}{m-1}}\right) ^{\frac{m-1}{m}}\left( \int _\varOmega |\nabla c_{\varepsilon }|^{2m}\right) ^{\frac{1}{m}}} \\ \le&\displaystyle { C_1\left( \int _\varOmega (n_{\varepsilon } +\varepsilon )^{\frac{m(p+1-m)}{m-1}}\right) ^{\frac{m-1}{m}}~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon })}. \end{aligned} \end{aligned}$$

(2.3.41)

Further, noticing that $\frac{2(mp-m^2+1)}{m(p+m-1)}<2$ due to $m>1$, an application of the Gagliardo–Nirenberg inequality then leads to

$$\begin{aligned} \begin{aligned}&\displaystyle {C_1\left( \int _\varOmega (n_{\varepsilon }+\varepsilon )^{\frac{m(p+1-m)}{m-1}}\right) ^{\frac{m-1}{m}}} \\ =&\displaystyle {C_1\Vert (n_{\varepsilon }+\varepsilon )^{\frac{m(p+1-m)}{m-1}}\Vert ^{\frac{2(p+1-m)}{m+p-1}}_{L^{\frac{2m(p+1-m)}{(m-1)(m+p-1)}}(\varOmega )}} \\ \le&C_2(\Vert \nabla (n_{\varepsilon }+\varepsilon )^{\frac{p+m-1}{2}}\Vert _{L^2(\varOmega )}^{\frac{mp-m^2+1}{m(p+1-m)}} \Vert (n_{\varepsilon }+\varepsilon )^{\frac{p+m-1}{2}}\Vert _{L^\frac{2}{p+m-1}(\varOmega )}^{\frac{m-1}{m(p+1-m)}} \\&+\Vert (n_{\varepsilon }+\varepsilon )^{\frac{p+m-1}{2}}\Vert _{L^\frac{2}{p+m-1}(\varOmega )})^{\frac{2(p+1-m)}{m+p-1}} \\ \le&\displaystyle {C_{3}(\Vert \nabla (n_{\varepsilon }+\varepsilon )^{\frac{p+m-1}{2}}\Vert _{L^2(\varOmega )}^{\frac{2(mp-m^2+1)}{m(p+m-1)}}+1)} \\ \le&\displaystyle {\frac{m(p-1)}{4}\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+p-3} |\nabla n_{\varepsilon }|^2+C_4} \end{aligned} \end{aligned}$$

(2.3.42)

for some $C_3>0$ and $C_4>0$. Therefore, combining (2.3.40), (2.3.41) with (2.3.42), we arrive at

$$ \displaystyle \frac{1}{{p}}\displaystyle \frac{d}{dt}\Vert n_{\varepsilon }+\varepsilon \Vert ^{{{p}}}_{L^{{p}}(\varOmega )}+ \displaystyle \frac{m(p-1)}{4}\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+p-3} |\nabla n_{\varepsilon }|^2\le \displaystyle {C_5,} $$

which along with the fact that for some $C_6>0$,

$$ \Vert n_{\varepsilon }+\varepsilon \Vert ^{{{p}}}_{L^{{p}}(\varOmega )}\le \displaystyle \frac{m(p-1)}{8}\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+p-3} |\nabla n_{\varepsilon }|^2+C_6, $$

and Lemma 2.2 implies that (2.3.39) holds.

Now we can rely on standard reasoning to obtain the following.

Lemma 2.12

Let $m>1$ and $\gamma \in (\frac{1}{2},1).$ Then one can find a positive constant C independent of $\varepsilon$, such that

$$ \Vert n_\varepsilon (\cdot ,t)\Vert _{L^\infty (\varOmega )} \le C ~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon }), $$

$$ \Vert c_\varepsilon (\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )} \le C ~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon }) $$

as well as

$$ \Vert A^\gamma u_\varepsilon (\cdot ,t)\Vert _{L^{2}(\varOmega )} \le C ~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon }). $$

Proof

Firstly, applying the variation-of-constants formula to the projected version of the third equation in (2.2.1), we derive that

$$ u_\varepsilon (\cdot , t) = e^{-tA}u_0 +\int _0^te^{-(t-\tau )A} \mathscr {P}[n_\varepsilon (\cdot ,t)\nabla \phi -\kappa (Y_{\varepsilon }u_{\varepsilon } \cdot \nabla )u_{\varepsilon }]d\tau ~~ \text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon }). $$

Let $h_{\varepsilon }=\mathscr {P}[n_\varepsilon (\cdot ,t)\nabla \phi -\kappa (Y_{\varepsilon }u_{\varepsilon } \cdot \nabla )u_{\varepsilon }]$. Then in view of the standard smoothing properties of the Stokes semigroup, we can conclude that for $\gamma \in ( \frac{1}{2}, 1)$ and $p_0\in (\frac{2}{3-2\gamma } ,2)$, there exists $C_{1} > 0$ such that

$$\begin{aligned} \begin{aligned}&\Vert A^\gamma u_{\varepsilon }(\cdot , t)\Vert _{L^2(\varOmega )} \\ \le&\displaystyle {\Vert A^\gamma e^{-tA}u_0\Vert _{L^2(\varOmega )} +\int _0^t\Vert A^\gamma e^{-(t-\tau )A}h_{\varepsilon }(\cdot ,\tau )d\tau \Vert _{L^2(\varOmega )}d\tau } \\ \le&\displaystyle {\Vert A^\gamma u_0\Vert _{L^2(\varOmega )} +C_{1}\int _0^t(t-\tau )^{-\gamma -\frac{1}{p_0}+\frac{1}{2}}e^{-\lambda (t-\tau )}\Vert h_{\varepsilon }(\cdot ,\tau )\Vert _{L^{p_0}(\varOmega )}d\tau } \end{aligned} \end{aligned}$$

(2.3.43)

for all $t\in (0,T_{max,\varepsilon })$.

In light of (2.3.39), for some positive constant $C_2$, it has

$$ \displaystyle {\Vert n_\varepsilon (\cdot , t)\Vert _{L^{p_0}(\varOmega )}\le C_2~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon })}. $$

So employing the Hölder inequality and the continuity of $\mathscr {P}$ in $L^p(\varOmega ;\mathbb {R}^2)$ (see Fujiwara and Morimoto 1977), there exist positive constants $C_{i}, (i=3,4,5,6),$ such that

$$\begin{aligned} \begin{aligned}&\Vert h_{\varepsilon }(\cdot ,t)\Vert _{L^{p_0}(\varOmega )} \\ \le&C_{3}\Vert (Y_{\varepsilon }u_{\varepsilon } \cdot \nabla )u_{\varepsilon }(\cdot ,t)\Vert _{L^{p_0}(\varOmega )}+ C_{3}\Vert n_\varepsilon (\cdot ,t)\Vert _{L^{p_0}(\varOmega )} \\ \le&C_{4}\Vert Y_{\varepsilon }u_{\varepsilon }\Vert _{L^{\frac{2p_0}{2-p_0}}(\varOmega )} \Vert \nabla u_{\varepsilon }(\cdot ,t)\Vert _{L^{2}(\varOmega )}+ C_{4} \\ \le&C_{5}\Vert \nabla Y_{\varepsilon }u_{\varepsilon }\Vert _{L^{2}(\varOmega )} \Vert \nabla u_{\varepsilon }(\cdot ,t)\Vert _{L^{2}(\varOmega )}+ C_{4} \\ \le&C_{6}\Vert \nabla u_{\varepsilon }(\cdot ,t)\Vert ^2_{L^{2}(\varOmega )}+ C_{4}~~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon }), \end{aligned} \end{aligned}$$

(2.3.44)

where we have used the fact that $W^{1,2}(\varOmega )\hookrightarrow L^\frac{2p_0}{2-p_0}(\varOmega )$. Collecting (2.3.43), (2.3.21) and (2.3.44), we conclude that

$$ \Vert A^\gamma u_{\varepsilon }(\cdot , t)\Vert _{L^2(\varOmega )} \le C_7 ~~\text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon }) $$

which together with the fact that $D(A^\gamma )$ is continuously embedded into $L^\infty (\varOmega )$ by $\gamma >\frac{1}{2}$ yields

$$\begin{aligned} \Vert u_{\varepsilon }(\cdot , t)\Vert _{L^\infty (\varOmega )}\le C_{8}~~ \text{ for } \text{ all }~~ t\in (0,T_{max,\varepsilon }). \end{aligned}$$

(2.3.45)

Further, in view of (2.3.25), one may use (2.1.11), $m>1$ and the smoothing properties of the Neumann heat semigroup $(e^{t\varDelta })_{t\ge 0}$ to obtain that there exists $C_{9} > 0$ such that

$$\begin{aligned} \Vert \nabla c_{\varepsilon }(\cdot , t)\Vert _{L^{{\infty }}(\varOmega )}\le C_{9} ~~ \text{ for } \text{ all }~~~ t\in (0,T_{max,\varepsilon }). \end{aligned}$$

(2.3.46)

Moreover, the boundedness of $n_{\varepsilon }$ can be archived by the well-known Moser–Alikakos iteration procedure (see, e.g., Lemma A.1 in Tao and Winkler 2012a). Indeed, by (2.3.45) and (2.3.46), we see that the hypotheses of Lemma A.1 in Tao and Winkler (2012a) are valid provided that the parameter p in Lemma 2.11 is appropriately large. The proof of Lemma 2.12 is complete.

With all the above regularization properties of $n_{\varepsilon }$, $c_{\varepsilon }$ and $u_{\varepsilon }$ at hand, we can show the existence of global bounded solutions to the regularized system (2.2.1).

Lemma 2.13

Let $m> 1$ and $\gamma \in (\frac{1}{2},1)$ and $(n_\varepsilon , c_\varepsilon , u_\varepsilon , P_\varepsilon )_{\varepsilon \in (0,1)}$ be classical solutions of (2.2.1) constructed in Lemma 2.2 on $[0, T_{max,\varepsilon })$. Then the solution is global on $[0,\infty )$. Moreover, one can find $C > 0$ independent of $\varepsilon \in (0, 1)$ such that

$$ \Vert n_\varepsilon (\cdot ,t)\Vert _{L^\infty (\varOmega )} \le C ~~\text{ for } \text{ all }~~ t\in (0,\infty ) $$

and

$$ \Vert c_\varepsilon (\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )} \le C ~~\text{ for } \text{ all }~~ t\in (0,\infty ) $$

as well as

$$ \Vert A^\gamma u_\varepsilon (\cdot ,t)\Vert _{L^{2}(\varOmega )} \le C ~~\text{ for } \text{ all }~~ t\in (0,\infty ). $$

Then, with the help of Lemma 2.13, we can straightforwardly deduce the uniform Hölder properties of $c_\varepsilon ,\nabla c_\varepsilon$ and $u_\varepsilon$ by the standard parabolic regularity theory as the proof of Lemmas 3.18–3.19 in Winkler (2015b) (see also Zheng 2016).

Lemma 2.14

Let $m> 1$. Then one can find $\mu \in (0, 1)$ such that for some $C > 0$

$$ \Vert c_\varepsilon (\cdot ,t)\Vert _{C^{\mu ,\frac{\mu }{2}}(\varOmega \times [t,t+1])} \le C ~~\text{ for } \text{ all }~~ t\in (0,\infty ) $$

as well as

$$ \Vert u_\varepsilon (\cdot ,t)\Vert _{C^{\mu ,\frac{\mu }{2}}(\varOmega \times [t,t+1])} \le C ~~\text{ for } \text{ all }~~ t\in (0,\infty ), $$

and for any $\tau > 0$, there exists $C(\tau ) > 0$ fulfilling

$$ \Vert \nabla c_\varepsilon (\cdot ,t)\Vert _{C^{\mu ,\frac{\mu }{2}}(\varOmega \times [t,t+1])} \le C(\tau ) ~~\text{ for } \text{ all }~~ t\in (\tau ,\infty ). $$

2.3.2 Global Boundedness of Weak Solutions

Based on the above lemmas, the weak solution of (2.1.6)–(2.1.8) can be obtained as the limitation of classical solutions to the systems (2.2.1). Applying the idea of Zheng (2016) (see also Liu and Wang 2016 and Winkler 2015b), we first state the definition of the solution as follows.

Definition 2.2

Let $T > 0$ and $(n_0, c_0, u_0)$ fulfill (2.1.11). Then a triple of functions (n, c, u) is called a weak solution of (2.1.6)–(2.1.8) in $\varOmega \times (0,T)$ if the following conditions are satisfied:

$$ \left\{ \begin{aligned} n\in L_{loc}^1(\bar{\varOmega }\times [0,T)), \\ c \in L_{loc}^1([0,T); W^{1,1}(\varOmega )), \\ u \in L_{loc}^1([0,T); W^{1,1}(\varOmega )), \end{aligned}\right. $$

where $n\ge 0$ and $c\ge 0$ in $\varOmega \times (0, T)$ as well as $\nabla \cdot u = 0$ in the distributional sense in $\varOmega \times (0, T)$, moreover, $n^m~\in ~~ L^1_{loc}(\bar{\varOmega }\times [0, \infty )), cu,~nu$ and $n\nabla c$ belong to $L^1_{loc}(\bar{\varOmega }\times [0, \infty );\mathbb {R}^{2})$ and $u\bigotimes u\in L^1_{loc}(\bar{\varOmega }\times [0, \infty );\mathbb {R}^{2\times 2});$ and

$$ \displaystyle {-\int _0^{T}\int _{\varOmega }n\varphi _t-\int _{\varOmega }n_0\varphi (\cdot ,0)} =\displaystyle {\int _0^T\int _{\varOmega }n^m\varDelta \varphi +\int _0^T\int _{\varOmega }n\nabla c\cdot \nabla \varphi } +\displaystyle {\int _0^T\int _{\varOmega }nu\cdot \nabla \varphi } $$

for any $\varphi \in C_0^{\infty } (\bar{\varOmega }\times [0, T))$ satisfying $\frac{\partial \varphi }{\partial \nu }= 0$ on $\partial \varOmega \times (0, T)$, as well as

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

for any $\varphi \in C_0^{\infty } (\bar{\varOmega }\times [0, T))$ and

$$\begin{aligned}&\displaystyle {-\int _0^{T}\int _{\varOmega }u\varphi _t-\int _{\varOmega }u_0\varphi (\cdot ,0) } \\ =&\displaystyle {\kappa \int _0^T\int _{\varOmega } u\otimes u\cdot \nabla \varphi -\int _0^T\int _{\varOmega }\nabla u\cdot \nabla \varphi - \int _0^T\int _{\varOmega }n\nabla \phi \cdot \varphi } \end{aligned}$$

for any $\varphi \in C_0^{\infty } (\bar{\varOmega }\times [0, T);\mathbb {R}^2)$ fulfilling $\nabla \varphi \equiv 0$ in $\varOmega \times (0, T)$.

If for each $T>0$, (n, c, u) :$\varOmega \times (0,\infty )\longrightarrow \mathbb {R}^4$ is a weak solution of (2.1.6)–(2.1.8) in $\varOmega \times (0, T)$, then we call (n, c, u) a global weak solution of (2.1.6)–(2.1.8).

In order to apply the Aubin–Lions Lemma (Simon 1986), we will need the regularity of the time derivative of bounded solutions. Employing almost exactly the same arguments as that in the proof of Lemmas 3.22–3.23 in Winkler (2015b) (the minor necessary changes are left as an exercise to the reader), and taking advantage of Lemma 2.13, we conclude the following lemma.

Lemma 2.15

Let $m> 1$ and let $\varsigma > \max \{m,2(m - 1 )\}$. Then for every $T > 0$ and $\varepsilon \in (0,1)$, one can find $C(T)>0$ independent of $\varepsilon$ such that $\int _0^T\Vert \partial _t(n_{\varepsilon }+\varepsilon )^\varsigma (\cdot ,t)\Vert _{(W^{2,2}_0(\varOmega ))^*}dt \le C(T)$ as well as $\int _{0}^T\int _{\varOmega } |\nabla (n_{\varepsilon }+\varepsilon )^{\varsigma }|^2\le C(T).$

At this position, the main result can be proved as follows.

Proof of Theorem 2.1.  In conjunction with Lemmas 2.13, 2.11 and the Aubin–Lions compactness lemma (see Simon 1986), one can infer the existence of a sequence of numbers $\varepsilon = \varepsilon _j \searrow 0$ along which

$$\begin{aligned} n_\varepsilon \longrightarrow n ~~\text{ a.e. } \text{ in }~~ \varOmega \times (0,\infty ), \end{aligned}$$

(2.3.47)

$$\begin{aligned} \nabla n^m_\varepsilon \rightharpoonup \nabla n^m ~~\text{ in }~~ L^2_{loc}(\varOmega \times [0,\infty )), \end{aligned}$$

(2.3.48)

$$\begin{aligned} c_\varepsilon \rightarrow c ~~\text{ in }~~ C^0_{loc}(\bar{\varOmega }\times [0,\infty )), \end{aligned}$$

(2.3.49)

$$\begin{aligned} \nabla c_\varepsilon \rightarrow \nabla c ~~\text{ in }~~ C^0_{loc}(\bar{\varOmega }\times (0,\infty )), \end{aligned}$$

(2.3.50)

$$\begin{aligned} \nabla c_\varepsilon \rightharpoonup \nabla c ~~\text{ weakly } \text{ star } \text{ in }~~ L^{\infty }(\varOmega \times (0,\infty )) \end{aligned}$$

(2.3.51)

as well as

$$\begin{aligned} u_\varepsilon \rightarrow u ~~\text{ in }~~ C^0_{loc}(\bar{\varOmega }\times [0,\infty )) \end{aligned}$$

(2.3.52)

and

$$\begin{aligned} D u_\varepsilon \rightharpoonup Du ~~\text{ weakly } \text{ star } \text{ in }~~L^{\infty }(\varOmega \times (0,\infty )) \end{aligned}$$

(2.3.53)

holds for some limit $(n,c,u) \in (L^\infty (\varOmega \times (0,\infty )))^4$ with nonnegative n and c. Indeed, Lemma 2.15 implies that for each $T > 0,$ $(n_{\varepsilon }^\varsigma )_{\varepsilon \in (0,1)}$ is bounded in $L^2((0, T);W^{1,2}(\varOmega ))$. By using Aubin–Lions lemma, one then obtains $n_{\varepsilon }^\varsigma \rightarrow z^\varsigma$ for some nonnegative measurable $z:\varOmega \times (0,\varOmega )\rightarrow \mathbb {R}$. Further by Lemma 2.11 and the Egorov theorem, one has (2.3.47) and (2.3.48).

Due to these convergence properties (see (2.3.47)–(2.3.53)), by applying the standard arguments, we may take $\varepsilon = \varepsilon _j\searrow 0$ in each term of the natural weak formulation of (2.2.1) separately. Then we can verify that (n, c, u) can be complemented by some pressure function P in such a way that (n, c, u, P) is a weak solution of (2.1.6)–(2.1.8). Finally, we can infer from the boundedness of $(n_{\varepsilon },c_{\varepsilon },u_{\varepsilon })$ and the Banach–Alaoglu theorem that (n, c, u) is bounded.

2.4 Global Existence of Solutions to a Three-Dimensional Keller–Segel–Navier–Stokes System

2.4.1 A Priori Estimates for Approximate Solutions

In this subsection, we are going to establish an iteration step to develop the main ingredients of our result. The iteration depends on a series of a priori estimates. We first recall some properties of $F_\varepsilon$ and $F'_\varepsilon$, which play an important role in the proof of Theorem 2.2.

Lemma 2.16

Assume $F_{\varepsilon }$ is given by (2.2.9). Then

$$\begin{aligned} 0\le F'_{\varepsilon }(s)=\frac{1}{1+\varepsilon s}\le 1~\text{ for } \text{ all }~s \ge 0~\text{ and }~\varepsilon > 0 \end{aligned}$$

(2.4.1)

as well as

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0^{+}}F_{\varepsilon }(s)=s,~\lim _{\varepsilon \rightarrow 0^{+}}F'_{\varepsilon }(s)=1~\text{ for } \text{ all }~s \ge 0 \end{aligned}$$

(2.4.2)

and

$$\begin{aligned} 0\le F_{\varepsilon }(s)\le s~\text{ for } \text{ all }~s \ge 0. \end{aligned}$$

(2.4.3)

Proof

Recalling (2.2.9), by tedious and simple calculations, we can derive (2.4.1)–(2.4.3).

The proof of this lemma is very similar to that of Lemmas 2.2 and 2.6 of Tao and Winkler (2015b) (see also Lemma 3.2 of Wang 2017), so we omit it here.

Lemma 2.17

There exists $\lambda > 0$ independent of $\varepsilon$ such that the solution of (2.2.8) satisfies

$$\begin{aligned} \int _{\varOmega }{n_{\varepsilon }}+\int _{\varOmega }{c_{\varepsilon }}\le \lambda ~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon }). \end{aligned}$$

(2.4.4)

Lemma 2.18

Let $\alpha >\frac{1}{3}$. Then there exists $C>0$ independent of $\varepsilon$ such that the solution of (2.2.8) satisfies

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{\varOmega } n_{\varepsilon }^{2\alpha }+\int _{\varOmega } c_{\varepsilon }^2+\int _{\varOmega } | {u_{\varepsilon }}|^2\le C~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon }).}\\ \end{aligned} \end{aligned}$$

(2.4.5)

Moreover, for $T\in (0, T_{max,\varepsilon })$, one can find a constant $C > 0$ independent of $\varepsilon$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{0}^T\int _{\varOmega } \left[ n_{\varepsilon }^{2\alpha -2} |\nabla {n_{\varepsilon }}|^2+ |\nabla {c_{\varepsilon }}|^2+ |\nabla {u_{\varepsilon }}|^2\right] \le C.}\\ \end{aligned} \end{aligned}$$

(2.4.6)

Proof

The proof consists of two cases.

Case (I) $2\alpha \ne 1$: We first obtain from $\nabla \cdot u_\varepsilon =0$ in $\varOmega \times (0, T_{max,\varepsilon })$ and straightforward calculations that

$$\begin{aligned} \begin{aligned}&\displaystyle {\mathrm{sign}(2\alpha -1)\frac{1}{{2\alpha }}\frac{d}{dt}\Vert n_{\varepsilon } \Vert ^{{{2\alpha }}}_{L^{{2\alpha }}(\varOmega )}}\\&\displaystyle {+ \mathrm{sign}(2\alpha -1)(2\alpha -1)\int _{\varOmega } n_{\varepsilon }^{2\alpha -2} |\nabla n_{\varepsilon }|^2}\\ =&\displaystyle {- \int _{\varOmega }\mathrm{sign}(2\alpha -1) n_{\varepsilon }^{2\alpha -1}\nabla \cdot (n_{\varepsilon }F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\cdot \nabla c_{\varepsilon })}\\ \le&\displaystyle {\mathrm{sign}(2\alpha -1)(2\alpha -1) \int _{\varOmega } n_{\varepsilon }^{2\alpha -2}n_{\varepsilon }F'_{\varepsilon }(n_{\varepsilon })|S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })||\nabla n_{\varepsilon }||\nabla c_{\varepsilon }|} \end{aligned} \end{aligned}$$

(2.4.7)

for all $t\in (0, T_{max,\varepsilon }).$ Therefore, from (2.4.1), in light of (2.1.5) and (2.2.9), we can estimate the right-hand side of (2.4.7) as follows:

$$\begin{aligned} \begin{aligned}&\displaystyle {\mathrm{sign}(2\alpha -1)(2\alpha -1) \int _{\varOmega } n_{\varepsilon }^{2\alpha -2}n_{\varepsilon }F'_{\varepsilon }(n_{\varepsilon })|S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })||\nabla n_{\varepsilon }||\nabla c_{\varepsilon }|}\\ \le&\displaystyle {\mathrm{sign}(2\alpha -1)(2\alpha -1) \int _{\varOmega } n_{\varepsilon }^{2\alpha -2}n_{\varepsilon }C_S(1 + n_{\varepsilon })^{-\alpha }|\nabla n_{\varepsilon }||\nabla c_{\varepsilon }|}\\ \le&\displaystyle {\mathrm{sign}(2\alpha -1)\frac{2\alpha -1}{2}\int _{\varOmega } n_{\varepsilon }^{2\alpha -2} |\nabla n_{\varepsilon }|^2}\\&\displaystyle {+\frac{|2\alpha -1|}{2}C_S^2\int _{\varOmega } n_{\varepsilon }^{2\alpha -2}n_{\varepsilon }^2(1 + n_{\varepsilon })^{-2\alpha }|\nabla c_{\varepsilon }|^2}\\ \le&\displaystyle {\mathrm{sign}(2\alpha -1)\frac{2\alpha -1}{2}\int _{\varOmega } n_{\varepsilon }^{2\alpha -2} |\nabla n_{\varepsilon }|^2}\\&\displaystyle {+\frac{|2\alpha -1|}{2}C_S^2\int _{\varOmega }|\nabla c_{\varepsilon }|^2~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon })} \end{aligned} \end{aligned}$$

(2.4.8)

by using Young’s inequality, where in the last inequality we have used the fact that $n_{\varepsilon }^{2\alpha -2}n_{\varepsilon }^2(1 + n_{\varepsilon })^{-2\alpha }\le 1$ for all $\varepsilon \ge 0$ and $(x,t)\in \varOmega \times (0, T_{max,\varepsilon }).$ Inserting (2.4.8) into (2.4.7), we conclude that

$$\begin{aligned} \begin{aligned}&\displaystyle {\mathrm{sign}(2\alpha -1)\frac{1}{{2\alpha }}\frac{d}{dt}\Vert {n_{\varepsilon }}\Vert ^{{{2\alpha }}}_{L^{{2\alpha }}(\varOmega )}+ \mathrm{sign}(2\alpha -1)\frac{2\alpha -1}{2}\int _{\varOmega } n_{\varepsilon }^{2\alpha -2} |\nabla n_{\varepsilon }|^2}\\ \le&\displaystyle {\frac{|2\alpha -1|}{2}C_S^2\int _{\varOmega }|\nabla c_{\varepsilon }|^2~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon }).} \end{aligned} \end{aligned}$$

(2.4.9)

To track the time evolution of $c_\varepsilon$, taking ${c_{\varepsilon }}$ as the test function for the second equation of (2.2.8) and using $\nabla \cdot u_\varepsilon =0$ and (2.4.3) together with Hölder’s inequality yields

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{1}{{2}}\displaystyle \frac{d}{dt}\Vert {c_{\varepsilon }}\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}+ \int _{\varOmega } |\nabla c_{\varepsilon }|^2+ \int _{\varOmega } | c_{\varepsilon }|^2 \\ =&\displaystyle {\int _{\varOmega } F_{\varepsilon }(n_{\varepsilon })c_{\varepsilon }}\\ \le&\displaystyle {\int _{\varOmega } n_{\varepsilon }c_{\varepsilon }}\\ \le&\displaystyle {\Vert n_{\varepsilon }\Vert _{L^{\frac{6}{5}}(\varOmega )}\Vert c_{\varepsilon }\Vert _{L^{6}(\varOmega )}~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon }).} \end{aligned} \end{aligned}$$

(2.4.10)

By applying Sobolev embedding $W^{1,2}(\varOmega )\hookrightarrow L^6(\varOmega )$ in the three-dimensional setting, in view of (2.4.4), there exist positive constants $C_1$ and $C_2$ such that

$$\begin{aligned} \begin{aligned} \displaystyle \Vert c_{\varepsilon }\Vert _{L^{6}(\varOmega )}^2\le&\displaystyle {C_1\Vert \nabla c_{\varepsilon }\Vert _{L^{2}(\varOmega )}^2+C_1\Vert c_{\varepsilon }\Vert _{L^{1}(\varOmega )}^2}\\ \le&\displaystyle {C_1\Vert \nabla c_{\varepsilon }\Vert _{L^{2}(\varOmega )}^2+C_2~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon }).}\\ \end{aligned} \end{aligned}$$

(2.4.11)

Thus, by means of Young’s inequality and (2.4.11), we proceed to estimate

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{1}{{2}}\displaystyle \frac{d}{dt}\Vert {c_{\varepsilon }}\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}+ \int _{\varOmega } |\nabla c_{\varepsilon }|^2+ \int _{\varOmega } | c_{\varepsilon }|^2 \\ \le&\displaystyle {\frac{1}{2C_1}\Vert c_{\varepsilon }\Vert _{L^{6}(\varOmega )}^2+\frac{C_1}{2}\Vert n_{\varepsilon } \Vert _{L^{\frac{6}{5}}(\varOmega )}^2}\\ \le&\displaystyle {\frac{1}{2}\Vert \nabla c_{\varepsilon }\Vert _{L^{2}(\varOmega )}^2+\frac{C_1}{2}\Vert n_{\varepsilon } \Vert _{L^{\frac{6}{5}}(\varOmega )}^2+C_3 ~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon })} \end{aligned} \end{aligned}$$

(2.4.12)

and some positive constant $C_3$ independent of $\varepsilon$. Therefore,

$$\begin{aligned} \begin{aligned} \displaystyle \frac{1}{{2}}\displaystyle \frac{d}{dt}\Vert {c_{\varepsilon }}\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}+\frac{1}{2} \int _{\varOmega } |\nabla c_{\varepsilon }|^2+ \int _{\varOmega } | c_{\varepsilon }|^2\le&\displaystyle {\frac{C_1}{2}\Vert n_{\varepsilon } \Vert _{L^{\frac{6}{5}}(\varOmega )}^2+C_3 ~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon }).}\\ \end{aligned} \end{aligned}$$

(2.4.13)

To estimate $\Vert n_{\varepsilon }\Vert _{L^{\frac{6}{5}(\varOmega )}}$ for all $t\in (0, T_{max,\varepsilon })$, we should notice that $\alpha >\frac{1}{3}$ ensures that $\frac{2}{6\alpha -1}<2$, so that in light of (2.4.4), the Gagliardo–Nirenberg inequality and Young’s inequality allow us to estimate that

$$\begin{aligned} \begin{aligned}&\displaystyle \Vert n_{\varepsilon } \Vert _{L^{\frac{6}{5}}(\varOmega )}^2\\ =&\displaystyle {\Vert n_{\varepsilon }^{\alpha }\Vert _{L^{\frac{6}{5\alpha }}(\varOmega )}^{\frac{2}{\alpha }}}\\ \le&\displaystyle {C_4(\Vert \nabla n_{\varepsilon }^{\alpha }\Vert _{L^{2}(\varOmega )}^{\frac{2}{6\alpha -1}}\Vert n_{\varepsilon }^{\alpha }\Vert _{L^{\frac{1}{\alpha }}(\varOmega )}^{\frac{2}{\alpha }-\frac{2}{6\alpha -1}}+\Vert n_{\varepsilon }^{\alpha }\Vert _{L^{\frac{1}{\alpha }}(\varOmega )}^{\frac{2}{\alpha }})}\\ \le&\displaystyle {\frac{1}{4}\frac{1}{C_1\alpha ^2C_S^2}\Vert \nabla n_{\varepsilon }^{\alpha }\Vert _{L^{2}(\varOmega )}^{2}+C_5~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon })}\\ \end{aligned} \end{aligned}$$

(2.4.14)

with some positive constants $C_4$ and $C_5$ independent of $\varepsilon$. This together with (2.4.13) contributes to

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{1}{{2}}\displaystyle \frac{d}{dt}\Vert {c_{\varepsilon }}\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}+\frac{1}{2} \int _{\varOmega } |\nabla c_{\varepsilon }|^2+ \int _{\varOmega } | c_{\varepsilon }|^2 \\ \le&\displaystyle {\frac{1}{8}\frac{1}{\alpha ^2C_S^2}\Vert \nabla n_{\varepsilon }^{\alpha }\Vert _{L^{2}(\varOmega )}^{2}+C_6 ~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon })} \end{aligned} \end{aligned}$$

(2.4.15)

and some positive constant $C_6$. Taking an evident linear combination of the inequalities provided by (2.4.9) and (2.4.15), one can obtain

$$\begin{aligned} \begin{aligned} \displaystyle&\mathrm{sign}(2\alpha -1)\displaystyle \frac{1}{{2\alpha }}\displaystyle \frac{d}{dt}\Vert { n_{\varepsilon } }\Vert ^{{{2\alpha }}}_{L^{{2\alpha }}(\varOmega )}+ |2\alpha -1|C_S^2\displaystyle \frac{d}{dt}\Vert {c_{\varepsilon }}\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}\\ \displaystyle&+\displaystyle \frac{|2\alpha -1|}{2}C_S^2 \int _{\varOmega } |\nabla c_{\varepsilon }|^2+ 2|2\alpha -1|C_S^2\int _{\varOmega } | c_{\varepsilon }|^2\\ \displaystyle&+\left( \mathrm{sign}(2\alpha -1)\displaystyle \frac{2\alpha -1}{2}-\frac{1}{4}|2\alpha -1|\right) \displaystyle \int _{\varOmega } n_{\varepsilon }^{2\alpha -2} |\nabla n_{\varepsilon }|^2\\ \le&\displaystyle {C_7 ~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon })}\\ \end{aligned} \end{aligned}$$

(2.4.16)

and some positive constant $C_7.$ Since $\mathrm{sign}(2\alpha -1)\displaystyle \frac{2\alpha -1}{2}=\displaystyle \frac{|2\alpha -1|}{2},$ (2.4.16) implies that

$$\begin{aligned} \begin{aligned} \displaystyle&\mathrm{sign}(2\alpha -1)\displaystyle \frac{1}{{2\alpha }}\displaystyle \frac{d}{dt}\Vert { n_{\varepsilon } }\Vert ^{{{2\alpha }}}_{L^{{2\alpha }}(\varOmega )}+ |2\alpha -1|C_S^2\displaystyle \frac{d}{dt}\Vert {c_{\varepsilon }}\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}\\ \displaystyle&+\displaystyle \frac{|2\alpha -1|}{2}C_S^2 \int _{\varOmega } |\nabla c_{\varepsilon }|^2+ 2|2\alpha -1|C_S^2\int _{\varOmega } | c_{\varepsilon }|^2\\&+\displaystyle \frac{|2\alpha -1|}{4}\displaystyle \int _{\varOmega } n_{\varepsilon }^{2\alpha -2} |\nabla n_{\varepsilon }|^2\\ \le&\displaystyle {C_7 ~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon }).}\\ \end{aligned} \end{aligned}$$

(2.4.17)

If $2\alpha >1$, then $\mathrm{sign}(2\alpha -1)=1>0,$ thus, integrating (2.4.17) over time, we can obtain

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{\varOmega } n_{\varepsilon }^{2\alpha }+\int _{\varOmega } c_{\varepsilon }^2\le C_8~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon })}\\ \end{aligned} \end{aligned}$$

(2.4.18)

and

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{0}^T\int _{\varOmega } \left[ n_{\varepsilon }^{2\alpha -2} |\nabla {n_{\varepsilon }}|^2+ |\nabla {c_{\varepsilon }}|^2\right] \le C_8(T+1)~~\text{ for } \text{ all }~ T\in (0, T_{max,\varepsilon })}\\ \end{aligned} \end{aligned}$$

(2.4.19)

and some positive constant $C_8.$ If $2\alpha <1$, then $\mathrm{sign}(2\alpha -1)=-1<0$; hence, in view of (2.4.4), integrating (2.4.17) over time and employing Hölder’s inequality, we also conclude that there exists a positive constant $C_9$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{\varOmega } n_{\varepsilon }^{2\alpha }+\int _{\varOmega } c_{\varepsilon }^2\le C_9~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon })}\\ \end{aligned} \end{aligned}$$

(2.4.20)

and

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{0}^T\int _{\varOmega } \left[ n_{\varepsilon }^{2\alpha -2} |\nabla {n_{\varepsilon }}|^2+ |\nabla {c_{\varepsilon }}|^2\right] \le C_9(T+1)~\text{ for } \text{ all }~ T\in (0, T_{max,\varepsilon }).}\\ \end{aligned} \end{aligned}$$

(2.4.21)

Case (II) $2\alpha =1$: Using the first equation of (2.2.8) and (2.2.9), integrating by parts, and applying (2.1.5) and (2.4.1), we obtain

$$ \begin{aligned}&\frac{d}{dt}\int _{\varOmega } n_{\varepsilon } \ln n_{\varepsilon } \\ =&\displaystyle {\int _{\varOmega }n_{\varepsilon t} \ln n_{\varepsilon }+\int _{\varOmega }n_{\varepsilon t}} \\ =&\displaystyle {\int _{\varOmega }\varDelta n_{\varepsilon } \ln n_{\varepsilon }- \int _{\varOmega }\ln n_{\varepsilon }\nabla \cdot (n_{\varepsilon }F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon }) \cdot \nabla c_{\varepsilon })} \\ \le&-\int _{\varOmega } \frac{|\nabla n_{\varepsilon }|^2}{n_{\varepsilon }}+ \int _{\varOmega } C_S(1 + n_{\varepsilon })^{-\alpha }\frac{n_{\varepsilon }}{ n_{\varepsilon } }|\nabla n_{\varepsilon }||\nabla c_{\varepsilon }|~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon }), \end{aligned} $$

which combined with Young’s inequality and $2\alpha =1$ implies that

$$ \frac{d}{dt}\int _{\varOmega } n_{\varepsilon } \ln n_{\varepsilon } +\frac{1}{2}\int _{\varOmega } \frac{|\nabla n_{\varepsilon }|^2}{n_{\varepsilon }} \le \frac{1}{2}C_S^2\int _{\varOmega }|\nabla c_{\varepsilon }|^2~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon }). $$

However, since $2\alpha =1$ yields $\alpha >\frac{1}{3}$, by employing almost exactly the same arguments as in the proof of (2.4.10)–(2.4.16) (with the minor necessary changes being left as an easy exercise to the reader), we conclude an estimate of

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{\varOmega } n_{\varepsilon } \ln n_{\varepsilon } +\int _{\varOmega } c_{\varepsilon }^2\le C_{10}~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon })}\\ \end{aligned} \end{aligned}$$

(2.4.22)

and

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{0}^T\int _{\varOmega } \left[ \frac{|\nabla n_{\varepsilon }|^2}{n_{\varepsilon }}+ |\nabla {c_{\varepsilon }}|^2\right] \le C_{10}(T+1)~\text{ for } \text{ all }~ T\in (0, T_{max,\varepsilon }).}\\ \end{aligned} \end{aligned}$$

(2.4.23)

Now, multiplying the third equation of (2.2.8) by $u_\varepsilon$, integrating by parts, and using $\nabla \cdot u_{\varepsilon }=0$ give

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\int _{\varOmega }{|u_{\varepsilon }|^2}+\int _{\varOmega }{|\nabla u_{\varepsilon }|^2}= \int _{\varOmega }n_{\varepsilon }u_{\varepsilon }\cdot \nabla \phi ~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon }). \end{aligned}$$

(2.4.24)

Here, we use Hölder’s inequality, Young’s inequality and the continuity of the embedding $W^{1,2}(\varOmega )\hookrightarrow L^6(\varOmega )$ to find $C_{11}$ and $C_{12}> 0$ such that

$$\begin{aligned} \begin{aligned} \displaystyle \int _{\varOmega }n_{\varepsilon }u_{\varepsilon }\cdot \nabla \phi \le&\displaystyle {\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}\Vert n_{\varepsilon } \Vert _{L^{\frac{6}{5}}(\varOmega )}\Vert u_{\varepsilon }\Vert _{L^{6}(\varOmega )}}\\ \le&\displaystyle {C_{11}\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}\Vert n_{\varepsilon } \Vert _{L^{\frac{6}{5}}(\varOmega )}\Vert \nabla u_{\varepsilon }\Vert _{L^{2}(\varOmega )}}\\ \le&\displaystyle {\frac{1}{2}\Vert \nabla u_{\varepsilon }\Vert _{L^{2}(\varOmega )}^2+C_{12}\Vert n_{\varepsilon } \Vert _{L^{\frac{6}{5}}(\varOmega )}^2~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon }).}\\ \end{aligned} \end{aligned}$$

(2.4.25)

Next, in view of (2.4.4) and $\alpha >\frac{1}{3}$, (2.4.14) and Young’s inequality along with the Gagliardo–Nirenberg inequality yield

$$\begin{aligned} \begin{aligned} \displaystyle {\int _{\varOmega }n_{\varepsilon }u_{\varepsilon }\cdot \nabla \phi } \le&\displaystyle {\frac{1}{2}\Vert \nabla u_{\varepsilon }\Vert _{L^{2}(\varOmega )}^2+C_8\Vert \nabla n_{\varepsilon }^{\alpha }\Vert _{L^{2}(\varOmega )}^{\frac{2}{6\alpha -1}}\Vert n_{\varepsilon }^{\alpha }\Vert _{L^{\frac{1}{\alpha }}(\varOmega )}^{\frac{2}{\alpha }-\frac{2}{6\alpha -1}}}\\ \le&\displaystyle {\frac{1}{2}\Vert \nabla u_{\varepsilon }\Vert _{L^{2}(\varOmega )}^2+\Vert \nabla n_{\varepsilon }^{\alpha }\Vert _{L^{2}(\varOmega )}^{2}+C_{13}~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon })}\\ \end{aligned} \end{aligned}$$

(2.4.26)

and some positive constant $C_{13}.$ Now, inserting (2.4.25) and (2.4.26) into (2.4.24) and using (2.4.19) and (2.4.23), one has

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{\varOmega } |u_{\varepsilon }|^2\le C_{14}~~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon })}\\ \end{aligned} \end{aligned}$$

(2.4.27)

and

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{0}^T\int _{\varOmega } |\nabla {u_{\varepsilon }}|^2\le C_{14}(T+1)~~\text{ for } \text{ all }~ T\in (0, T_{max,\varepsilon })}\\ \end{aligned} \end{aligned}$$

(2.4.28)

and some positive constant $C_{14}.$ Finally, collecting (2.4.18)–(2.4.21), (2.4.22)–(2.4.23) and (2.4.27)–(2.4.28), we can get (2.4.5) and (2.4.6).

With the help of Lemma 2.18, based on the Gagliardo–Nirenberg inequality and an application of well-known arguments from parabolic regularity theory, we can derive the following lemmas.

Lemma 2.19

Let $\alpha >\frac{1}{3}$. Then there exists $C>0$ independent of $\varepsilon$ such that, for each $T\in (0, T_{max,\varepsilon })$, the solution of (2.2.8) satisfies

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{0}^T\int _{\varOmega }\left[ |\nabla n_{\varepsilon }|^{\frac{3\alpha +1}{2}}+ n_{\varepsilon }^{\frac{6\alpha +2}{3}}\right] \le C(T+1)~\text{ if }~\frac{1}{3}<\alpha \le \frac{1}{2},} \end{aligned} \end{aligned}$$

(2.4.29)

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{0}^T\int _{\varOmega }\left[ |\nabla n_{\varepsilon }|^{\frac{10\alpha }{3+2\alpha }}+ n_{\varepsilon }^{\frac{10\alpha }{3}}\right] \le C(T+1)~\text{ if }~\frac{1}{2}<\alpha <1,}\\ \end{aligned} \end{aligned}$$

(2.4.30)

as well as

$$\begin{aligned} \int _{0}^T\int _{\varOmega } \left[ |\nabla {n_{\varepsilon }}|^2+ n_{\varepsilon }^{\frac{10}{3}}\right] \le C(T+1)~\text{ if }~\alpha \ge 1 \end{aligned}$$

(2.4.31)

and

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{0}^T\left\{ \int _{\varOmega }[c_{\varepsilon }^{\frac{10}{3}}+ |u_{\varepsilon }|^{\frac{10}{3}}]+\Vert u_{\varepsilon }\Vert ^2_{L^6(\varOmega )}\right\} \le C(T+1).}\\ \end{aligned} \end{aligned}$$

(2.4.32)

Proof

Case $\frac{1}{3}<\alpha \le \frac{1}{2}$: From (2.4.4), (2.4.5) and (2.4.6), in light of the Gagliardo–Nirenberg inequality, for some $C_1$ and $C_2> 0$ that are independent of $\varepsilon$, one may verify that

$$\begin{aligned} \begin{aligned}&\displaystyle \int _{0}^T\displaystyle \int _{\varOmega } n_{\varepsilon }^{\frac{6\alpha +2}{3}} \\ =&\displaystyle {\int _{0}^T\Vert { n_{\varepsilon }^{\alpha }}\Vert ^{{\frac{6\alpha +2}{3\alpha }}}_{L^{\frac{6\alpha +2}{3\alpha }}(\varOmega )}}\\ \le&\displaystyle {C_{1}\int _{0}^T\left( \Vert \nabla { n_{\varepsilon }^{\alpha }}\Vert ^{2}_{L^{2}(\varOmega )}\Vert { n_{\varepsilon }^{\alpha }}\Vert ^{{\frac{2}{3\alpha }}}_{L^{\frac{1}{\alpha }}(\varOmega )}+ \Vert { n_{\varepsilon }^{\alpha }}\Vert ^{{\frac{6\alpha +2}{3\alpha }}}_{L^{\frac{1}{\alpha }}(\varOmega )}\right) }\\ \le&\displaystyle {C_{2}(T+1)~\text{ for } \text{ all }~ T > 0.} \end{aligned} \end{aligned}$$

(2.4.33)

Therefore, employing Hölder’s inequality (with two exponents $\frac{4}{3\alpha +1}$ and $\frac{4}{3-3\alpha }$), we conclude that there exists a positive constant $C_3$ such that

$$\begin{aligned} \begin{aligned} \displaystyle \int _{0}^T\displaystyle \int _{\varOmega }|\nabla n_{\varepsilon }|^{\frac{3\alpha +1}{2}} \le&\displaystyle {\left[ \int _{0}^T\displaystyle \int _{\varOmega } n_{\varepsilon }^{2\alpha -2}|\nabla n_{\varepsilon }|^2\right] ^{\frac{3\alpha +1}{4}} \left[ \int _{0}^T\displaystyle \int _{\varOmega } n_{\varepsilon }^{\frac{6\alpha +2}{3}}\right] ^{\frac{3-3\alpha }{4}} }\\ \le&\displaystyle {C_{3}(T+1)~\text{ for } \text{ all }~ T > 0.}\\ \end{aligned} \end{aligned}$$

(2.4.34)

Case $\frac{1}{2}<\alpha <1$: Again by (2.4.4), (2.4.5) and (2.4.6) and the Gagliardo–Nirenberg inequality and Hölder’s inequality (with two exponents $\frac{3+2\alpha }{5\alpha }$ and $\frac{3+2\alpha }{3-3\alpha }$), we derive that there exist positive constants $C_4$, $C_5$ and $C_6$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle \int _{0}^T\displaystyle \int _{\varOmega } n_{\varepsilon }^{\frac{10\alpha }{3}}\\ =&\displaystyle {\int _{0}^T\Vert { n_{\varepsilon }^{\alpha }}\Vert ^{{\frac{10}{3 }}}_{L^{\frac{10}{3 }}(\varOmega )}}\\ \le&\displaystyle {C_{4}\int _{0}^T\left( \Vert \nabla { n_{\varepsilon }^{\alpha }}\Vert ^{2}_{L^{2}(\varOmega )}\Vert { n_{\varepsilon }^{\alpha }} \Vert ^{{\frac{4 }{3}}}_{L^{2}(\varOmega )}+ \Vert { n_{\varepsilon }^{\alpha }}\Vert ^{{\frac{10\alpha }{3}}}_{L^{2}(\varOmega )}\right) }\\ \le&\displaystyle {C_{5}(T+1)~\text{ for } \text{ all }~ T > 0}\\ \end{aligned} \end{aligned}$$

(2.4.35)

and

$$\begin{aligned} \begin{aligned} \displaystyle \int _{0}^T\displaystyle \int _{\varOmega }|\nabla n_{\varepsilon }|^{\frac{10\alpha }{3+2\alpha }} \le&\displaystyle {\left[ \int _{0}^T\displaystyle \int _{\varOmega } n_{\varepsilon }^{2\alpha -2}|\nabla n_{\varepsilon }|^2\right] ^{\frac{5\alpha }{3+2\alpha }} \left[ \int _{0}^T\displaystyle \int _{\varOmega } n_{\varepsilon }^{\frac{10\alpha }{3}}\right] ^{\frac{3-3\alpha }{3+2\alpha }} }\\ \le&\displaystyle {C_{6}(T+1)~\text{ for } \text{ all }~ T > 0.}\\ \end{aligned} \end{aligned}$$

(2.4.36)

Case $\alpha \ge 1$: Multiplying the first equation in (2.2.8) by $n_{\varepsilon }$, in view of (2.2.9) and using $\nabla \cdot u_\varepsilon =0$, we derive

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{2}}\frac{d}{dt}\Vert { n_{\varepsilon } }\Vert ^{{{{2}}}}_{L^{{{2}}}(\varOmega )}+ \int _{\varOmega } |\nabla n_{\varepsilon }|^2}\\ =&\displaystyle {- \int _{\varOmega } n_{\varepsilon }\nabla \cdot (n_{\varepsilon }F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\cdot \nabla c_{\varepsilon })}\\ \le&\displaystyle { \int _{\varOmega } n_{\varepsilon }F'_{\varepsilon }(n_{\varepsilon })|S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })||\nabla n_{\varepsilon }||\nabla c_{\varepsilon }|~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon }).} \end{aligned} \end{aligned}$$

(2.4.37)

Recalling (2.1.5) and (2.2.9) and using $\alpha \ge 1$, via Young’s inequality, we derive

$$\begin{aligned} \begin{aligned}&\displaystyle \int _{\varOmega } n_{\varepsilon }F'_{\varepsilon }(n_{\varepsilon })|S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })||\nabla n_{\varepsilon }||\nabla c_{\varepsilon }|\\ \le&\displaystyle {C_S\int _{\varOmega } |\nabla n_{\varepsilon }||\nabla c_{\varepsilon }|}\\ \le&\displaystyle {\frac{1}{2}\int _{\varOmega } |\nabla n_{\varepsilon }|^2+\frac{C_S^2}{2}\int _{\varOmega } |\nabla c_{\varepsilon }|^2~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon }).} \end{aligned} \end{aligned}$$

(2.4.38)

Here, we have used the fact that

$$ n_{\varepsilon }F'_{\varepsilon }(n_{\varepsilon })|S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })|\le C_S n_{\varepsilon }(1 + n_{\varepsilon })^{-1}\le C_S $$

by using (2.1.5). Therefore, collecting (2.4.37), (2.4.38) and using (2.4.6), we conclude that

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{\varOmega } n_{\varepsilon }^{2}\le C_{7}~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon })}\\ \end{aligned} \end{aligned}$$

(2.4.39)

and

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{0}^{ T}\int _{\varOmega } |\nabla {n_{\varepsilon }}|^2\le C_{7}(T+1).}\\ \end{aligned} \end{aligned}$$

(2.4.40)

Hence, from (2.4.39)–(2.4.40) and (2.4.5)–(2.4.6), in light of the Gagliardo–Nirenberg inequality, we derive that there exist positive constants $C_{i}$, $(i=8,\cdots , 17)$ such that

$$\begin{aligned} \begin{aligned} \displaystyle \int _{0}^T\displaystyle \int _{\varOmega } n_{\varepsilon }^{\frac{10}{3}} \le&\displaystyle {C_{8}\int _{0}^T\left( \Vert \nabla { n_{\varepsilon }}\Vert ^{2}_{L^{2}(\varOmega )}\Vert { n_{\varepsilon } }\Vert ^{{\frac{4}{3}}}_{L^{2}(\varOmega )}+ \Vert { n_{\varepsilon } }\Vert ^{{\frac{10}{3}}}_{L^{2}(\varOmega )}\right) }\\ \le&\displaystyle {C_{9}(T+1)~\text{ for } \text{ all }~ T > 0,}\\ \end{aligned} \end{aligned}$$

(2.4.41)

$$\begin{aligned} \begin{aligned} \displaystyle \int _{0}^T\displaystyle \int _{\varOmega } c_{\varepsilon }^{\frac{10}{3}} \le&\displaystyle {C_{10}\int _{0}^T\left( \Vert \nabla { c_{\varepsilon }}\Vert ^{2}_{L^{2}(\varOmega )}\Vert { c_{\varepsilon }}\Vert ^{{\frac{4}{3}}}_{L^{2}(\varOmega )}+ \Vert { c_{\varepsilon }}\Vert ^{{\frac{10}{3}}}_{L^{2}(\varOmega )}\right) }\\ \le&\displaystyle {C_{11}(T+1)~\text{ for } \text{ all }~ T > 0}\\ \end{aligned} \end{aligned}$$

(2.4.42)

as well as

$$\begin{aligned} \begin{aligned} \displaystyle \int _{0}^T\displaystyle \int _{\varOmega } |u_{\varepsilon }|^{\frac{10}{3}} \le&\displaystyle {C_{14}\int _{0}^T\left( \Vert \nabla { u_{\varepsilon }}\Vert ^{2}_{L^{2}(\varOmega )}\Vert { u_{\varepsilon }}\Vert ^{{\frac{4}{3}}}_{L^{2}(\varOmega )}+ \Vert { u_{\varepsilon }}\Vert ^{{\frac{10}{3}}}_{L^{2}(\varOmega )}\right) }\\ \le&\displaystyle {C_{15}(T+1)~\text{ for } \text{ all }~ T > 0}\\ \end{aligned} \end{aligned}$$

(2.4.43)

and

$$\begin{aligned} \begin{aligned} \displaystyle \int _{0}^T\displaystyle \Vert u_{\varepsilon }\Vert ^{2}_{L^6(\varOmega )} \le&\displaystyle {C_{16}\int _{0}^T\Vert \nabla { u_{\varepsilon }}\Vert ^{2}_{L^{2}(\varOmega )}}\\ \le&\displaystyle {C_{17}(T+1)~\text{ for } \text{ all }~ T > 0,} \end{aligned} \end{aligned}$$

(2.4.44)

where in the last inequality we have used the embedding $W^{1,2}_{0,\sigma } (\varOmega ) \hookrightarrow L^6 (\varOmega )$ and the Poincaré inequality. Finally, combining (2.4.33)–(2.4.36) with (2.4.40)–(2.4.44), we can obtain the results.

Lemma 2.20

Let $\frac{1}{3}<\alpha \le \frac{8}{21}$. Then there exist $\gamma =\frac{2\alpha +\frac{2}{3}}{\alpha +1}\in (1,2)$ and $C > 0$ independent of $\varepsilon$ such that, for each $T\in (0, T_{max,\varepsilon })$, the solution of (2.2.8) satisfies

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{0}^T\Vert n_{\varepsilon }\Vert ^{\frac{2\gamma }{2-\gamma }}_{L^\frac{6\gamma }{6-\gamma }(\varOmega )}\le C(T+1).}\\ \end{aligned} \end{aligned}$$

(2.4.45)

Proof

To this end, we first prove that for all $p\in (1,6\alpha )$, there exists a positive constant $C_1$ independent of $\varepsilon$ such that, for each $T\in (0, T_{max,\varepsilon })$, the solution of (2.2.8) satisfies

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{0}^T\Vert n_{\varepsilon }\Vert ^{\frac{2p(\alpha -\frac{1}{6})}{p-1}}_{L^p(\varOmega )}\le C_1(T+1).}\\ \end{aligned} \end{aligned}$$

(2.4.46)

In fact, by (2.4.4) and (2.4.6), we derive that for some positive constants $C_{2}$ and $C_3$ independent of $\varepsilon$ such that

$$ \begin{aligned}&\displaystyle \int _{0}^T\Vert n_{\varepsilon }\Vert ^{\frac{2p(\alpha -\frac{1}{6})}{p-1}}_{L^p(\varOmega )} \\ =&\displaystyle {\int _{0}^T\Vert { n_{\varepsilon }^{\alpha }}\Vert ^{{\frac{2p}{p-1 }\cdot \frac{6\alpha -1}{6\alpha }}}_{L^{\frac{p}{\alpha }}(\varOmega )}}\\ \le&\displaystyle {C_{2}\int _{0}^T\left( \Vert \nabla { n_{\varepsilon }^{\alpha }}\Vert ^{2}_{L^{2}(\varOmega )}\Vert { n_{\varepsilon }^{\alpha }}\Vert ^{{\frac{2p}{p-1 }\cdot \frac{6\alpha -1}{6\alpha }-2}}_{L^{\frac{1}{\alpha }}(\varOmega )}+ \Vert { n_{\varepsilon }^{\alpha }}\Vert ^{{\frac{2p}{p-1 }\cdot \frac{6\alpha -1}{6\alpha }}}_{L^{\frac{1}{\alpha }}(\varOmega )}\right) }\\ \le&\displaystyle {C_{3}(T+1)~\text{ for } \text{ all }~ T > 0,}\\ \end{aligned} $$

which implies that (2.4.46) holds. Next, by $\alpha \in (\frac{1}{3},\frac{8}{21}],$ we may choose $\gamma =\frac{2\alpha +\frac{2}{3}}{\alpha +1}$ such that

$$\begin{aligned} 1<\gamma <\min \{\frac{6\alpha }{\alpha +1},2\} \end{aligned}$$

(2.4.47)

as well as

$$\begin{aligned} p:=\frac{6\gamma }{6-\gamma }\in (1,6\alpha ) \end{aligned}$$

(2.4.48)

and

$$\begin{aligned} \frac{2p(\alpha -\frac{1}{6})}{p-1}=\frac{12\gamma (\alpha -\frac{1}{6})}{7\gamma -6}>\frac{2\gamma }{2-\gamma }. \end{aligned}$$

(2.4.49)

Collecting (2.4.46)–(2.4.49), one can derive (2.4.45) by the Young inequality.

2.4.2 Global Solvability of the Approximate System

The main task of this subsection is to prove the global solvability of the regularized problem (2.2.8). To this end, first, we need to establish some $\varepsilon$-dependent estimates for $n_{\varepsilon }$, $c_{\varepsilon }$ and $u_{\varepsilon }$.

Lemma 2.21

Let $\alpha >\frac{1}{3}$. Then there exists $C=C(\varepsilon )>0$ depending on $\varepsilon$ such that the solution of (2.2.8) satisfies

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{\varOmega }n^{2\alpha +2}_{\varepsilon }+\int _{\varOmega } | \nabla {u_{\varepsilon }}|^2\le C~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon }).}\\ \end{aligned} \end{aligned}$$

(2.4.50)

In addition, for each $T\in (0, T_{max,\varepsilon }]$ with $T < \infty$, one can find a constant $C > 0$ depending on $\varepsilon$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{0}^{ T}\int _{\varOmega } \left[ n_{\varepsilon }^{2\alpha } |\nabla {n_{\varepsilon }}|^2+ |\varDelta {u_{\varepsilon }}|^2\right] \le C.}\\ \end{aligned} \end{aligned}$$

(2.4.51)

Proof

In view of (2.2.9), we derive

$$F'_{\varepsilon }(n_{\varepsilon })\le \displaystyle \frac{1}{\varepsilon n_{\varepsilon }},$$

so that, by multiplying the first equation in (2.2.8) by $n_{\varepsilon }^{1+2\alpha }$, using $\nabla \cdot u_\varepsilon =0$, and applying the same argument as in the proof of (2.4.7)–(2.4.21), one can obtain that there exist positive constants $C_1$ and $C_2$ depending on $\varepsilon$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{\varOmega }n^{2\alpha +2}_{\varepsilon }\le C_1~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon })}\\ \end{aligned} \end{aligned}$$

(2.4.52)

and

$$ \int _{0}^{ T}\int _{\varOmega } n_{\varepsilon }^{2\alpha } |\nabla {n_{\varepsilon }}|^2\le C_2~~\text{ for } \text{ all }~T\in (0, T_{max,\varepsilon }]~~ \text{ with }~~T < \infty . $$

Now, from $D(1 + \varepsilon A) :=W^{2,2}(\varOmega ) \cap W_{0,\sigma }^{1,2}(\varOmega )\hookrightarrow L^\infty (\varOmega )$ and (2.4.5), it follows that, for some $C_3> 0$ and $C_4 > 0$,

$$\begin{aligned} \Vert Y_{\varepsilon }u_{\varepsilon }\Vert _{L^\infty (\varOmega )}=\Vert (I+\varepsilon A)^{-1}u_{\varepsilon }\Vert _{L^\infty (\varOmega )}\le C_3\Vert u_{\varepsilon }(\cdot ,t)\Vert _{L^2(\varOmega )}\le C_4~\text{ for } \text{ all }~t\in (0,T_{max,\varepsilon }). \end{aligned}$$

(2.4.53)

Next, testing the projected Stokes equation $u_{\varepsilon t} +Au_{\varepsilon } = \mathscr {P}[-\kappa (Y_{\varepsilon }u_{\varepsilon } \cdot \nabla )u_{\varepsilon }+n_{\varepsilon }\nabla \phi ]$ by $Au_{\varepsilon }$, we derive

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{2}}\frac{d}{dt}\Vert A^{\frac{1}{2}}u_{\varepsilon }\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}+ \int _{\varOmega }|Au_{\varepsilon }|^2 }\\ =&\displaystyle { \int _{\varOmega }Au_{\varepsilon }\mathscr {P}(-\kappa (Y_{\varepsilon }u_{\varepsilon } \cdot \nabla )u_{\varepsilon })+ \int _{\varOmega }\mathscr {P}(n_{\varepsilon }\nabla \phi ) Au_{\varepsilon }}\\ \le&\displaystyle { \frac{1}{2}\int _{\varOmega }|Au_{\varepsilon }|^2+\kappa ^2\int _{\varOmega } |(Y_{\varepsilon }u_{\varepsilon } \cdot \nabla )u_{\varepsilon }|^2+ \Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}\int _{\varOmega }n_{\varepsilon }^2~\text{ for } \text{ all }~t\in (0,T_{max,\varepsilon }).}\\ \end{aligned} \end{aligned}$$

(2.4.54)

However, in light of the Gagliardo–Nirenberg inequality, Young’s inequality and (2.4.53), there exists a positive constant $C_5$ such that

$$\begin{aligned} \begin{aligned} \kappa ^2\displaystyle \int _{\varOmega } |(Y_{\varepsilon }u_{\varepsilon } \cdot \nabla )u_{\varepsilon }|^2\le&\displaystyle { \kappa ^2\Vert Y_{\varepsilon }u_{\varepsilon }\Vert ^2_{L^\infty (\varOmega )}\int _{\varOmega }|\nabla u_{\varepsilon }|^2}\\ \le&\displaystyle { \kappa ^2\Vert Y_{\varepsilon }u_{\varepsilon }\Vert ^2_{L^\infty (\varOmega )}\int _{\varOmega }|\nabla u_{\varepsilon }|^2}\\ \le&\displaystyle { C_5\int _{\varOmega }|\nabla u_{\varepsilon }|^2~\text{ for } \text{ all }~t\in (0,T_{max,\varepsilon }).}\\ \end{aligned} \end{aligned}$$

(2.4.55)

Here, we have used the well-known fact that $\Vert A(\cdot )\Vert _{L^{2}(\varOmega )}$ defines a norm equivalent to $\Vert \cdot \Vert _{W^{2,2}(\varOmega )}$ on D(A) (see Theorem 2.c2.2-1.3 of Sohr 2001). Now, recall that $\Vert A^{\frac{1}{2}}u_{\varepsilon }\Vert ^{{{2}}}_{L^{{2}}(\varOmega )} = \Vert \nabla u_{\varepsilon }\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}.$ Substituting (2.4.55) into (2.4.54) yields

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{2}}\frac{d}{dt}\Vert \nabla u_{\varepsilon }\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}+ \int _{\varOmega }|\varDelta u_{\varepsilon }|^2 } \\ \le&\displaystyle {C_6\int _{\varOmega }|\nabla u_{\varepsilon }|^2+ \Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}\int _{\varOmega } n_{\varepsilon }^2~\text{ for } \text{ all }~t\in (0,T_{max,\varepsilon }).}\\ \end{aligned} \end{aligned}$$

(2.4.56)

Since $\alpha >\frac{1}{3}$ yields $2\alpha +2>\frac{8}{3}>2,$ by collecting (2.4.52) and (2.4.56) and performing some basic calculations, we can get the results.

Lemma 2.22

Under the assumptions of Theorem 2.2, one can find that there exists $C=C(\varepsilon )> 0$ depending on $\varepsilon$ such that

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

(2.4.57)

and

$$\begin{aligned} \int _0^{T}\int _{\varOmega }{|\varDelta c_{\varepsilon }|^2}\le C~\text{ for } \text{ all }~ T\in (0, T_{max,\varepsilon }]~~ \text{ with }~~T < \infty . \end{aligned}$$

(2.4.58)

Proof

First, testing the second equation in (2.2.8) against $-\varDelta c_{\varepsilon }$, employing Young’s inequality, and using (2.4.3) yields

$$\begin{aligned} \begin{aligned} \displaystyle {\frac{1}{{2}}\frac{d}{dt} \Vert \nabla c_{\varepsilon }\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}}=&\displaystyle {\int _{\varOmega } -\varDelta c_{\varepsilon }(\varDelta c_{\varepsilon }-c_{\varepsilon }+F_{\varepsilon }(n_{\varepsilon })-u_{\varepsilon }\cdot \nabla c_{\varepsilon })} \\ =&\displaystyle {-\int _{\varOmega } |\varDelta c_{\varepsilon }|^2-\int _{\varOmega } |\nabla c_{\varepsilon }|^{2}-\int _\varOmega F_{\varepsilon }(n_{\varepsilon })\varDelta c_{\varepsilon }-\int _\varOmega (u_{\varepsilon }\cdot \nabla c_{\varepsilon })\varDelta c_{\varepsilon }}\\ \le&\displaystyle {-\frac{1}{4}\int _{\varOmega } |\varDelta c_{\varepsilon }|^2-\int _{\varOmega } |\nabla c_{\varepsilon }|^{2}+ \int _\varOmega n_{\varepsilon }^2+\int _\varOmega |u_{\varepsilon }\cdot \nabla c_{\varepsilon }||\varDelta c_{\varepsilon }|} \end{aligned} \end{aligned}$$

(2.4.59)

for all $t\in (0,T_{max,\varepsilon })$. Next, one needs to estimate the last term on the right-hand side of (2.4.59). Indeed, in view of Sobolev’s embedding theorem ($W^{1,2}(\varOmega )\hookrightarrow L^6(\varOmega )$) and applying (2.4.50) and (2.4.5), we derive from Hölder’s inequality, the Gagliardo–Nirenberg inequality, and Young’s inequality that there exist positive constants $C_1$, $C_2$, $C_3$ and $C_4$ such that

$$\begin{aligned} \begin{aligned} \displaystyle {\int _\varOmega |u_{\varepsilon }\cdot \nabla c_{\varepsilon }||\varDelta c_{\varepsilon }|} \le&\displaystyle {\Vert u_{\varepsilon }\Vert _{L^{6}(\varOmega )}\Vert \nabla c_{\varepsilon }\Vert _{L^{3}(\varOmega )}}\Vert \varDelta c_{\varepsilon }\Vert _{L^{2}(\varOmega )}\\ \le&\displaystyle {C_1\Vert \nabla c_{\varepsilon }\Vert _{L^{3}(\varOmega )}}\Vert \varDelta c_{\varepsilon }\Vert _{L^{2}(\varOmega )}\\ \le&\displaystyle {C_2(\Vert \varDelta c_{\varepsilon }\Vert ^{\frac{3}{4}}_{L^2(\varOmega )}\Vert c_{\varepsilon }\Vert ^{\frac{1}{4}}_{L^2(\varOmega )}+\Vert c_{\varepsilon }\Vert ^{2}_{L^2(\varOmega )})\Vert \varDelta c_{\varepsilon }\Vert _{L^{2}(\varOmega )}}\\ \le&\displaystyle {C_3(\Vert \varDelta c_{\varepsilon }\Vert ^{\frac{7}{4}}_{L^2(\varOmega )}+\Vert \varDelta c_{\varepsilon }\Vert _{L^{2}(\varOmega )})}\\ \le&\displaystyle {\frac{1}{4}\Vert \varDelta c_{\varepsilon }\Vert ^{2}_{L^2(\varOmega )}+C_4~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon }).} \end{aligned} \end{aligned}$$

(2.4.60)

Inserting (2.4.60) into (2.4.59) and using (2.4.50), one obtains (2.4.57) and (2.4.58). This completes the proof of Lemma 2.22.

Lemma 2.23

Let $\alpha >\frac{1}{3}$. Assume that the hypothesis of Theorem 2.2 holds. Then there exists a positive constant $C=C(\varepsilon )$ depending on $\varepsilon$ such that, for any $3<q<6,$ the solution of (2.2.8) from Lemma 2.3 satisfies

$$\begin{aligned} \begin{aligned} \Vert A^\gamma u_{\varepsilon }(\cdot , t)\Vert _{L^2(\varOmega )}\le&\displaystyle {C~ \text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon })}\\ \end{aligned} \end{aligned}$$

(2.4.61)

as well as

$$\begin{aligned} \begin{aligned} \Vert u_{\varepsilon }(\cdot , t)\Vert _{L^{\infty }(\varOmega )}\le&\displaystyle {C~ \text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon })}\\ \end{aligned} \end{aligned}$$

(2.4.62)

and

$$\begin{aligned} \begin{aligned} \Vert \nabla c_{\varepsilon }(\cdot , t)\Vert _{L^{q}(\varOmega )}\le&\displaystyle {C~ \text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon }),}\\ \end{aligned} \end{aligned}$$

(2.4.63)

where $\gamma$ is the same as in (2.1.13).

Proof

Let $h_{\varepsilon }(x,t)=\mathscr {P}[n_{\varepsilon }\nabla \phi -\kappa (Y_{\varepsilon }u_{\varepsilon } \cdot \nabla )u_{\varepsilon } ]$. Because $\alpha >\frac{1}{3}$, then along with (2.4.50), and (2.4.53), there exist positive constants $q_0>\frac{3}{2}$ and $C_{1}$ such that

$$\begin{aligned} \Vert n_{\varepsilon }(\cdot ,t)\Vert _{L^{q_0}(\varOmega )}\le C_{1} ~\text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon }) \end{aligned}$$

(2.4.64)

and

$$\begin{aligned} \Vert h_{\varepsilon }(\cdot ,t)\Vert _{L^{q_0}(\varOmega )}\le C_{1} ~\text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon }). \end{aligned}$$

(2.4.65)

Hence, because $q_0>\frac{3}{2}$, we pick an arbitrary $\gamma \in (\frac{3}{4}, 1)$ and, then, $-\gamma -\frac{3}{2}(\frac{1}{q_0}-\frac{1}{2})>-1$. Therefore, in view of the smoothing properties of the Stokes semigroup Giga (1986), we derive that, for some $\lambda$, $C_{2} > 0,$ and $C_{3} > 0$,

$$\begin{aligned} \begin{aligned}&\Vert A^\gamma u_{\varepsilon }(\cdot , t)\Vert _{L^2(\varOmega )} \\ \le&\displaystyle {\Vert A^\gamma e^{-tA}u_0\Vert _{L^2(\varOmega )} +\int _0^t\Vert A^\gamma e^{-(t-\tau )A}h_{\varepsilon }(\cdot ,\tau )d\tau \Vert _{L^2(\varOmega )}d\tau }\\ \le&\displaystyle {\Vert A^\gamma u_0\Vert _{L^2(\varOmega )} +C_{2}\int _0^t(t-\tau )^{-\gamma -\frac{3}{2}(\frac{1}{q_0}-\frac{1}{2})}e^{-\lambda (t-\tau )}\Vert h_{\varepsilon }(\cdot ,\tau )\Vert _{L^{q_0}(\varOmega )}d\tau }\\ \le&\displaystyle {C_{3}~ \text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon }).}\\ \end{aligned} \end{aligned}$$

(2.4.66)

Observe that $\gamma >\frac{3}{4},$ and $D(A^\gamma )$ is continuously embedded into $L^\infty (\varOmega )$. Therefore, we derive that there exists a positive constant $C_{4}$ such that

$$\begin{aligned} \Vert u_{\varepsilon }(\cdot , t)\Vert _{L^\infty (\varOmega )}\le C_{4}~ \text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon }) \end{aligned}$$

(2.4.67)

from (2.4.66). However, from (2.4.57), with the help of Sobolev’s embedding theorem, it follows that, for any fixed $\tilde{q}\in (3,6)$,

$$\begin{aligned} \begin{aligned} \Vert c_{\varepsilon }(\cdot , t)\Vert _{L^{\tilde{q}}(\varOmega )}\le C_{5}~ \text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon }). \end{aligned} \end{aligned}$$

(2.4.68)

Now, involving the variation-of-constants formula for $c_{\varepsilon }$ and applying $\nabla \cdot u_{\varepsilon }=0$ in $x\in \varOmega , t>0$, we have

$$\begin{aligned} c_{\varepsilon }(t)=e^{t(\varDelta -1)}c_0 +\int _{0}^{t}e^{(t-s)(\varDelta -1)}(F_\varepsilon (n_{\varepsilon }(s))+\nabla \cdot (u_{\varepsilon }(s) c_{\varepsilon }(s)) ds,~ t\in (0, T_{max,\varepsilon }), \end{aligned}$$

(2.4.69)

so that, for any $3<q <\min \{\frac{3q_0}{(3-q_0)_{+}},\tilde{q}\}$, we have

$$\begin{aligned} \begin{aligned}&\displaystyle {\Vert \nabla c_{\varepsilon }(\cdot , t)\Vert _{L^{q}(\varOmega )}}\\ \le&\displaystyle {\Vert \nabla e^{t(\varDelta -1)} c_0\Vert _{L^{q}(\varOmega )}+ \int _{0}^t\Vert \nabla e^{(t-s)(\varDelta -1)}F_\varepsilon (n_{\varepsilon }(s))\Vert _{L^q(\varOmega )}ds}\\&\displaystyle {+\int _{0}^t\Vert \nabla e^{(t-s)(\varDelta -1)}\nabla \cdot (u_{\varepsilon }(s) c_{\varepsilon }(s))\Vert _{L^q(\varOmega )}ds.}\\ \end{aligned} \end{aligned}$$

(2.4.70)

To address the right-hand side of (2.4.70), in view of (2.1.13), we first use Lemma 2.4 to get

$$\begin{aligned} \begin{aligned} \Vert \nabla e^{t(\varDelta -1)} c_0\Vert _{L^{q}(\varOmega )}\le&\displaystyle {C_{6}~ \text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon }).}\\ \end{aligned} \end{aligned}$$

(2.4.71)

Since (2.4.64) and (2.4.68) yields

$$ -\frac{1}{2}-\frac{3}{2}\left( \frac{1}{q_0}-\frac{1}{q}\right) >-1, $$

together with this and (2.4.3), by using Lemma 2.4 again, the second term of the right-hand side is estimated as

$$ \begin{aligned}&\displaystyle {\int _{0}^t\Vert \nabla e^{(t-s)(\varDelta -1)}F_\varepsilon (n_{\varepsilon }(s))\Vert _{L^{q}(\varOmega )}ds}\\ \le&\displaystyle {C_{7}\int _{0}^t[1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{q})}] e^{-(t-s)}\Vert n_{\varepsilon }(s)\Vert _{L^{q_0}(\varOmega )}ds}\\ \le&\displaystyle {C_{8}~ \text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon }).}\\ \end{aligned} $$

Finally, we will address the third term on the right-hand side of (2.4.70). To this end, we choose $0< \iota < \frac{1}{2}$ satisfying $\frac{1}{2} + \frac{3}{2}(\frac{1}{\tilde{q}}-\frac{1}{q}) <\iota$ and $\tilde{\kappa }\in (0, \frac{1}{2}-\iota )$. In view of Hölder’s inequality, we derive from Lemma 2.4, (2.4.68) and (2.4.67) that there exist constants $C_{9}$, $C_{10}$, $C_{11}$ and $C_{12}$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{0}^t\Vert \nabla e^{(t-s)(\varDelta -1)}\nabla \cdot (u_{\varepsilon }(s) c_{\varepsilon }(s))\Vert _{L^{\tilde{q}}(\varOmega )}ds}\\ \le&\displaystyle {C_{9}\int _{0}^t\Vert (-\varDelta +1)^\iota e^{(t-s)(\varDelta -1)}\nabla \cdot (u_\varepsilon (s) c_\varepsilon (s))\Vert _{L^{q}(\varOmega )}ds}\\ \le&\displaystyle {C_{10}\int _{0}^t(t-s)^{-\iota -\frac{1}{2}-\tilde{\kappa }} e^{-\lambda (t-s)}\Vert u_\varepsilon (s) c_\varepsilon (s)\Vert _{L^{\tilde{q}}(\varOmega )}ds}\\ \le&\displaystyle {C_{11}\int _{0}^t(t-s)^{-\iota -\frac{1}{2}-\tilde{\kappa }} e^{-\lambda (t-s)}\Vert u_\varepsilon (s)\Vert _{L^{\infty }(\varOmega )}\Vert c_\varepsilon (s)\Vert _{L^{\tilde{q}}(\varOmega )}ds}\\ \le&\displaystyle {C_{12}~ \text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon }).}\\ \end{aligned} \end{aligned}$$

(2.4.72)

Here, we have used the fact that

$$ \begin{aligned}\displaystyle \int _{0}^t(t-s)^{-\iota -\frac{1}{2}-\tilde{\kappa }} e^{-\lambda (t-s)}ds\le&\displaystyle {\int _{0}^{\infty }\sigma ^{-\iota -\frac{1}{2}-\tilde{\kappa }} e^{-\lambda \sigma }d\sigma <+\infty .}\\ \end{aligned} $$

Finally, collecting (2.4.70)–(2.4.72), we can obtain that there exists a positive constant $C_{13}$ such that

$$\begin{aligned} \int _{\varOmega }|\nabla {c_{\varepsilon }}(t)|^{q}\le C_{13}~\text{ for } \text{ all }~ t\in (0, T_{max,\varepsilon })~\text{ and } \text{ some }~q\in \left( 3,\min \left\{ \frac{3q_0}{(3-q_0)_{+}},\tilde{q}\right\} \right) . \end{aligned}$$

(2.4.73)

The proof of Lemma 2.23 is complete.

Then we can establish global existence in the approximate problem (2.2.8) by using Lemmas 2.21 and 2.22.

Lemma 2.24

Let $\alpha >\frac{1}{3}$. Then, for all $\varepsilon \in (0,1),$ the solution of (2.2.8) is global in time.

Proof

Assume that $T_{max,\varepsilon }$ is finite for some $\varepsilon \in (0,1)$. Fix $T\in (0, T_{max,\varepsilon })$, and let $M(T):=\sup _{t\in (0,T)}\Vert n_{\varepsilon }(\cdot ,t)\Vert _{L^\infty (\varOmega )}$ and $\tilde{h}_{\varepsilon }:=F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_{\varepsilon }+u_\varepsilon$. Then, by Lemma 2.23, (2.1.5) and (2.4.1), there exists $C_1 > 0$ such that

$$\begin{aligned} \begin{aligned} \Vert \tilde{h}_{\varepsilon }(\cdot , t)\Vert _{L^{q}(\varOmega )}\le&\displaystyle {C_1~ \text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon })~\text{ and } \text{ some } ~3<q<6.}\\ \end{aligned} \end{aligned}$$

(2.4.74)

Hence, because $\nabla \cdot u_{\varepsilon }=0$, we can derive

$$\begin{aligned} n_{\varepsilon }(t)=e^{(t-t_0)\varDelta }n_{\varepsilon }(\cdot ,t_0)-\int _{t_0}^{t}e^{(t-s)\varDelta }\nabla \cdot (n_{\varepsilon }(\cdot ,s)\tilde{h}_{\varepsilon }(\cdot ,s)) ds,~ t\in (t_0, T) \end{aligned}$$

(2.4.75)

by means of an associate variation-of-constants formula for n, where $t_0 := (t-1)_{+}$. If $t\in (0,1]$, by virtue of the maximum principle, we can derive

$$\begin{aligned} \begin{aligned} \Vert e^{(t-t_0)\varDelta }n_{\varepsilon }(\cdot ,t_0)\Vert _{L^{\infty }(\varOmega )}\le&\displaystyle {\Vert n_0\Vert _{L^{\infty }(\varOmega )},}\\ \end{aligned} \end{aligned}$$

(2.4.76)

while if $t > 1$ then, with the help of the $L^p$–$L^q$ estimates for the Neumann heat semigroup and Lemma 2.17, we conclude that

$$\begin{aligned} \begin{aligned} \Vert e^{(t-t_0)\varDelta }n_{\varepsilon }(\cdot ,t_0)\Vert _{L^{\infty }(\varOmega )}\le&\displaystyle {C_2(t-t_0)^{-\frac{3}{2}}\Vert n_{\varepsilon }(\cdot ,t_0)\Vert _{L^{1}(\varOmega )}\le C_3.}\\ \end{aligned} \end{aligned}$$

(2.4.77)

Finally, we fix an arbitrary $p\in (3,q)$ and then once more invoke known smoothing properties of the Stokes semigroup (see P. 201 of Giga 1986) and Hölder’s inequality to find $C_4 > 0$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle \int _{t_0}^t\Vert e^{(t-s)\varDelta }\nabla \cdot (n_{\varepsilon }(\cdot ,s)\tilde{h}_{\varepsilon }(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds\\ \le&\displaystyle C_4\int _{t_0}^t(t-s)^{-\frac{1}{2}-\frac{3}{2p}}\Vert n_{\varepsilon }(\cdot ,s)\tilde{h}_{\varepsilon }(\cdot ,s)\Vert _{L^p(\varOmega )}ds\\ \le&\displaystyle C_4\int _{t_0}^t(t-s)^{-\frac{1}{2}-\frac{3}{2p}}\Vert n_{\varepsilon }(\cdot ,s)\Vert _{L^{\frac{pq}{q-p}}(\varOmega )}\Vert \tilde{h}_{\varepsilon }(\cdot ,s)\Vert _{L^{q}(\varOmega )}ds\\ \le&\displaystyle C_4\int _{t_0}^t(t-s)^{-\frac{1}{2}-\frac{3}{2p}}\Vert u_{\varepsilon }(\cdot ,s)\Vert _{L^{\infty }(\varOmega )}^b\Vert u_{\varepsilon }(\cdot ,s)\Vert |_{L^1(\varOmega )}^{1-b}\Vert \tilde{h}_{\varepsilon }(\cdot ,s)\Vert _{L^{q}(\varOmega )}ds\\ \le&\displaystyle C_5M^b(T)~\text{ for } \text{ all }~ t\in (0, T),\\ \end{aligned} \end{aligned}$$

(2.4.78)

where $b:=\frac{pq-q+p}{pq}\in (0,1)$ and

$$ C_5:=C_4C_1^{2-b}\int _{0}^{1}\sigma ^{-\frac{1}{2}-\frac{3}{2p}}d\sigma . $$

Since $p>3$, we conclude that $-\frac{1}{2}-\frac{3}{2p}>-1$. In combination with (2.4.75)–(2.4.78) and using the definition of M(T),  we obtain $C_6 > 0$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle M(T)\le C_6+C_6M^b(T)~\text{ for } \text{ all }~ T\in (0, T_{max,\varepsilon }).\\ \end{aligned} \end{aligned}$$

(2.4.79)

Hence, in view of $b<1$, with some basic calculation, since $T\in (0, T_{max,\varepsilon })$ was arbitrary, we can obtain there exists a positive constant $C_7$ such that

$$\begin{aligned} \begin{aligned} \Vert n_{\varepsilon }(\cdot , t)\Vert _{L^{\infty }(\varOmega )}\le&\displaystyle {C_7~ \text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon }).}\\ \end{aligned} \end{aligned}$$

(2.4.80)

To prove the boundedness of $\Vert \nabla c_{\varepsilon }(\cdot , t)\Vert _{L^\infty (\varOmega )}$, we rewrite the variation-of-constants formula for $c_{\varepsilon }$ in the form

$$c_{\varepsilon }(\cdot , t) = e^{t(\varDelta -1) }c_0 +\int _{0}^te^{(t-s)(\varDelta -1)}[F_{\varepsilon }(n_{\varepsilon })(s)-u_{\varepsilon }(s)\cdot \nabla c_{\varepsilon }(s)]ds~ \text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon }).$$

Now, we choose $\theta \in (\frac{1}{2}+\frac{3}{2q},1),$ where $3<q<6$ (see (2.4.73)), then the domain of the fractional power $D((-\varDelta + 1)^\theta )\hookrightarrow W^{1,\infty }(\varOmega )$ (see Horstmann and Winkler 2005). Hence, in view of $L^p$–$L^q$ estimates associated with the heat semigroup, (2.4.62), (2.4.63) and (2.4.3), we derive that there exist positive constants $\lambda$, $C_{8}$, $C_{9}$, $C_{10}$ and $C_{11}$ such that

$$\begin{aligned} \begin{aligned}&\Vert c_{\varepsilon }(\cdot , t)\Vert _{W^{1,\infty }(\varOmega )}\\ \le&\displaystyle {C_{8}\Vert (-\varDelta +1)^\theta c_{\varepsilon }(\cdot , t)\Vert _{L^{q}(\varOmega )}}\\ \le&\displaystyle {C_{9}t^{-\theta }e^{-\lambda t}\Vert c_0\Vert _{L^{q}(\varOmega )}+C_{9}\int _{0}^t(t-s)^{-\theta }e^{-\lambda (t-s)} \Vert (F_{\varepsilon }(n_{\varepsilon })-u_{\varepsilon } \cdot \nabla c_{\varepsilon })(s)\Vert _{L^{q}(\varOmega )}ds}\\ \le&\displaystyle {C_{10}+C_{10}\int _{0}^t(t-s)^{-\theta }e^{-\lambda (t-s)}[\Vert n_{\varepsilon }(s)\Vert _{L^q(\varOmega )}+\Vert u_{\varepsilon }(s)\Vert _{L^\infty (\varOmega )} \Vert \nabla c_{\varepsilon }(s)\Vert _{L^q(\varOmega )}]ds}\\ \le&\displaystyle {C_{11}~ \text{ for } \text{ all }~ t\in (0,T_{max,\varepsilon }).} \end{aligned} \end{aligned}$$

(2.4.81)

Here, we have used Hölder’s inequality as well as

$$\int _{0}^t(t-s)^{-\theta }e^{-\lambda (t-s)}\le \int _{0}^{\infty }\sigma ^{-\theta }e^{-\lambda \sigma }d\sigma <+\infty .$$

In view of (2.4.61), (2.4.80) and (2.4.81), we apply Lemma 2.3 to reach a contradiction.

2.4.3 Regularity Property of Time Derivatives

In preparation of an Aubin–Lions-type compactness argument, we will rely on an additional regularity estimate for $n_\varepsilon F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_\varepsilon$, $u_\varepsilon \cdot \nabla c_\varepsilon$, $n_\varepsilon u_\varepsilon$ and $c_\varepsilon u_\varepsilon$.

Lemma 2.25

Let $\alpha >\frac{1}{3}$, and assume that (2.1.13) holds. Then one can find $C > 0$ independent of $\varepsilon$ such that, for all $T\in (0,\infty ),$

$$\begin{aligned} \begin{aligned}&\displaystyle \int _0^T\int _{\varOmega }\left[ |n_\varepsilon F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_\varepsilon |^{\frac{3\alpha +1}{2}} +|n_\varepsilon u_\varepsilon |^{\frac{2\alpha +\frac{2}{3}}{\alpha +1}}\right] \\ \le&C(T+1), \qquad \text{ if }~\frac{1}{3}<\alpha \le \frac{8}{21},\\ \end{aligned} \end{aligned}$$

(2.4.82)

$$\begin{aligned} \begin{aligned}&\displaystyle \int _0^T\int _{\varOmega }\left[ |n_\varepsilon F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_\varepsilon |^{\frac{3\alpha +1}{2}} +|n_\varepsilon u_\varepsilon |^{\frac{10(3\alpha +1)}{9(\alpha +2)}}\right] \\ \le&C(T+1), \qquad \text{ if }~\frac{8}{21}<\alpha \le \frac{1}{2},\\ \end{aligned} \end{aligned}$$

(2.4.83)

$$\begin{aligned} \begin{aligned}&\displaystyle \int _0^T\int _{\varOmega }\left[ |n_\varepsilon F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_\varepsilon |^{\frac{10\alpha }{3+2\alpha }}+|n_\varepsilon u_\varepsilon |^{\frac{10\alpha }{3(\alpha +1)}} \right] ,\\ \le&C(T+1)\qquad \text{ if }~\frac{1}{2}<\alpha <1\\ \end{aligned} \end{aligned}$$

(2.4.84)

as well as

$$\begin{aligned} \begin{aligned} \displaystyle \int _0^T\int _{\varOmega }\left[ |n_\varepsilon F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_\varepsilon |^{2}+|n_\varepsilon u_\varepsilon |^{\frac{5}{3}}\right] \le C(T+1),~\text{ if }~\alpha \ge 1\\ \end{aligned} \end{aligned}$$

(2.4.85)

and

$$\begin{aligned} \displaystyle \int _0^T\int _{\varOmega }\left[ |u_\varepsilon \cdot \nabla c_\varepsilon |^{\frac{5}{4}} +|c_\varepsilon u_\varepsilon |^{\frac{5}{3}}\right] \le C(T+1). \end{aligned}$$

(2.4.86)

Proof

First, by (2.1.5), (2.4.1) and (2.2.10), we derive

$$n_\varepsilon F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\le C_Sn_{\varepsilon }^{(1-\alpha )_{+}}$$

with $(1-\alpha )_{+}=\max \{0,1-\alpha \}.$ Case $\frac{8}{21}<\alpha \le \frac{1}{2}$: It is not difficult to verify that

$$\frac{2}{3\alpha +1}=\frac{1}{2}+\frac{3}{6\alpha +2}(1-\alpha )$$

and

$$\frac{9(\alpha +2)}{10(3\alpha +1)}=\frac{3}{10}+\frac{3}{6\alpha +2},$$

so that, recalling (2.4.29), (2.4.44) and Hölder’s inequality, we can obtain (2.4.83). While if $\frac{1}{3}<\alpha \le \frac{8}{21}$, in light of (2.4.6), (2.4.29), (2.4.32), (2.4.45), an employment of the Hölder and Young inequalities to shows that

$$ \begin{aligned}&\displaystyle \int _0^T\int _{\varOmega }\left[ |n_\varepsilon F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_\varepsilon |^{\frac{3\alpha +1}{2}} +|n_\varepsilon u_\varepsilon |^{\gamma }\right] \\ \le&\displaystyle C_1\left[ \int _{0}^T\displaystyle \int _{\varOmega } n_\varepsilon ^{\frac{6\alpha +2}{3}}\right] ^{\frac{3-3\alpha }{4}} \left[ \int _{0}^T\displaystyle \int _{\varOmega } |\nabla c_\varepsilon |^{2}\right] ^{\frac{3\alpha +1}{4}}\\&\displaystyle +C_1 \int _0^T\Vert n_\varepsilon \Vert ^{\gamma }_{L^{\frac{6\gamma }{6-\gamma }}(\varOmega )}\Vert u_\varepsilon \Vert ^{\gamma }_{L^6(\varOmega )}\\ \le&C_2(T+1), \end{aligned} $$

where $\gamma =\frac{2\alpha +\frac{2}{3}}{\alpha +1}$ is given by Lemma 2.20.

Other cases can be proved very similarly. Therefore, we omit their proofs.

To prepare our subsequent compactness properties of $(n_\varepsilon , c_\varepsilon ,u_\varepsilon )$ by means of the Aubin–Lions lemma (see Simon 1986), we use Lemmas 2.17–2.19 to obtain the following regularity property with respect to the time variable.

Lemma 2.26

Let $\alpha >\frac{1}{3}$, and assume that (2.1.13) holds. Then there exists $C>0$ independent of $\varepsilon$ such that

$$\begin{aligned} \displaystyle \int _0^T\Vert \partial _tn_\varepsilon (\cdot ,t)\Vert _{(W^{2,4}(\varOmega ))^*}dt \le C(T+1)~\text{ for } \text{ all }~ T\in (0,\infty ) \end{aligned}$$

(2.4.87)

as well as

$$\begin{aligned} \displaystyle \int _0^T\Vert \partial _tc_\varepsilon (\cdot ,t)\Vert _{(W^{1,5}(\varOmega ))^*}^{\frac{5}{4}}dt \le C(T+1)~\text{ for } \text{ all }~ T\in (0,\infty ) \end{aligned}$$

(2.4.88)

and

$$\begin{aligned} \displaystyle \int _0^T\Vert \partial _tu_\varepsilon (\cdot ,t)\Vert _{(W^{1,5}_{0,\sigma }(\varOmega ))^*}^{\frac{5}{4}}dt \le C(T+1)~\text{ for } \text{ all }~ T\in (0,\infty ). \end{aligned}$$

(2.4.89)

Proof

Firstly, testing the first equation of (2.2.8) by certain $\varphi \in C^{\infty }(\bar{\varOmega })$, we have

$$ \begin{aligned}&\displaystyle \left| \int _{\varOmega }(n_{\varepsilon ,t})\varphi \right| \\ =&\displaystyle {\left| \int _{\varOmega }\left[ \varDelta n_{\varepsilon }-\nabla \cdot (n_{\varepsilon }F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_{\varepsilon })-u_{\varepsilon }\cdot \nabla n_{\varepsilon }\right] \varphi \right| } \\ =&\displaystyle {\left| \int _\varOmega \left[ -\nabla n_{\varepsilon }\cdot \nabla \varphi +n_{\varepsilon }F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_{\varepsilon }\cdot \nabla \varphi + n_{\varepsilon }u_{\varepsilon }\cdot \nabla \varphi \right] \right| }\\ \le&\displaystyle {\left| \int _\varOmega \left[ |\nabla n_{\varepsilon }|+|n_{\varepsilon }F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_{\varepsilon }|+ |n_{\varepsilon }u_{\varepsilon }|\right] \right| \Vert \varphi \Vert _{W^{1,\infty }(\varOmega )}} \end{aligned} $$

for all $t>0$.

Observe that the embedding $W^{2,4 }(\varOmega )\hookrightarrow W^{1,\infty }(\varOmega )$, so that, in view of $\alpha >\frac{1}{3}$, Lemmas 2.19 and 2.25, we deduce from the Young inequality that for some $C_1$ and $C_2$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle \int _0^T\Vert \partial _{t}n_{\varepsilon }(\cdot ,t)\Vert _{(W^{2,4 }(\varOmega ))^*}dt\\ \le&\displaystyle {C_1\left\{ \int _0^T\int _{\varOmega } |\nabla n_{\varepsilon }|^{r_1}+\int _0^T\int _{\varOmega }|n_\varepsilon F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_\varepsilon |^{r_1}+\int _0^T\int _{\varOmega }|n_\varepsilon u_\varepsilon |^{r_2}+T\right\} }\\ \le&\displaystyle {C_2(T+1)~~\text{ for } \text{ all }~~ T > 0, }\\ \end{aligned} \end{aligned}$$

(2.4.90)

where

$$ r_1= \left\{ \begin{aligned} \frac{3\alpha +1}{2}~~\text{ if }~\frac{1}{3}<\alpha \le \frac{1}{2},\\ \frac{10\alpha }{3+2\alpha }~~\text{ if }~\frac{1}{2}<\alpha <1,\\ 2~~\text{ if }~\alpha \ge 1\\ \end{aligned}\right. $$

and

$$ r_2= \left\{ \begin{aligned} \frac{2\alpha +\frac{2}{3}}{\alpha +1}~~\text{ if }~\frac{1}{3}<\alpha \le \frac{8}{21},\\ \frac{10(3\alpha +1)}{9(\alpha +2)}~~\text{ if }~\frac{8}{21}<\alpha \le \frac{1}{2},\\ \frac{10\alpha }{3(\alpha +1)}~~\text{ if }~\frac{1}{2}<\alpha <1,\\ \frac{5}{3}~~\text{ if }~\alpha \ge 1,\\ \end{aligned}\right. $$

Likewise, given any $\varphi \in C^\infty (\bar{\varOmega })$, we may test the second equation in (2.2.8) against $\varphi$ to conclude that

$$ \begin{aligned}&\displaystyle \left| \int _{\varOmega }\partial _{t}c_{\varepsilon }(\cdot ,t)\varphi \right| \\=&\displaystyle {\left| \int _{\varOmega }\left[ \varDelta c_{\varepsilon }-c_{\varepsilon }+n_{\varepsilon }-u_{\varepsilon }\cdot \nabla c_{\varepsilon }\right] \cdot \varphi \right| } \\ =&\displaystyle {\left| -\int _\varOmega \nabla c_{\varepsilon }\cdot \nabla \varphi -\int _\varOmega c_{\varepsilon }\varphi +\int _\varOmega n_{\varepsilon } \varphi +\int _\varOmega c_{\varepsilon }u_\varepsilon \cdot \nabla \varphi \right| }\\ \le&\displaystyle {\left\{ \Vert \nabla c_{\varepsilon }\Vert _{L^{{\frac{5}{4}}}(\varOmega )}+\Vert c_{\varepsilon } \Vert _{L^{\frac{5}{4}}(\varOmega )}+\Vert n_{\varepsilon } \Vert _{L^{\frac{5}{4}}(\varOmega )}+\Vert c_{\varepsilon }u_\varepsilon \Vert _{L^{\frac{5}{4}}(\varOmega )}\right\} \Vert \varphi \Vert _{W^{1,5}(\varOmega )}}\\ \end{aligned} $$

for all $t>0$. Thus, from Lemmas 2.19 and 2.25 again, in light of $\alpha >\frac{1}{3}$, we invoke the Young inequality again and obtain that there exist positive constant $C_{3}$ and $C_{4}$ such that

$$ \begin{aligned}&\displaystyle \int _0^T\Vert \partial _{t}c_{\varepsilon }(\cdot ,t)\Vert ^{\frac{5}{4}}_{(W^{1,5}(\varOmega ))^*}dt\\ \le&\displaystyle {C_{3}\left( \int _0^T\int _\varOmega |\nabla c_{\varepsilon }|^{2}+\int _0^T\int _\varOmega n_{\varepsilon }^{r_3}+\int _0^T\int _\varOmega c_\varepsilon ^{\frac{10}{3}}+\int _0^T\int _\varOmega |u_\varepsilon |^{\frac{10}{3}}+T\right) }\\ \le&\displaystyle {C_{4}(T+1) ~~\text{ for } \text{ all }~~ T>0}\\ \end{aligned} $$

with

$$\begin{aligned} r_3= \left\{ \begin{aligned} \frac{6\alpha +2}{3}~~\text{ if }~\frac{1}{3}<\alpha \le \frac{1}{2},\\ \frac{10\alpha }{3}~~\text{ if }~\frac{1}{2}<\alpha <1,\\ \frac{10}{3}~~\text{ if }~\alpha \ge 1.\\ \end{aligned}\right. \end{aligned}$$

(2.4.91)

Finally, for any given $\varphi \in C^{\infty }_{0,\sigma } (\varOmega ;\mathbb {R}^3)$, we infer from the third equation in (2.2.8) that for all $t>0$

$$ \displaystyle \left| \int _{\varOmega }\partial _{t}u_{\varepsilon }(\cdot ,t)\varphi \right| = \displaystyle \left| -\int _\varOmega \nabla u_{\varepsilon }\cdot \nabla \varphi -\kappa \int _\varOmega (Y_{\varepsilon }u_{\varepsilon }\otimes u_{\varepsilon })\cdot \nabla \varphi +\int _\varOmega n_{\varepsilon }\nabla \phi \cdot \varphi \right| . $$

Now, by virtue of (2.4.6), Lemmas 2.19 and 2.25, we also get that there exist positive constants $C_{5},C_{6}$ and $C_{7}$ such that

$$ \begin{aligned}&\displaystyle \int _0^T\Vert \partial _{t}u_{\varepsilon }(\cdot ,t)\Vert _{(W^{1,5}_{0,\sigma }(\varOmega ))^*}^{\frac{5}{4}}dt\\ \le&\displaystyle {C_{5}\left( \int _0^T\int _\varOmega |\nabla u_{\varepsilon }|^{\frac{5}{4}} +\int _0^T\int _\varOmega |Y_{\varepsilon }u_{\varepsilon }\otimes u_{\varepsilon }|^{\frac{5}{4}}+\int _0^T\int _\varOmega n_\varepsilon ^{\frac{5}{4}}\right) }\\F \le&\displaystyle {C_{6}\left( \int _0^T\int _\varOmega |\nabla u_{\varepsilon }|^{2}+\int _0^T\int _\varOmega |Y_{\varepsilon }u_\varepsilon |^{2} +\int _0^T\int _\varOmega |u_\varepsilon |^{\frac{10}{3}}+\int _0^T\int _\varOmega n_{\varepsilon }^{r_3}+T\right) }\\ \le&\displaystyle {C_{7}(T+1) ~~\text{ for } \text{ all }~~ T>0,} \end{aligned} $$

which implies (2.4.89). Here $r_3$ is the same as (2.4.91).

2.4.4 Global Existence of Weak Solutions

Based on the above lemmas and by extracting suitable subsequences in a standard way, we can prove Theorem 2.2.

Lemma 2.27

Let (2.1.4), (2.1.5) and (2.1.13) hold, and suppose that $\alpha >\frac{1}{3}.$ There exists $(\varepsilon _j)_{j\in \mathbb {N}}\subset (0, 1)$ such that $\varepsilon _j\searrow 0$ as $j\rightarrow \infty$ and such that as $\varepsilon = \varepsilon _j\searrow 0$ we have

$$\begin{aligned} n_\varepsilon \rightarrow n ~\text{ a.e. }~\text{ in }~\varOmega \times (0,\infty )~\text{ and } \text{ in }~ L_{loc}^{r}(\bar{\varOmega }\times [0,\infty ))~\text{ with }~r= \left\{ \begin{aligned}&\frac{3\alpha +1}{2}~\text{ if }~\frac{1}{3}<\alpha \le \frac{1}{2},\\&\frac{10\alpha }{3+2\alpha }~\text{ if }~\frac{1}{2}<\alpha <1,\\&2~\text{ if }~\alpha \ge 1, \end{aligned}\right. \end{aligned}$$

(2.4.92)

$$\begin{aligned} \nabla n_\varepsilon \rightharpoonup \nabla n ~\text{ in }~ L_{loc}^{r}(\bar{\varOmega }\times [0,\infty ))~\text{ with }~r= \left\{ \begin{aligned}&\frac{3\alpha +1}{2}~\text{ if }~\frac{1}{3}<\alpha \le \frac{1}{2},\\&\frac{10\alpha }{3+2\alpha }~\text{ if }~\frac{1}{2}<\alpha <1,\\&2~\text{ if }~\alpha \ge 1, \end{aligned}\right. \end{aligned}$$

(2.4.93)

$$\begin{aligned} c_\varepsilon \rightarrow c ~\text{ in }~ L^{2}_{loc}(\bar{\varOmega }\times [0,\infty ))~\text{ and }~\text{ a.e. }~\text{ in }~\varOmega \times (0,\infty ), \end{aligned}$$

(2.4.94)

$$\begin{aligned} \nabla c_\varepsilon \rightarrow \nabla c ~\text{ a.e. }~\text{ in }~\varOmega \times (0,\infty ), \end{aligned}$$

(2.4.95)

$$\begin{aligned} u_\varepsilon \rightarrow u~\text{ in }~ L_{loc}^2(\bar{\varOmega }\times [0,\infty ))~\text{ and }~\text{ a.e. }~\text{ in }~\varOmega \times (0,\infty ) \end{aligned}$$

(2.4.96)

as well as

$$\begin{aligned} \nabla c_\varepsilon \rightharpoonup \nabla c~\begin{aligned} \text{ in }~ L_{loc}^{2}(\bar{\varOmega }\times [0,\infty )) \end{aligned} \end{aligned}$$

(2.4.97)

and

$$\begin{aligned} \nabla u_\varepsilon \rightharpoonup \nabla u ~ \text{ in }~L^{2}_{loc}(\bar{\varOmega }\times [0,\infty )) \end{aligned}$$

(2.4.98)

and

$$\begin{aligned} u_\varepsilon \rightharpoonup u ~ \text{ in }~L^{\frac{10}{3}}_{loc}(\bar{\varOmega }\times [0,\infty )) \end{aligned}$$

(2.4.99)

with some triple (n, c, u) that is a global weak solution of (2.1.3) in the sense of Definition 2.1.

Proof

First, from Lemma 2.19 and (2.4.87), we derive that there exists a positive constant $C_0$ such that

$$\begin{aligned} \begin{aligned} \Vert n_{\varepsilon }\Vert _{L^{r}_{loc}([0,\infty ); W^{1,r}(\varOmega ))}\le C_0(T+1)~\text{ and }~\Vert \partial _{t}n_{\varepsilon }\Vert _{L^{1}_{loc}([0,\infty ); (W^{2,4}(\varOmega ))^*)}\le C_0(T+1), \end{aligned} \end{aligned}$$

(2.4.100)

where r is given by (2.4.92). Hence, from (2.4.100) and the Aubin–Lions lemma (see, e.g., Simon 1986), we conclude that

$$\begin{aligned} (n_{\varepsilon })_{\varepsilon \in (0,1)}~\text{ is } \text{ strongly } \text{ precompact } \text{ in }~L^{r}_{loc}(\bar{\varOmega }\times [0,\infty )), \end{aligned}$$

(2.4.101)

so that, there exists a sequence $(\varepsilon _j)_{j\in \mathbb {N}}\subset (0, 1)$ such that $\varepsilon =\varepsilon _j\searrow 0$ as $j\rightarrow \infty$ and

$$\begin{aligned} n_\varepsilon \rightarrow n ~\text{ a.e. }~\text{ in }~\varOmega \times (0,\infty )~\text{ and } \text{ in }~ L_{loc}^{r}(\bar{\varOmega }\times [0,\infty ))~~~\text{ as }~~\varepsilon =\varepsilon _j\searrow 0, \end{aligned}$$

(2.4.102)

where r is the same as (2.4.92). Now, in view of Lemmas 2.18, 2.19, 2.25 and 2.26, employing the same arguments as in the proof of (2.4.100)–(2.4.102), we can derive (2.4.92)–(2.4.94) and (2.4.96)–(2.4.99) hold. Next, let $g_\varepsilon (x, t) := -c_\varepsilon +F_{\varepsilon }(n_{\varepsilon })-u_{\varepsilon }\cdot \nabla c_{\varepsilon }.$ With this notation, the second equation of (2.2.8) can be rewritten in component form as

$$\begin{aligned} c_{\varepsilon t}-\varDelta c_{\varepsilon } = g_\varepsilon . \end{aligned}$$

(2.4.103)

Case $\frac{1}{3}<\alpha \le \frac{1}{2}$: Observe that

$$\frac{5}{4}<\frac{4}{3}<\min \left\{ \frac{6\alpha +2}{3},\frac{10}{3}\right\} ~\text{ for }~\frac{1}{3}<\alpha \le \frac{1}{2}.$$

Thus, recalling (2.4.29), (2.4.32) and (2.4.86) and applying Hölder’s inequality, we conclude that, for any $\varepsilon \in (0,1)$, $g_\varepsilon$ is bounded in $L^{\frac{5}{4}} (\varOmega \times (0, T))$, and we may invoke the standard parabolic regularity theory to (2.4.103) and infer that $(c_{\varepsilon })_{\varepsilon \in (0,1)}$ is bounded in $L^{\frac{5}{4}} ((0, T); W^{2,\frac{5}{4}}(\varOmega ))$. Hence, by virtue of (2.4.88) and the Aubin–Lions lemma, we derive the relative compactness of $(c_{\varepsilon })_{\varepsilon \in (0,1)}$ in $L^{\frac{5}{4}} ((0, T); W^{1,\frac{5}{4}}(\varOmega ))$. We can pick an appropriate subsequence that is still written as $(\varepsilon _j )_{j\in \mathbb {N}}$ such that $\nabla c_{\varepsilon _j} \rightarrow z_1$ in $L^{\frac{5}{4}} (\varOmega \times (0, T))$ for all $T\in (0, \infty )$ and some $z_1\in L^{\frac{5}{4}} (\varOmega \times (0, T))$ as $j\rightarrow \infty$. Therefore, by (2.4.88), we can also derive that $\nabla c_{\varepsilon _j} \rightarrow z_1$ a.e. in $\varOmega \times (0, \infty )$ as $j \rightarrow \infty$. In view of (2.4.97) and Egorov’s theorem, we conclude that $z_1=\nabla c$ and hence (2.4.95) holds. Next, we pay attention to the case $\frac{1}{2}<\alpha <1$: By straightforward calculations, and using relation $\frac{1}{2}<\alpha <1$, one has

$$\frac{5}{4}<\frac{5}{3}<\min \left\{ \frac{10\alpha }{3},\frac{10}{3}\right\} .$$

Consequently, based on (2.4.30), (2.4.32) and (2.4.86), it follows from Hölder’s inequality that

$$\begin{aligned} c_{\varepsilon t}-\varDelta c_{\varepsilon }=g_\varepsilon ~\text{ is } \text{ bounded } \text{ in } ~L^{\frac{5}{4}} (\varOmega \times (0, T))~~\text{ for } \text{ any }~~\varepsilon \in (0,1). \end{aligned}$$

(2.4.104)

Employing almost exactly the same arguments as in the proof of the case $\frac{1}{3}<\alpha \le \frac{1}{2}$, and taking advantage of (2.4.104), we conclude the estimate (2.4.97). The proof of case $\alpha \ge 1$ is similar to that of case $\frac{1}{3}<\alpha \le \frac{1}{2}$, so we omit it.

In the following proof, we shall prove that (n, c, u) is a weak solution of problem (2.1.3) in Definition 2.1. In fact, by $\alpha >\frac{1}{3}$, we conclude that

$$ r>1, $$

where r is given by (2.4.92). Therefore, with the help of (2.4.92)–(2.4.94) and (2.4.96)–(2.4.98), we can derive (2.2.3). Now, by the nonnegativity of $n_\varepsilon$ and $c_\varepsilon$, we obtain $n \ge 0$ and $c\ge 0$. Next, from (2.4.98) and $\nabla \cdot u_{\varepsilon } = 0$, we conclude that $\nabla \cdot u = 0$ a.e. in $\varOmega \times (0, \infty )$. However, in view of (2.4.83), (2.4.84) and (2.4.85), we conclude that

$$\begin{aligned} n_\varepsilon F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_\varepsilon \rightharpoonup z_2 ~\text{ in }~ L^{r}(\varOmega \times (0,T)) \end{aligned}$$

(2.4.105)

as $\varepsilon = \varepsilon _j\searrow 0$ for each $T\in (0,\infty )$, where r is given by (2.4.92). However, it follows from (2.1.4), (2.2.10), (2.4.2), (2.4.92), (2.4.94) and (2.4.95) that

$$\begin{aligned} n_\varepsilon F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_\varepsilon \rightarrow nS(x, n, c)\nabla c~\text{ a.e. }~\text{ in }~\varOmega \times (0,\infty )~\text{ as }~\varepsilon = \varepsilon _j\searrow 0. \end{aligned}$$

(2.4.106)

Again by Egorov’s theorem, we gain $z_2=nS(x, n, c)\nabla c$, and therefore (2.4.105) can be rewritten as

$$\begin{aligned} n_\varepsilon F'_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_\varepsilon \rightharpoonup nS(x, n, c)\nabla c ~\text{ in }~ L^{r}(\varOmega \times (0,T)) \end{aligned}$$

(2.4.107)

as $\varepsilon = \varepsilon _j\searrow 0~$ for each $T\in (0,\infty )$, which together with $r>1$ implies the integrability of $nS(x, n, c)\nabla c$ in (2.2.4) as well. It is not difficult to check that

$$\begin{aligned}&{\frac{2\alpha +\frac{2}{3}}{\alpha +1}}>1~\text{ if }~\frac{1}{3}<\alpha \le \frac{8}{21}, \\&{\frac{10(3\alpha +1)}{9(\alpha +2)}}>1~\text{ if }~\frac{8}{21}<\alpha \le \frac{1}{2}, \\&{\frac{10\alpha }{3(\alpha +1)}} >1~\text{ if }~\frac{1}{2}<\alpha <1. \end{aligned}$$

Thereupon, recalling (2.4.83), (2.4.84) and (2.4.85), we infer that, for each $T\in (0, \infty ),$ when $\varepsilon = \varepsilon _j\searrow 0$,

$$\begin{aligned} n_\varepsilon u_{\varepsilon }\rightharpoonup z_3 ~\text{ in }~ L^{\tilde{r}}(\varOmega \times (0,T))~\text{ with }~\tilde{r} = \left\{ \begin{aligned}&{\frac{2\alpha +\frac{2}{3}}{\alpha +1}}~\text{ if }~\frac{1}{3}<\alpha \le \frac{8}{21}, \\&\frac{10(3\alpha +1)}{9(\alpha +2)}~\text{ if }~\frac{8}{21}<\alpha \le \frac{1}{2},\\&\frac{10\alpha }{3(\alpha +1)}~\text{ if }~\frac{1}{2}<\alpha <1,\\&\frac{5}{3}~\text{ if }~\alpha \ge 1. \end{aligned}\right. \end{aligned}$$

(2.4.108)

(2.4.108) together with (2.4.92) and (2.4.96) implies

$$\begin{aligned} n_\varepsilon u_\varepsilon \rightarrow nu~\text{ a.e. }~\text{ in }~\varOmega \times (0,\infty )~\text{ as }~\varepsilon = \varepsilon _j\searrow 0. \end{aligned}$$

(2.4.109)

(2.4.108) along with (2.4.109) and Egorov’s theorem guarantees that $z_3=nu$, whereupon we derive from (2.4.108) that

$$\begin{aligned} n_\varepsilon u_{\varepsilon }\rightharpoonup nu ~\text{ in }~ L^{\tilde{r}}(\varOmega \times (0,T))~\text{ with }~\tilde{r}= \left\{ \begin{aligned}&{\frac{2\alpha +\frac{2}{3}}{\alpha +1}}~\text{ if }~\frac{1}{3}<\alpha \le \frac{8}{21},\\&\frac{10(3\alpha +1)}{9(\alpha +2)}~\text{ if }~\frac{8}{21}<\alpha \le \frac{1}{2},\\&\frac{10\alpha }{3(\alpha +1)}~\text{ if }~\frac{1}{2}<\alpha <1,\\&\frac{5}{3}~\text{ if }~\alpha \ge 1 \end{aligned}\right. \end{aligned}$$

(2.4.110)

as $\varepsilon = \varepsilon _j\searrow 0,$ for each $T\in (0, \infty )$.

As a straightforward consequence of (2.4.94) and (2.4.96), it holds that

$$\begin{aligned} c_\varepsilon u_\varepsilon \rightarrow cu ~ \text{ in }~ L^{1}_{loc}(\bar{\varOmega }\times (0,\infty ))~\text{ as }~\varepsilon =\varepsilon _j\searrow 0. \end{aligned}$$

(2.4.111)

Thus, the integrability of nu and cu in (2.2.4) is verified by (2.4.94) and (2.4.96).

Next, by (2.4.96) and the fact that $\Vert Y_{\varepsilon }\varphi \Vert _{L^2(\varOmega )} \le \Vert \varphi \Vert _{L^2(\varOmega )}(\varphi \in L^2_{\sigma }(\varOmega ))$ and $Y_{\varepsilon }\varphi \rightarrow \varphi$ in $L^2(\varOmega )$ as $\varepsilon \searrow 0$, we can get that there exists a positive constant $C_1$ such that, for any $\varepsilon \in (0,1)$,

$$ \begin{aligned} \left\| Y_{\varepsilon }u_{\varepsilon }(\cdot ,t)-u(\cdot ,t)\right\| _{L^2(\varOmega )} \le&\displaystyle {\left\| Y_{\varepsilon }[u_{\varepsilon }(\cdot ,t)-u(\cdot ,t)]\right\| _{L^2(\varOmega )}+ \left\| Y_{\varepsilon }u(\cdot ,t)-u(\cdot ,t)\right\| _{L^2(\varOmega )}}\\ \le&\displaystyle {\left\| u_{\varepsilon }(\cdot ,t)-u(\cdot ,t)\right\| _{L^2(\varOmega )}+ \left\| Y_{\varepsilon }u(\cdot ,t)-u(\cdot ,t)\right\| _{L^2(\varOmega )}}\\ \rightarrow&\displaystyle {0~\text{ as }~\varepsilon =\varepsilon _j\searrow 0}\\ \end{aligned} $$

and

$$ \begin{aligned} \left\| Y_{\varepsilon }u_{\varepsilon }(\cdot ,t)-u(\cdot ,t)\right\| _{L^2(\varOmega )}^2 \le&\displaystyle {\left( \Vert Y_{\varepsilon }u_{\varepsilon }(\cdot ,t)|\Vert _{L^2(\varOmega )}+\Vert u(\cdot ,t)|\Vert _{L^2(\varOmega )}\right) ^2}\\ \le&\displaystyle {\left( \Vert u_{\varepsilon }(\cdot ,t)|\Vert _{L^2(\varOmega )}+\Vert u(\cdot ,t)|\Vert _{L^2(\varOmega )}\right) ^2}\\ \le&\displaystyle {C_1}\\ \end{aligned} $$

for all $t\in (0,\infty )/ N$ with some null set $N\subset (0,\infty )$, and thus by the dominated convergence theorem, we can find that

$$ \begin{aligned} \displaystyle \int _{0}^T\Vert Y_{\varepsilon }u_{\varepsilon }(\cdot ,t)-u(\cdot ,t)\Vert _{L^2(\varOmega )}^2dt\rightarrow 0 ~\text{ as }~\varepsilon =\varepsilon _j\searrow 0 ~\text{ for } \text{ all }~T>0. \end{aligned} $$

Therefore,

$$\begin{aligned} Y_\varepsilon u_\varepsilon \rightarrow u ~\text{ in }~ L_{loc}^2([0,\infty ); L^2(\varOmega )). \end{aligned}$$

(2.4.112)

Now, combining (2.4.96) with (2.4.112), we derive

$$\begin{aligned} \begin{aligned} Y_{\varepsilon }u_{\varepsilon }\otimes u_{\varepsilon }\rightarrow u \otimes u ~\text{ in }~L^1_{loc}(\bar{\varOmega }\times [0,\infty ))~\text{ as }~\varepsilon =\varepsilon _j\searrow 0. \end{aligned} \end{aligned}$$

(2.4.113)

Therefore, the integrability of $nS(x,n,c)\nabla c$, nu, cu and $u\otimes u$ in (2.2.4) is verified by (2.4.107), (2.4.110), (2.4.111) and (2.4.113). Finally, for any fixed $T\in (0, \infty )$, applying (2.4.92), one can get

$$\begin{aligned} \begin{aligned}&\displaystyle \int _0^T\left\| F_{\varepsilon }(n_{\varepsilon }(\cdot ,t))-n(\cdot ,t)\right\| _{L^r(\varOmega )}^rdt \\ \le&\displaystyle \displaystyle \int _0^T\left\| F_{\varepsilon }(n_{\varepsilon }(\cdot ,t))-F_{\varepsilon }(n(\cdot ,t))\right\| _{L^r(\varOmega )}^rdt \\&+\displaystyle \int _0^T\left\| F_{\varepsilon }(n(\cdot ,t))-n(\cdot ,t)\right\| _{L^r(\varOmega )}^rdt \\ \le&\displaystyle \Vert F'_{\varepsilon }\Vert _{L^\infty (\varOmega \times (0,\infty ))}\displaystyle \int _0^T\left\| n_{\varepsilon }(\cdot ,t)-n(\cdot ,t)\right\| _{L^r(\varOmega )}^rdt \\&+\displaystyle \int _0^T\left\| F_{\varepsilon }(n(\cdot ,t))-n(\cdot ,t)\right\| _{L^r(\varOmega )}^rdt, \end{aligned} \end{aligned}$$

(2.4.114)

where r is the same as in (2.4.92). Besides that, we also deduce from (2.4.3) and $r>1$ that

$$ \begin{aligned} \left\| F_{\varepsilon }(n(\cdot ,t))-n(\cdot ,t)\right\| _{L^r(\varOmega \times (0,T))}^r \le&\displaystyle {2^r\Vert n(\cdot ,t)\Vert }\\ \end{aligned} $$

for each $t\in (0,T)$, which together with (2.4.92) shows the integrability of

$$\left\| F_{\varepsilon }(n(\cdot ,t))-n(\cdot ,t)\right\| _{L^r(\varOmega )}^r $$

on (0, T). Thereupon, by virtue of (2.4.2), we infer from the dominated convergence theorem that

$$\begin{aligned} \begin{aligned} \displaystyle \int _0^T\left\| F_{\varepsilon }(n)-n\right\| _{L^r(\varOmega )}^rdt\rightarrow 0 ~\text{ as }~\varepsilon =\varepsilon _j\searrow 0 \end{aligned} \end{aligned}$$

(2.4.115)

for each $T\in (0, \infty )$. Inserting (2.4.115) into (2.4.114) and using (2.4.92) and (2.4.1), we can see clearly that

$$\begin{aligned} \begin{aligned} F_{\varepsilon }(n)\rightarrow n ~\text{ in }~L^r_{loc}(\bar{\varOmega }\times [0,\infty ))~\text{ as }~\varepsilon =\varepsilon _j\searrow 0. \end{aligned} \end{aligned}$$

(2.4.116)

Finally, according to (2.4.92)–(2.4.94), (2.4.96)–(2.4.98), (2.4.107), (2.4.110)– (2.4.113) and (2.4.116), we may pass to the limit in the respective weak formulations associated with the regularized system (2.2.8) and obtain the integral identities (2.2.5)–(2.2.7).