Blow-up prevention by nonlinear diffusion in a 2D Keller-Segel-Navier-Stokes system with rotational flux

TL;DR

This paper proves global bounded solutions for a 2D Keller-Segel-Navier-Stokes system with nonlinear diffusion and rotational flux, extending previous results and identifying the optimal parameter conditions for solution existence.

Contribution

It establishes the existence of global bounded solutions for the system with optimal conditions on the diffusion parameter, improving upon prior work.

Findings

Global existence of bounded solutions in 2D bounded domains.

Optimal condition on the diffusion exponent m for solution existence.

Use of entropy-like estimates for analysis.

Abstract

This paper investigates the following Keller-Segel-Navier-Stokes system with nonlinear diffusion and rotational flux $$\begin{align}\begin{cases} &n_t+u\cdot\nabla n=\Delta n^m-\nabla\cdot(nS(x, n, c)\nabla c),\quad &x\in \Omega, t>0, \\ &c_t+u\cdot\nabla c=\Delta c-c+n,\quad &x\in \Omega, t>0, \\ &u_t+\kappa (u\cdot\nabla)u+\nabla P=\Delta u+n\nabla \phi,\quad &x\in \Omega, t>0, \\ &\nabla\cdot u=0,\quad &x\in \Omega, t>0, \end{cases}\end{align}$$ where $\kappa\in \mathbb{R},\phi\in W^{2,\infty}(\Omega)$ and $S$ is a given function with values in $\mathbb{R}^{2\times2}$ which fulfills $$ |S(x,n,c)| \leq C_S $$ with some $C _S > 0$. Systems of this type describe chemotaxis-fluid interaction in cases when the evolution of the chemoattractant is essentially dominated by production through cells. If $m>1$ and $\Omega\subset \mathbb{R}^2$ is a {\bf bounded} domain with smooth boundary, then for all reasonably regular initial data, a corresponding initial-boundary value problem for $(KSNF)$ possesses a global and bounded (weak) solution, which significantly improves previous results of several authors. Moreover, the {\bf optimal condition} on the parameter $m$ for global existence is obtained. Our approach underlying the derivation of main result is based on an entropy-like estimate involving the functional %Our main tool is consideration of the energy functional $$\int_{\Omega}(n_{\varepsilon} +\varepsilon)^{m}+\int_{\Omega}|\nabla c_\varepsilon|^{2},$$ where $n_\varepsilon$ and $c_\varepsilon$ are components of the solutions to (2.1) below.

Equations

$$\left\{\begin{array}[]{lll}&n_{t}+u\cdot\nabla n=\Delta n^{m}-\nabla\cdot(nS(x,n,c)\nabla c),&x\in\Omega,t>0,\\ &c_{t}+u\cdot\nabla c=\Delta c-c+n,&x\in\Omega,t>0,\\ &u_{t}+\kappa(u\cdot\nabla)u+\nabla P=\Delta u+n\nabla\phi,&x\in\Omega,t>0,\\ &\nabla\cdot u=0,&x\in\Omega,t>0,\end{array}\right.$$

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

$${\int }_{\Omega }(n_{\epsilon }+\epsilon {)}^m+{\int }_{\Omega }|\nabla c_{\epsilon }{|}^2,$$

$$\left\{\begin{array}[]{lll}&n_{t}+u\cdot\nabla n=\Delta n-\nabla\cdot(n\chi(c)\nabla c),&x\in\Omega,t>0,\\ &c_{t}+u\cdot\nabla c=\Delta c-nf(c),&x\in\Omega,t>0,\\ &u_{t}+\kappa(u\cdot\nabla)u+\nabla P=\Delta u+n\nabla\phi,&x\in\Omega,t>0,\\ &\nabla\cdot u=0,&x\in\Omega,t>0\end{array}\right.$$

$$\left\{\begin{array}[]{lll}&n_{t}+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\nabla c),&x\in\Omega,t>0,\\ &c_{t}+u\cdot\nabla c=\Delta c-nf(c),&x\in\Omega,t>0,\\ &u_{t}+\kappa(u\cdot\nabla)u+\nabla P=\Delta u+n\nabla\phi,&x\in\Omega,t>0,\\ &\nabla\cdot u=0,&x\in\Omega,t>0\end{array}\right.$$

$$\left\{\begin{array}[]{lll}&n_{t}+u\cdot\nabla n=\Delta n^{m}-\nabla\cdot(nS(x,n,c)\nabla c),&x\in\Omega,t>0,\\ &c_{t}+u\cdot\nabla c=\Delta c-nf(c),&x\in\Omega,t>0,\\ &u_{t}+\kappa(u\cdot\nabla)u+\nabla P=\Delta u+n\nabla\phi,&x\in\Omega,t>0,\\ &\nabla\cdot u=0,&x\in\Omega,t>0,\end{array}\right.$$

$$\left\{\begin{array}[]{lll}&n_{t}+u\cdot\nabla n=\Delta n^{m}-\nabla\cdot(nS(x,n,c)\nabla c),&x\in\Omega,t>0,\\ &c_{t}+u\cdot\nabla c=\Delta c-c+n,&x\in\Omega,t>0,\\ &u_{t}+\kappa(u\cdot\nabla)u+\nabla P=\Delta u+n\nabla\phi,&x\in\Omega,t>0,\\ &\nabla\cdot u=0,&x\in\Omega,t>0.\end{array}\right.$$

$${\int }_{\Omega }{n}^m+{\int }_{\Omega }|\nabla c{|}^2,$$

$$n(x,0)=n_0(x),c(x,0)=c_0(x),u(x,0)=u_0(x),x\in \Omega ,$$

$$\left(nS(x,n,c)\nabla c\right)\cdot \nu =\nabla c\cdot \nu =0,u=0,x\in \partial \Omega ,t>0,$$

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

$$|S(x,n,c)|\le C_S\text{mbox}forall(x,n,c)\in \Omega \times [0,\infty {)}^2$$

$$\varphi \in W^{2,\infty }(\Omega )$$

$$\left\{\begin{array}[]{ll}\displaystyle{n_{0}\in C^{\kappa}(\bar{\Omega})~{}~{}\mbox{for certain}~{}~{}\kappa>0~{}~{}\mbox{with}~{}~{}n_{0}\geq 0~{}~{}\mbox{in}~{}~{}\Omega},\\ \displaystyle{c_{0}\in W^{2,\infty}(\Omega)~{}~{}\mbox{with}~{}~{}c_{0},w_{0}\geq 0~{}~{}\mbox{in}~{}~{}\bar{\Omega},}\\ \displaystyle{u_{0}\in D(A),}\\ \end{array}\right.$$

$$\Vert n(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert c(\cdot ,t){\Vert }_{W^{1,\infty }(\Omega )}+\Vert u(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}\le C\text{mbox}forallt>0$$

$$\left\{\begin{array}[]{lll}&n_{\varepsilon t}+u_{\varepsilon}\cdot\nabla n_{\varepsilon}=\Delta(n_{\varepsilon}+\varepsilon)^{m}-\nabla\cdot(n_{\varepsilon}S_{\varepsilon}(x,n_{\varepsilon},c_{\varepsilon})\nabla c_{\varepsilon}),&x\in\Omega,\;t>0,\\ &c_{\varepsilon t}+u_{\varepsilon}\cdot\nabla c_{\varepsilon}=\Delta c_{\varepsilon}-c_{\varepsilon}+n_{\varepsilon},&x\in\Omega,\;t>0,\\ &u_{\varepsilon t}+\nabla P_{\varepsilon}=\Delta u_{\varepsilon}-\kappa(Y_{\varepsilon}u_{\varepsilon}\cdot\nabla)u_{\varepsilon}+n_{\varepsilon}\nabla\phi,&x\in\Omega,\;t>0,\\ &\nabla\cdot u_{\varepsilon}=0,&x\in\Omega,\;t>0,\\ &\displaystyle{\nabla n_{\varepsilon}\cdot\nu=\nabla c_{\varepsilon}\cdot\nu=0,u_{\varepsilon}=0},&x\in\partial\Omega,\;t>0,\\ &\displaystyle{n_{\varepsilon}(x,0)=n_{0}(x),c_{\varepsilon}(x,0)=c_{0}(x),\;u_{\varepsilon}(x,0)=u_{0}(x)},&x\in\Omega,\end{array}\right.$$

$$S_{\epsilon }(x,n,c):={\rho }_{\epsilon }(x){\chi }_{\epsilon }(u)S(x,n,c),x\in \bar{\Omega },n\ge 0,c\ge 0,$$

$${\rho }_{\epsilon }\in C_0^{\infty }(\Omega )\text{mbox}suchthat0\le {\rho }_{\epsilon }\le 1\text{mbox}in\Omega \text{mbox}and{\rho }_{\epsilon }\nearrow 1\text{mbox}in\Omega \text{mbox}as\epsilon \searrow 0,$$

$${\chi }_{\epsilon }\in C_0^{\infty }([0,\infty ))\text{mbox}suchthat0\le {\chi }_{\epsilon }\le 1\text{mbox}in[0,\infty )\text{mbox}and{\chi }_{\epsilon }\nearrow 1\text{mbox}in[0,\infty )\text{mbox}as\epsilon \searrow 0,$$

$$Y_{\epsilon }w:=(1+\epsilon A{)}^{-1}w\text{mbox}forallw\in L_{\sigma }^2(\Omega )$$

$$\left\{\begin{array}[]{ll}n_{\varepsilon}\in C^{0}(\bar{\Omega}\times[0,T_{max,\varepsilon}))\cap C^{2,1}(\bar{\Omega}\times(0,T_{max,\varepsilon})),\\ c_{\varepsilon}\in C^{0}(\bar{\Omega}\times[0,T_{max,\varepsilon}))\cap C^{2,1}(\bar{\Omega}\times(0,T_{max,\varepsilon}))\cap\bigcap_{p>1}L^{\infty}([0,T_{max,\varepsilon});W^{1,p}(\Omega)),\\ u_{\varepsilon}\in C^{0}(\bar{\Omega}\times[0,T_{max,\varepsilon}))\cap C^{2,1}(\bar{\Omega}\times(0,T_{max,\varepsilon}))\cap\bigcap_{\gamma\in(0,1)}C^{0}([0,T_{max,\varepsilon});D(A^{\gamma})),\\ P_{\varepsilon}\in C^{1,0}(\bar{\Omega}\times(0,T_{max,\varepsilon})).\end{array}\right.$$

$$\underset{t\nearrow T_{max,\epsilon }}{\mathrm{limsup}}[\Vert n_{\epsilon }(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert c_{\epsilon }(\cdot ,t){\Vert }_{W^{1,\infty }(\Omega )}+\Vert A^{\gamma }{u}_{\epsilon }(\cdot ,t){\Vert }_{L^2(\Omega )}]=\infty$$

$$y^{\prime }(t)+Ay(t)\le h(t)\text{mbox}fora.e.t\in (0,T)$$

$${\int }_t^{t+\sigma }h(s)ds\le B\text{mbox}forallt\in (0,T-\sigma ).$$

$$y(t)\le \max \{y_0+B,\frac{B}{A\tau }+2B\}\text{mbox}forallt\in (0,T).$$

$${\int }_{\Omega }{n}_{\epsilon }={\int }_{\Omega }{n}_0\text{mbox}forallt\in (0,T_{max,\epsilon })$$

$${\int }_{\Omega }{c}_{\epsilon }\le \max \{{\int }_{\Omega }{n}_0,{\int }_{\Omega }{c}_0\}\text{mbox}forallt\in (0,T_{max,\epsilon }).$$

$${\int }_{\Omega }{n}_{\epsilon }+{\int }_{\Omega }(n_{\epsilon }+\epsilon {)}^{m-1}+{\int }_{\Omega }{c}_{\epsilon }^2+{\int }_{\Omega }|u_{\epsilon }{|}^2\le C\text{mbox}forallt\in (0,T_{max,\epsilon }).$$

$${\int }_t^{t+\tau }{\int }_{\Omega }\left[(n_{\epsilon }+\epsilon {)}^{2m-4}|\nabla n_{\epsilon }{|}^2+|\nabla c_{\epsilon }{|}^2+|\nabla u_{\epsilon }{|}^2\right]\le C,$$

$$\Vert c_{\epsilon }(\cdot ,t){\Vert }_{L^q(\Omega )}\le C\text{mbox}forallt\in (0,T_{max,\epsilon }).$$

Full text

[TABLE]

where $\kappa\in\mathbb{R},\phi\in W^{2,\infty}(\Omega)$ and $S$ is a given function with values in $\mathbb{R}^{2\times 2}$ which fulfills

[TABLE]

with some $C_{S}>0$. Systems of this type describe chemotaxis-fluid interaction in cases when the evolution of the chemoattractant is essentially dominated by production through cells. If $m>1$ and $\Omega\subset\mathbb{R}^{2}$ is a bounded domain with smooth boundary, then for all reasonably regular initial data, a corresponding initial-boundary value problem for $(KSNF)$ possesses a global and bounded (weak) solution, which significantly improves previous results of several authors. Moreover, the optimal condition on the parameter $m$ for global existence is obtained. Our approach underlying the derivation of main result is based on an entropy-like estimate involving the functional

[TABLE]

where $n_{\varepsilon}$ and $c_{\varepsilon}$ are components of the solutions to (2.1) below.

Key words: Navier-Stokes system; Keller-Segel model; Global existence; Nonlinear diffusion

2010 Mathematics Subject Classification:  35K55, 35Q92, 35Q35, 92C17

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 [6]). For example, the fruit fly Drosophila melanogaster navigates up gradients of attractive odours during food location, and male moths follow pheromone gradients released by the female during mate location. In 1970 Keller and Segel [8] proposed a mathematical model describing chemotactic aggregation of cellular slime molds. But in their model, they did not take into account the relationship between cells and their environment. So the model can be used to describe that bacterial chemotaxis was viewed as locomotion in an otherwise quiescent fluid. Yet suspensions of aerobic bacteria often develop flows from the interplay of chemotaxis and buoyancy. Tuval and his cooperator [17] described the above biological phenomena and proposed the mathematical model consisting of oxygen diffusion and consumption, chemotaxis, and fluid dynamics

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in a domain $\Omega\subset\mathbb{R}^{N}(N\geq 1)$, where $n$, $c$, $u$, and $P$ denote, respectively, the density of cells, chemical concentration, velocity field and pressure of the fluid. The coefficient $\kappa$ is related to the strength of nonlinear fluid convection, $\phi$ stands for the potential of the gravitational field within which the cells are driven through buoyant forces, the function $\chi(c)$ measures the chemotactic sensitivity, and $f(c)$ represents the oxygen consumption rate. Some modeling approaches suggested that an adequate description of bacterial motion near surfaces of their surrounding fluid should involve rotational components in the cross-diffusive flux (see [23, 24]), so the natural generalizations of chemotaxis-fluid systems should model the evolution of the cell density, as the following form

[TABLE]

where $S$ stands for the chemotactic sensitivity. Moreover, since the diffusion of bacteria (or, more generally, of cells) in a viscous fluid is more like movement in a porous medium, the authors in [2] extended the above model to one with a porous medium-type diffusion

[TABLE]

where $m>1$. Concerning the framework where the chemical is produced by the cells instead of consumed, then the corresponding chemotaxis-fluid model is then the quasilinear Keller-Segel-Navier-Stokes system of the form (see [1, 6])

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Due to the presence of the tensor-valued sensitivity as well as the strongly nonlinear term $(u\cdot\nabla)u$ and lower regularity for $n$, the mathematical analysis of (1.1) regarding global and bounded solutions is far from trivial. Some simplified cases of the system (1.1) have been studied. When $\kappa=0$, which is corresponding to the chemotaxis-Stokes system, the results focused on the global existence and boundedness of the solutions, for example, Wang and Xiang ([19]) dealt with the case $m=1$ in $2$-dimensional space; while for $m\not=1$, Li, Wang and Xiang ([9]), Peng and Xiang ([12]) considered the problem with the spatial dimension $N=2$ and $N=3$, respectively. When $\kappa\not=0$, $m=1$ and $|S(x,n,c)|\leq C_{S}(1+n)^{-\alpha}$ for some $C_{S}\geq 0$ and $\alpha>0$, Wang, Winkler and Xiang ([18]) and Ke and Zheng ([7]) considered the global existence of the solution for the case $N=2$ and $N=3$, respectively. But till now, as far as we know, it is still not clearly that in the case that $\kappa\not=0$ and $\alpha=0$, whether the solution of the system (1.1) is bounded or not. At the same time, we also noticed that when dealing with the problem of $\kappa=0$ and $\alpha=0$, or $\kappa\not=0$ and $\alpha>0$, Li, Wang and Xiang ([9]) and Wang, Winkler and Xiang ([18]) both added the assumption that the domain is convex. Whether the convexity of the domain is necessary also arouses our interest. By considering the key energy functional

[TABLE]

we can obtain the global existence and boundedness of the solution for the system (1.1), which corresponding to the case that $\kappa\not=0$ and $\alpha=0$, in a more general non-convex domain.

In this paper, we shall subsequently consider the chemotaxis-Navier-Stokes system (1.1) along with the initial data

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and under the boundary conditions

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in a bounded domain $\Omega\subset\mathbb{R}^{2}$ with smooth boundary, where we assume that the chemotactic sensitivity tensor $S(x,n,c)$ be satisfied

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and

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with some $C_{S}>0$. Throughout this paper, we assume that

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and the initial data $(n_{0},c_{0},u_{0})$ fulfills

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where $A$ denotes the Stokes operator with domain $D(A):=W^{2,{2}}(\Omega)\cap W^{1,{2}}_{0}(\Omega)\cap L^{2}_{\sigma}(\Omega)$, and $L^{2}_{\sigma}(\Omega):=\{\varphi\in L^{2}(\Omega)|\nabla\cdot\varphi=0\}$. (see [14]).

Within the above frameworks, our main result concerning global existence and boundedness of solutions to (1.1)-(1.3) is as follows.

Theorem 1.1**.**

Let $m>1$, $\Omega\subset\mathbb{R}^{2}$ be a bounded domain with smooth boundary, and assume (1.4)-(1.7) hold. Then the problem (1.1)-(1.3) admits a global-in-time weak solution $(n,c,u,P)$, which is uniformly bounded in the sense that

[TABLE]

with some positive constant $C$.

Remark 1.1**.**

(i) If $u\equiv 0$, Theorem 1.1 is (partly) coincides with Theorem 4.1 of [20], which is optimal according to the fact that the 2D fluid-free system admits a global bounded classical solution for $m>1$ as mentioned by [15] (see also [20]).

(ii) Theorem 1.1 extends the results of Li, Wang and Xiang [9], who proved the possibility of boundedness in the case that $\Omega$ is a bounded convex domain $\Omega\subset\mathbb{R}^{2}$ with smooth boundary, $\kappa=0$ and $S$ satisfies (1.4) as well as (1.5) with some $m>1$.

This paper is organized as follows. In Section 2, we do some preliminary works and propose a approximate problem. In Section 3, we use some iteration technique to establish the necessary a priori estimates. Finally, in Section 4, we obtain the global existence and boundedness of the solutions for the system (1.1)-(1.3) in a bounded domain.

2 Preliminaries

In order to construct a weak solutions by an approximation procedure, we construct the approximate problems as follows

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where

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[TABLE]

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and

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is a standard Yosida approximation.

By the well-established fixed-point arguments (see Lemma 2.1 of [22], [21] and Lemma 2.1 of [11]), we could show the local solvability of system (2.1).

Lemma 2.1**.**

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded domain with smooth boundary, and assume (1.4)-(1.7) 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 (2.1) in $\Omega\times[0,T_{max,\varepsilon})$. Here

[TABLE]

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

[TABLE]

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

Lemma 2.2**.**

([16]) 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 and such that

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with some nonnegative function $h\in L^{1}_{loc}([0,T))$ satisfying

[TABLE]

Then

[TABLE]

3 Some basic priori estimates

In order to establish the global solvability of system (2.1), in this section, we plan to derive some estimates for the approximate system (2.1), which plays a significant role in obtaining the main result. Let us first state two basic estimates on $n_{\varepsilon}$ and $c_{\varepsilon}$.

Lemma 3.1**.**

([7]) The solution of (2.1) satisfies

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as well as

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According to Lemma 3.1, we can obtain the following energy-type equality, which was also used in Lemma 3.3 in [7] (see also [26, 18]).

Lemma 3.2**.**

Let $m>1$. Then there exists $C>0$ independent of $\varepsilon$ such that the solution of (2.1) satisfies

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Moreover, for all $t\in(0,T_{max,\varepsilon}-\tau)$, it holds that one can find a constant $C>0$ independent of $\varepsilon$ such that

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

In order to obtain the boundedness of $n_{\varepsilon}$, we need to give higher norm estimates on $c_{\varepsilon}$.

Lemma 3.3**.**

Let $(n_{\varepsilon},c_{\varepsilon},u_{\varepsilon})$ be the solution of (2.2) and $\tau=\min\{1,\frac{1}{6}T_{max,\varepsilon}\}$. Then for any $q>2$, there exists $C:=C(q,K)$ independent of $\varepsilon$ such that

[TABLE]

Proof.

Let $p>3+4(m-1)$. Multiplying the second equation in $(\ref{1.1fghyuisda})$ by ${c^{p-1}_{\varepsilon}}$, using the fact $\nabla\cdot u_{\varepsilon}=0$, and applying the integration by parts, we have

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by the Hölder inequality. Now, due to the Gagliardo–Nirenberg inequality and (3.1), for some positive constants $\kappa_{0}$ and $\kappa_{1}$, we derive

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So that, in light of (3.5) and the Young inequality, we derive that for all $t\in(0,T_{max,\varepsilon})$,

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where we have used the fact that $\frac{p-2(m-1)-1}{p-2(m-1)}+\frac{1}{p-2(m-1)}=1.$ In view of $p>3+4(m-1)$, again, from the Young inequality, there exist positive constants $C_{3}$ and $C_{4}$ such that

[TABLE]

In the following, we will estimate the integrals on the right-hand side of (3.6). In view of the Gagliardo-Nirenberg inequality, for some $C_{4},C_{5}$ and $C_{6}>0$ which are independent of $\varepsilon$, we may derive from (3.3) that

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where $\tau=\min\{1,\frac{1}{6}T_{max,\varepsilon}\}.$ Therefore, (3.4) holds by applying Lemma 2.2 and the Hölder inequality. ∎

Based on Lemma 3.2 and Lemma 3.3, we can get a series of important estimates of $n_{\varepsilon}$ and $c_{\varepsilon}$.

Lemma 3.4**.**

Let $m>1$. Then the solution of (2.1) satisfies

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and

[TABLE]

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

Proof.

Multiplying the first equation of $(\ref{1.1fghyuisda})$ by ${(n_{\varepsilon}+\varepsilon)^{m-1}}$, integrating the product in $\Omega$, and noticing $\nabla\cdot u_{\varepsilon}=0$, one obtains

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by using (1.5). Then, by using the Young inequality, we have

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On the other hand, in view of Lemma 3.2 and invoking the Gagliardo–Nirenberg inequality, we infer with some $\gamma_{0}>0$ and $\gamma_{1}>0$ that

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We then achieve, with the help of the above inequality, that

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Here, the Young inequality allows to be written as

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where

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and

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In light of (3.4), there exist positive constants $l_{0}>\frac{1}{m-1}$ and $C_{2}$, such that

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Next, with the help of the Gagliardo–Nirenberg inequality and (3.12), we derive that

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with some positive constants $C_{3}$ and $C_{4}$, where

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This, together with the Young inequality and $a\frac{4m}{2m-1}<2$ (due to $l_{0}>\frac{1}{m-1}$), yields

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Taking $-\Delta{c_{\varepsilon}}$ as the test function for the second equation of (2.1), and using the Young inequality, it yields that for all $t\in(0,T_{max,\varepsilon})$

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where we have used the fact that

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Meanwhile, we can further use Gagliardo-Nirenberg inequality and the elliptic regularity ([4]) to conclude that for some $C_{6}>0$,

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This, together with the Cauchy-Schwarz inequality and the Young inequality, yields

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Applying the Cauchy-Schwarz inequality, one obtain

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From (3.14) and (3.15), we thus infer that

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Collecting (3.9), (3.13)–(3.17), we derive that for all $t\in(0,T_{max,\varepsilon})$,

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Moreover, it follows from the Young inequality and $m>1$, that

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By substituting (3.10) into (3.18) and using (3.11), we find that

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Therefore, we derive from the Young inequality that

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where we have used the fact that $2\int_{\Omega}n_{\varepsilon}^{{{m}}}\leq\frac{1}{\gamma_{1}}\frac{{m}({m-1})}{(2{m}-1)^{2}}\int_{\Omega}(n_{\varepsilon}+\varepsilon)^{2{m}}+C_{10}$, $m>1$ and the Young inequality. Now, again, from the Gagliardo–Nirenberg inequality, (3.3), and Lemma 3.2, there exist constants $\gamma_{3}>0$ and $\gamma_{4}>0$, such that

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where $\tau=\min\{1,\frac{1}{6}T_{max,\varepsilon}\}.$ Therefore, by (3.20), we conclude that

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Thus, for $t\in(0,T_{max,\varepsilon})$, if we write

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and

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(3.19) implies that

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where

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Next, by using estimates (3.21) and (3.3), one obtains

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and

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for all $t\in(0,T_{max,\varepsilon}-\tau)$. For given $t\in(0,T_{max,\varepsilon})$, using estimates (3.21) and (3.3) again, one can choose $t_{0}\geq 0$ such that $t_{0}\in[t-\tau,t)$ and

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This, together with (3.22) and the Gronwall lemma, yields

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Finally, collecting (3.22) and (3.23), it yields (3.7) and (3.8). ∎

Lemma 3.5**.**

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

[TABLE]

Proof.

Firstly, applying the Helmholtz projection to both sides of the first equation in (2.1), then multiplying the result identified by $Au_{\varepsilon}$, integrating by parts, and using the Young inequality, we find that

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Noticing that $\|Y_{\varepsilon}u_{\varepsilon}\|_{L^{2}(\Omega)}\leq\|u_{\varepsilon}\|_{L^{2}(\Omega)},$ it follows from the Gagliardo-Nirenberg inequality and the Cauchy-Schwarz inequality that with some $C_{1}>0$ and $C_{2}>0$

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Now, from the fact that $D(A^{\frac{1}{2}}):=W^{1,2}_{0}(\Omega;\mathbb{R}^{2})\cap L_{\sigma}^{2}(\Omega)$ and (3.2), it follows that

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Due to Theorem 2.1.1 in [14], $\|A(\cdot)\|_{L^{2}(\Omega)}$ defines a norm equivalent to $\|\cdot\|_{W^{2,2}(\Omega)}$ on $D(A)$. This, together with the Young inequality and estimates (3.27) and (3.26), yields

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which combining with (3.25) implies that

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

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where

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as well as

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and

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However, (3.3) along with (3.8) warrants that for some positive constant $\alpha_{0}$,

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and

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with $\tau=\min\{1,\frac{1}{6}T_{max,\varepsilon}\}.$ Now, (3.29) and (3.30) ensure that for all $t\in(0,T_{max,\varepsilon}-\tau)$

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and

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For given $t\in(0,T_{max,\varepsilon})$, applying (3.29) again, we can choose $t_{0}\geq 0$ such that $t_{0}\in[t-\tau,t)$ and

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which combined with (3.28) implies that

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by integration. The claimed inequality (3.24) thus results from (3.31). ∎

Lemma 3.6**.**

Let $m>1$. Then there exists a positive constant $C$ independent of $\varepsilon$ such that the solution of (2.1) satisfies

[TABLE]

Proof.

Considering the fact that $\nabla c_{\varepsilon}\cdot\nabla\Delta c_{\varepsilon}=\frac{1}{2}\Delta|\nabla c_{\varepsilon}|^{2}-|D^{2}c_{\varepsilon}|^{2}$, by a straightforward computation using the second equation in (2.1) and several integrations by parts, we find that

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for all $t\in(0,T_{max})$. Here, since $|\Delta c_{\varepsilon}|\leq\sqrt{2}|D^{2}c_{\varepsilon}|$, by utilizing the Young inequality, we can estimate

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and, similarly,

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

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and

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Observe that

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Let us take $r\in(0,\frac{1}{2})$. Due to Proposition 4.22 (ii) of [5], we have that $W^{r+\frac{1}{2},2}(\Omega)\hookrightarrow L^{2}(\partial\Omega)$ is compact, so that,

[TABLE]

Now, let us pick $a=\frac{2{m}+2r-1}{2{m}}$. By $r\in(0,\frac{1}{2})$ and $\beta>1$, it implies that $r+\frac{1}{2}\leq a<1$. Therefore, from the fractional Gagliardo–Nirenberg inequality and Lemma 3.4, for some positive constants $\delta_{0},\delta_{1}$ and $C_{1}$, we conclude

[TABLE]

Combining (3.38)–(3.40), using the Young inequality and the fact that $a\in(0,1)$, it yields

[TABLE]

Now, together with (3.33)–(3.37) and (3.41), we can derive that, for some positive constant $C_{4}$,

[TABLE]

We proceed to estimate the first term on the right-hand side of (3.42). By using the Young inequality, we conclude that

[TABLE]

and

[TABLE]

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 (3.7), we derive from the Gagliardo–Nirenberg inequality that for some positive constants $C_{7}$ and $C_{8}$

[TABLE]

which together with the Young inequality provides a constant $C_{9}$ such that

[TABLE]

Inserting (3.45) into (3.44), we derive that

[TABLE]

Substituting (3.43) and (3.46) into (3.42), we have

[TABLE]

Next, since $W^{1,2}(\Omega)\hookrightarrow L^{p}(\Omega)$ for any $p>1,$ the boundedness of $\|\nabla u_{\varepsilon}(\cdot,t)\|_{L^{2}(\Omega)}$ (see Lemma 3.5) implies that there exists a positive constant $C_{11}$ such that

[TABLE]

which together with (3.8) yields to (3.32) by using Lemma 2.2. This completes the proof of Lemma 3.6. ∎

Lemma 3.7**.**

Let $m>1$. Then for all $p>1,$ there exists a positive constant $C$ independent of $\varepsilon$, such that the solution of (2.1) from Lemma 2.1 satisfies

[TABLE]

Proof.

Let $p>\max\{1,m-1\}$. Taking ${(n_{\varepsilon}+\varepsilon)^{p-1}}$ as the test function for the first equation of $(\ref{1.1fghyuisda})$, combining with the second equation, and using (1.5), the Young inequality and the fact $\nabla\cdot u_{\varepsilon}=0$, we obtain, for all $t\in(0,T_{max,\varepsilon})$,

[TABLE]

which implies that

[TABLE]

for all $t\in(0,T_{max,\varepsilon})$. In the following, we will estimate the right-hand side of (3.48). In fact, due to $m>1$, we conclude from (3.32) that

[TABLE]

by using the Hölder inequality. These together with (3.2) and $m>1$ implies that

[TABLE]

by using the Gagliardo–Nirenberg inequality as well as the Young inequality and the fact that

[TABLE]

Inserting (3.49) into (3.48), we have

[TABLE]

Therefore, (3.47) holds by using Lemma 2.2 and some basic calculation. This completes the proof of Lemma 3.7. ∎

Lemma 3.8**.**

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

[TABLE]

[TABLE]

as well as

[TABLE]

and

[TABLE]

Proof.

Firstly, applying the variation-of-constants formula to the projected version of the third equation in (2.1), we derive that

[TABLE]

Now, picking $h_{\varepsilon}=\mathcal{P}[n_{\varepsilon}(\cdot,t)\nabla\phi-\kappa(Y_{\varepsilon}u_{\varepsilon}\cdot\nabla)u_{\varepsilon}]$, then, in view of the standard smoothing properties of the Stokes semigroup, we derive that for all $t\in(0,T_{max,\varepsilon})$ and $\gamma\in(\frac{1}{2},1)$, there exist $C_{1}>0$ and $C_{2}>0$ such that

[TABLE]

by using (1.7), where $p_{0}\in(1,2)$ satisfies that

[TABLE]

In light of (3.47), for some positive constant $C_{3}$, it has

[TABLE]

Employing the Hölder inequality and the continuity of $\mathcal{P}$ in $L^{p}(\Omega;\mathbb{R}^{2})$ (see [3]), there exist positive constants $C_{4},C_{5},C_{6}$ and $C_{7}$ such that

[TABLE]

where we have used the fact that $W^{1,2}(\Omega)\hookrightarrow L^{\frac{2p_{0}}{2-p_{0}}}(\Omega)$ and the boundedness of $\|\nabla u_{\varepsilon}(\cdot,t)\|_{L^{2}(\Omega)}.$ Collecting (3.50), (3.51) and (3.52), we conclude that

[TABLE]

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

[TABLE]

In view of (3.53) and (3.32), we may use (1.7), the fact that $m>1$, and the smoothing properties of the Neumann heat semigroup $(e^{t\Delta})_{t\geq 0}$ to see that there exists $C_{11}>0$ such that

[TABLE]

Then, the boundedness of $n_{\varepsilon}$ can be obtained by the well-known Moser-Alikakos iteration procedure (see e.g. Lemma A.1 of [15]). Indeed, by using (3.53) and (3.54), we see that the hypotheses of Lemma A.1 of [15] are valid provided that we take the parameter $p$ in Lemma 3.7 appropriately large. Thus, we obtain

[TABLE]

The proof of Lemma 3.8 is completed. ∎

With all above regularization properties of each component $n_{\varepsilon}$, $c_{\varepsilon}$, $u_{\varepsilon}$ at hand, we can show the existence of global bounded solutions to the regularized system (2.1).

Lemma 3.9**.**

Let $m>1$ and $\gamma\in(\frac{1}{2},1).$. Let $(n_{\varepsilon},c_{\varepsilon},u_{\varepsilon},P_{\varepsilon})_{\varepsilon\in(0,1)}$ be classical solutions of (2.1) constructed in Lemma 2.1 on $[0,T_{max})$. Then the solution is global on $[0,\infty)$. Moreover, one can find $C>0$ independent of $\varepsilon\in(0,1)$ such that

[TABLE]

and

[TABLE]

as well as

[TABLE]

In addition, we also have

[TABLE]

Then, with the help of Lemma 3.9, 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 [21] (see also [25]).

Lemma 3.10**.**

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

[TABLE]

as well as

[TABLE]

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

[TABLE]

4 Prove of the main result

In this section, we will give the prove of the main result. Based on the above lemmas, we will construct a weak solution as the limit of classical solutions to approximating systems (2.1). Applying the idea of [25] (see also [21] and [10]), we first state the definition of the solution as follows.

Definition 4.1**.**

Let $T>0$ and $(n_{0},c_{0},u_{0})$ fulfills (1.7). Then a triple of functions $(n,c,u)$ is called a weak solution of (1.1)-(1.3) if the following conditions are satisfied

[TABLE]

where $n\geq 0$ and $c\geq 0$ in $\Omega\times(0,T)$ as well as $\nabla\cdot u=0$ in the distributional sense in $\Omega\times(0,T)$, moreover,

[TABLE]

and

[TABLE]

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

[TABLE]

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

[TABLE]

for any $\varphi\in C_{0}^{\infty}(\bar{\Omega}\times[0,T);\mathbb{R}^{2})$ fulfilling $\nabla\varphi\equiv 0$ in $\Omega\times(0,T)$. If for each $T>0$, $(n,c,u)$ :$\Omega\times(0,\infty)\longrightarrow\mathbb{R}^{4}$ is a weak solution of (1.1)-(1.3) in $\Omega\times(0,T)$, then we call $(n,c,u)$ a global weak solution of (1.1)-(1.3).

In order to use the Aubin-Lions Lemma (see e.g. [13]), 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 [21] (the minor necessary changes are left as an easy exercise to the reader), and taking advantage of Lemma 3.9, we conclude the following Lemma.

Lemma 4.1**.**

Let $m>1$ and $\varsigma>\max\{m,2(m-1)\}$. Then for all $\varepsilon\in(0,1)$, there exists a positive constant $C$ independent of $\varepsilon$ such that

[TABLE]

Moreover, let $\varsigma>\max\{m,2(m-1)\}$. Then for all $T>0$ and $\varepsilon\in(0,1)$, one can find $C(T)$ independent of $\varepsilon$ such that

[TABLE]

and

[TABLE]

Finally, we can prove the main result.

Proof of Theorem 1.1. In conjunction with Lemma 3.9 and the Aubin-Lions compactness lemma (see e.g. Simon [13]), we thus infer the existence of a sequence of numbers $\varepsilon=\varepsilon_{j}\searrow 0$ along which

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

as well as

[TABLE]

and

[TABLE]

holds for some limit $(n,c,u)\in(L^{\infty}(\Omega\times(0,\infty)))^{4}$ with nonnegative $n$ and $c$. On the other hand, Lemma 4.1 implies that for each $T>0,$ $(n_{\varepsilon}^{\varsigma})_{\varepsilon\in(0,1)}$ is bounded in $L^{2}((0,T);W^{1,2}(\Omega))$, so that, using Aubin-Lions lemma again, one may obtain $n_{\varepsilon}^{\varsigma}\rightarrow z^{\varsigma}$ for some nonnegative measurable $z:\Omega\times(0,\Omega)\rightarrow\mathbb{R}$. Thus, (4.1) and the Egorov theorem yields to $z=n$ necessarily, and thereby

[TABLE]

holds.

Due to these convergence properties (see (4.1)–(4.8)), applying standard arguments we may take $\varepsilon=\varepsilon_{j}\searrow 0$ in each term of the natural weak formulation of (2.1) separately to verify that in fact $(n,c,u)$ can be complemented by some pressure function $P$ in such a way that $(n,c,u,P)$ is a weak solution of (1.1)-(1.3). In the end, we can infer from the boundedness of $(n_{\varepsilon},c_{\varepsilon},u_{\varepsilon})$ and the Banach-Alaoglu theorem that $(n,c,u)$ is bounded. $\Box$

Acknowledgement: This work is partially supported by the Shandong Provincial Science Foundation for Outstanding Youth (No. ZR2018JL005) and the National Natural Science Foundation of China (No. 11601215).