Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion

TL;DR

This paper proves the existence of global weak solutions for a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion when the diffusion exponent exceeds 2, under certain boundary conditions and initial data.

Contribution

It establishes the first global weak solution existence result for the 3D Keller-Segel-Navier-Stokes system with nonlinear diffusion for m>2.

Findings

Global weak solutions exist for m>2

Solutions are valid for reasonably regular initial data

The results apply under Neumann and no-slip boundary conditions

Abstract

The coupled quasilinear Keller-Segel-Navier-Stokes system is considered under Neumann boundary conditions for $n$ and $c$ and no-slip boundary conditions for $u$ in three-dimensional bounded domains $\Omega\subseteq \mathbb{R}^3$ with smooth boundary, where $m>0,\kappa\in \mathbb{R}$ are given constants, $\phi\in W^{1,\infty}(\Omega)$. If $ m> 2$, then for all reasonably regular initial data, a corresponding initial-boundary value problem for $(KSNF)$ possesses a globally defined weak solution.

Equations

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

$$\left\{\begin{array}[]{ll}n_{t}=\nabla\cdot(D(n)\nabla n)-\chi\nabla\cdot(S(n)\nabla c),\\ \displaystyle{c_{t}=\Delta c-c+n}\end{array}\right.$$

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

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

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

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

$$\begin{array}[]{rl}&u\otimes u\in L^{1}_{loc}(\bar{\Omega}\times[0,\infty);\mathbb{R}^{3\times 3})~{}~{}\mbox{and}~{}~{}~{}n^{m}~{}\mbox{belong to}~{}~{}L^{1}_{loc}(\bar{\Omega}\times[0,\infty)),\\ &cu,~{}~{}nu~{}~{}\mbox{and}~{}~{}n|\nabla c|~{}\mbox{belong to}~{}~{}L^{1}_{loc}(\bar{\Omega}\times[0,\infty);\mathbb{R}^{3})\end{array}$$

$$\begin{array}[]{rl}\displaystyle{-\int_{0}^{T}\int_{\Omega}n\varphi_{t}-\int_{\Omega}n_{0}\varphi(\cdot,0)}=&\displaystyle{\int_{0}^{T}\int_{\Omega}n^{m}\Delta\varphi+\int_{0}^{T}\int_{\Omega}n\nabla c\cdot\nabla\varphi}\\ &+\displaystyle{\int_{0}^{T}\int_{\Omega}nu\cdot\nabla\varphi}\\ \end{array}$$

$$\begin{array}[]{rl}\displaystyle{-\int_{0}^{T}\int_{\Omega}c\varphi_{t}-\int_{\Omega}c_{0}\varphi(\cdot,0)}=&\displaystyle{-\int_{0}^{T}\int_{\Omega}\nabla c\cdot\nabla\varphi-\int_{0}^{T}\int_{\Omega}c\varphi+\int_{0}^{T}\int_{\Omega}n\varphi+\int_{0}^{T}\int_{\Omega}cu\cdot\nabla\varphi}\\ \end{array}$$

$$\begin{array}[]{rl}\displaystyle{-\int_{0}^{T}\int_{\Omega}u\varphi_{t}-\int_{\Omega}u_{0}\varphi(\cdot,0)-\kappa\int_{0}^{T}\int_{\Omega}u\otimes u\cdot\nabla\varphi}=&\displaystyle{-\int_{0}^{T}\int_{\Omega}\nabla u\cdot\nabla\varphi-\int_{0}^{T}\int_{\Omega}n\nabla\phi\cdot\varphi}\\ \end{array}$$

$$\varphi \in W^{1,\infty }(\Omega )$$

$$\left\{\begin{array}[]{ll}\displaystyle{n_{0}\in C^{\kappa}(\bar{\Omega})~{}~{}\mbox{for certain}~{}~{}\kappa>0~{}~{}\mbox{with}~{}~{}n_{0}\geq 0~{}~{}\mbox{in}~{}~{}\Omega},\\ \displaystyle{c_{0}\in W^{1,\infty}(\Omega)~{}~{}\mbox{with}~{}~{}c_{0}\geq 0~{}~{}\mbox{in}~{}~{}\bar{\Omega},}\\ \displaystyle{u_{0}\in D(A^{\gamma}_{r})~{}~{}\mbox{for~{}~{} some}~{}~{}\gamma\in(\frac{1}{2},1)~{}~{}\mbox{and any}~{}~{}{r}\in(1,\infty),}\\ \end{array}\right.$$

$$m>2.$$

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

$$\begin{array}[]{ll}Y_{\varepsilon}w:=(1+\varepsilon A)^{-1}w~{}~{}~{}~{}\mbox{for all}~{}~{}w\in L^{2}_{\sigma}(\Omega)\end{array}$$

$$\left\{\begin{array}[]{ll}n_{\varepsilon}\in C^{0}(\bar{\Omega}\times[0,T_{max,\varepsilon}))\cap C^{2,1}(\bar{\Omega}\times(0,T_{max,\varepsilon})),\\ c_{\varepsilon}\in C^{0}(\bar{\Omega}\times[0,T_{max,\varepsilon}))\cap C^{2,1}(\bar{\Omega}\times(0,T_{max,\varepsilon})),\\ u_{\varepsilon}\in C^{0}(\bar{\Omega}\times[0,T_{max,\varepsilon}))\cap C^{2,1}(\bar{\Omega}\times(0,T_{max,\varepsilon})),\\ P_{\varepsilon}\in C^{1,0}(\bar{\Omega}\times(0,T_{max,\varepsilon})),\\ \end{array}\right.$$

$$\Vert n_{\epsilon }(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert c_{\epsilon }(\cdot ,t){\Vert }_{W^{1,\infty }(\Omega )}+\Vert A^{\gamma }{u}_{\epsilon }(\cdot ,t){\Vert }_{L^2(\Omega )}\to \infty \text{mbox}ast\to T_{max,\epsilon },$$

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

$$\begin{array}[]{rl}&\displaystyle{\int_{\Omega}(n_{\varepsilon}+\varepsilon)^{m-1}+\int_{\Omega}c_{\varepsilon}^{2}+\int_{\Omega}|{u_{\varepsilon}}|^{2}\leq C~{}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max,\varepsilon}).}\\ \end{array}$$

$$\begin{array}[]{rl}&\displaystyle{\int_{0}^{T}\int_{\Omega}\left[(n_{\varepsilon}+\varepsilon)^{2m-4}|\nabla{n_{\varepsilon}}|^{2}+|\nabla{c_{\varepsilon}}|^{2}+|\nabla{u_{\varepsilon}}|^{2}\right]\leq C.}\\ \end{array}$$

$$\begin{array}[]{rl}\displaystyle\frac{1}{{2}}\displaystyle\frac{d}{dt}\|{c_{\varepsilon}}\|^{{{2}}}_{L^{{2}}(\Omega)}+\int_{\Omega}|\nabla c_{\varepsilon}|^{2}+\int_{\Omega}|c_{\varepsilon}|^{2}=&\displaystyle{\int_{\Omega}n_{\varepsilon}c_{\varepsilon}}\\ \leq&\displaystyle{\frac{1}{2}\int_{\Omega}n_{\varepsilon}^{2}+\frac{1}{2}\int_{\Omega}c_{\varepsilon}^{2}.}\\ \end{array}$$

$$\begin{array}[]{rl}\displaystyle\|n_{\varepsilon}\|_{L^{2}(\Omega)}^{2}\leq&\displaystyle{\|n_{\varepsilon}+\varepsilon\|_{L^{2}(\Omega)}^{2}}\\ =&\displaystyle{\|(n_{\varepsilon}+\varepsilon)^{m-1}\|_{L^{\frac{2}{m-1}}(\Omega)}^{\frac{2}{m-1}}}\\ \leq&\displaystyle{C_{1}\|\nabla(n_{\varepsilon}+\varepsilon)^{m-1}\|_{L^{2}(\Omega)}^{\frac{6}{6m-7}}\|(n_{\varepsilon}+\varepsilon)^{m-1}\|_{L^{\frac{1}{m-1}}(\Omega)}^{\frac{2}{m-1}-\frac{6}{6m-7}}}\\ \leq&\displaystyle{C_{2}(\|\nabla(n_{\varepsilon}+\varepsilon)^{m-1}\|_{L^{2}(\Omega)}^{\frac{6}{6m-7}}+1)}\\ \leq&\displaystyle{\delta_{1}\|\nabla(n_{\varepsilon}+\varepsilon)^{m-1}\|_{L^{2}(\Omega)}^{2}+C_{3}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max,\varepsilon})}\\ \end{array}$$

$$\begin{array}[]{rl}\displaystyle\displaystyle\frac{d}{dt}\|{c_{\varepsilon}}\|^{{{2}}}_{L^{{2}}(\Omega)}+2\int_{\Omega}|\nabla c_{\varepsilon}|^{2}+\int_{\Omega}c_{\varepsilon}^{2}\leq&\displaystyle{\delta_{1}\|\nabla(n_{\varepsilon}+\varepsilon)^{m-1}\|_{L^{2}(\Omega)}^{2}+C_{3}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max,\varepsilon})}\\ \end{array}$$

$$\begin{array}[]{rl}&\displaystyle{\frac{1}{{m-1}}\frac{d}{dt}\|{n_{\varepsilon}}+\varepsilon\|^{{{m-1}}}_{L^{{m-1}}(\Omega)}+m(m-2)\int_{\Omega}({n_{\varepsilon}}+\varepsilon)^{2m-4}|\nabla n_{\varepsilon}|^{2}}\\ \leq&\displaystyle{(m-2)\int_{\Omega}({n_{\varepsilon}}+\varepsilon)^{m-2}|\nabla n_{\varepsilon}||\nabla c_{\varepsilon}|}\\ \leq&\displaystyle{\frac{m(m-2)}{2}\int_{\Omega}({n_{\varepsilon}}+\varepsilon)^{2m-4}|\nabla n_{\varepsilon}|^{2}+\frac{(m-2)}{2m}\int_{\Omega}|\nabla c_{\varepsilon}|^{2}.}\end{array}$$

$$\frac{1}{2}\frac{d}{dt}{\int }_{\Omega }|u_{\epsilon }{|}^2+{\int }_{\Omega }|\nabla u_{\epsilon }{|}^2={\int }_{\Omega }{n}_{\epsilon }{u}_{\epsilon }\cdot \nabla \varphi \text{mbox}forallt\in (0,T_{max,\epsilon }).$$

$$\begin{array}[]{rl}\displaystyle\int_{\Omega}n_{\varepsilon}u_{\varepsilon}\cdot\nabla\phi\leq&\displaystyle{\|\nabla\phi\|_{L^{\infty}(\Omega)}\|n_{\varepsilon}\|_{L^{\frac{6}{5}}(\Omega)}\|\nabla u_{\varepsilon}\|_{L^{2}(\Omega)}}\\ \leq&\displaystyle{C_{4}\|n_{\varepsilon}\|_{L^{\frac{6}{5}}(\Omega)}\|\nabla u_{\varepsilon}\|_{L^{2}(\Omega)}}\\ \leq&\displaystyle{\frac{C_{4}^{2}}{2}\|n_{\varepsilon}\|_{L^{\frac{6}{5}}(\Omega)}^{2}+\frac{1}{2}\|\nabla u_{\varepsilon}\|_{L^{2}(\Omega)}^{2}}\\ \leq&\displaystyle{C_{5}\|n_{\varepsilon}\|_{L^{2}(\Omega)}^{2}+\frac{1}{2}\|\nabla u_{\varepsilon}\|_{L^{2}(\Omega)}^{2}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max,\varepsilon}),}\\ \end{array}$$

$$\begin{array}[]{rl}\displaystyle\int_{\Omega}n_{\varepsilon}u_{\varepsilon}\cdot\nabla\phi\leq&\displaystyle{\frac{\delta_{2}}{2}\|\nabla(n_{\varepsilon}+\varepsilon)^{m-1}\|_{L^{2}(\Omega)}^{2}+\frac{1}{2}\|\nabla u_{\varepsilon}\|_{L^{2}(\Omega)}^{2}+C_{6}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max,\varepsilon}),}\\ \end{array}$$

$$\frac{d}{dt}{\int }_{\Omega }|u_{\epsilon }{|}^2+{\int }_{\Omega }|\nabla u_{\epsilon }{|}^2\le {\delta }_2\Vert \nabla (n_{\epsilon }+\epsilon {)}^{m-1}{\Vert }_{L^2(\Omega )}^2+C_7\text{mbox}forallt\in (0,T_{max,\epsilon }).$$

$$\begin{array}[]{rl}&\displaystyle{\frac{d}{dt}\left(\|{c_{\varepsilon}}\|^{{{2}}}_{L^{{2}}(\Omega)}+\frac{L}{{m-1}}\|{n_{\varepsilon}}+\varepsilon\|^{{{m-1}}}_{L^{{m-1}}(\Omega)}+\int_{\Omega}|u_{\varepsilon}|^{2}\right)+\int_{\Omega}|\nabla u_{\varepsilon}|^{2}}\\ &\displaystyle{+(L\frac{m(m-2)}{(m-1)^{2}}-\delta_{2}-\delta_{1})\|\nabla(n_{\varepsilon}+\varepsilon)^{m-1}\|_{L^{2}(\Omega)}^{2}+(2-L\frac{(m-2)}{2m})\int_{\Omega}|\nabla c_{\varepsilon}|^{2}+\int_{\Omega}c_{\varepsilon}^{2}}\\ \leq&\displaystyle{C_{8}~{}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max,\varepsilon}),}\\ \end{array}$$

$$\begin{array}[]{rl}&\displaystyle{\int_{\Omega}c_{\varepsilon}^{\frac{8(m-1)}{3}}\leq C~{}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max,\varepsilon}).}\\ \end{array}$$

Full text

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is considered under Neumann boundary conditions for $n$ and $c$ and no-slip boundary conditions for $u$ in three-dimensional bounded domains $\Omega\subseteq\mathbb{R}^{3}$ with smooth boundary, where $m>0,\kappa\in\mathbb{R}$ are given constants, $\phi\in W^{1,\infty}(\Omega)$. If $m>2$, then for all reasonably regular initial data, a corresponding initial-boundary value problem for $(KSNF)$ possesses a globally defined weak solution.

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

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

1 Introduction

Chemotaxis is a biological process in which cells move toward a chemically more favorable environment (see Hillen and Painter [12]). In 1970, Keller and Segel (see Keller and Segel [17, 18]) proposed a mathematical model for chemotaxis phenomena through a system of parabolic equations (see e.g. Winkler et al. [1, 15, 39], Osaki and Yagi [24], Horstmann [13]). To describe chemotaxis of cell populations, the signal is produced by the cells, an important variant of the quasilinear chemotaxis model

[TABLE]

was initially proposed by Painter and Hillen ([25], see also Winkler et al. [1, 31]), where $n$ denotes the cell density and $c$ describes the concentration of the chemical signal secreted by cells. The function $S$ measures the chemotactic sensitivity, which may depend on $n,$ $D(n)$ is the diffusion function. The results about the chemotaxis model (1.1) appear to be rather complete, which dealt with the problem (1.1) whether the solutions are global bounded or blow-up (see Cieślak et al. [4, 5, 7], Hillen [12], Horstmann et al. [14], Ishida et al. [16], Kowalczyk [20], Winkler et al. [30, 43, 39]). In fact, Tao and Winkler ([30]), proved that the solutions of (1.1) are global and bounded provided that $\displaystyle\frac{S(n)}{D(n)}\leq c(n+1)^{\frac{2}{N}+\varepsilon}$ for all $n\geq 0$ with some $\varepsilon>0$ and $c>0$, and $D(n)$ satisfies some another technical conditions. For the more related works in this direction, we mention that a corresponding quasilinear version, the logistic damping or the signal is consumed by the cells has been deeply investigated by Cieślak and Stinner [5, 6], Tao and Winkler [30, 36, 43] and Zheng et al. [46, 47, 51, 52].

In various situations, however, the migration of bacteria is furthermore substantially affected by changes in their environment (see Winkler et al. [1, 32]). As in the quasilinear Keller-Segel system (1.1) where the chemoattractant is produced by cells, the corresponding chemotaxis–fluid model is then is then quasilinear Keller-Segel-Navier-Stokes system of the form

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where $n$ and $c$ are denoted as before, $u$ and $P$ stand for the velocity of incompressible fluid and the associated pressure, respectively. $\phi$ is a given potential function and $\kappa\in\mathbb{R}$ denotes the strength of nonlinear fluid convection. Problem (1.2) is proposed to describe chemotaxis–fluid interaction in cases when the evolution of the chemoattractant is essentially dominated by production through cells ([1, 12]).

If the signal is consumed, rather than produced, by the cells, Tuval et al. ([33]) proposed the following model

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Here $f(c)$ is the consumption rate of the oxygen by the cells. Approaches based on a natural energy functional, the (quasilinear) chemotaxis-(Navier-)Stokes system (1.3) has been studied in the last few years and the main focus is on the solvability result (see e.g. Chae, Kang and Lee [3], Duan, Lorz, Markowich [9], Liu and Lorz [22, 23], Tao and Winkler [32, 38, 40, 42], Zhang and Zheng [45] and references therein). For instance, if $\kappa=0$ in (1.3), the model is simplified to the chemotaxis-Stokes equation. In [37], Winkler showed the global weak solutions of (1.3) in bounded three-dimensional domains. Other variants of the model of (1.3) that include porous medium-type diffusion and $S$ being a chemotactic sensitivity tensor, one can see Winkler ([41]) and Zheng ([50]) and the references therein for details.

In contrast to problem (1.3), the mathematical analysis of the Keller-Segel-Stokes system (1.2) ($\kappa=0$) is quite few (Black [2], Wang et al. [21, 34, 35]). Among these results, Wang et al. ([34, 35]) proved the global boundedness of solutions to the two-dimensional and there-dimensional Keller-Segel-Stokes system (1.2) when $S$ is a tensor satisfying some dampening condition with respective to $n$. However, for the there-dimensional fully Keller-Segel-Navier-Stokes system (1.2) ($\kappa\in\mathbb{R}$), to the best our knowledge, there is no result on global solvability. Motivated by the above works, we will investigate the interaction of the fully quasilinear Keller-Segel-Navier-Stokes in this paper. Precisely, we shall consider the following initial-boundary problem

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where $\Omega\subseteq\mathbb{R}^{3}$ is a bounded domain with smooth boundary.

In this paper, one of a key role in our approach is based on pursuing the time evolution of a coupled functional of the form $\int_{\Omega}n^{m-1}(\cdot,t)+\int_{\Omega}{c}^{2}(\cdot,t)+\int_{\Omega}|{u}(\cdot,t)|^{2}$ (see Lemma 3.2) which is a new (natural gradient-like energy functional) estimate of (1.4).

This paper is organized as follows. In Section 2, we firstly give the definition of weak solutions to (1.4), the regularized problems of (1.4) and state the main results of this paper and prove the local existence of classical solution to appropriately regularized problems of (1.4). Section 3 and Section 4 will be devoted to an analysis of regularized problems of (1.4). On the basis of the compactness properties thereby implied, in Section 5 we shall finally pass to the limit along an adequate sequence of numbers $\varepsilon=\varepsilon_{j}\searrow 0$ and thereby verify the main results.

2 Preliminaries and main results

Due to the strongly nonlinear term $(u\cdot\nabla)u$ and $\Delta n^{m},$ the problem (1.4) has no classical solutions in general, and thus we consider its weak solutions in the following sense. We first specify the notion of weak solution to which we will refer in the sequel.

Definition 2.1**.**

Let $T>0$ and $(n_{0},c_{0},u_{0})$ fulfills (2.7). Then a triple of functions $(n,c,u)$ is called a weak solution of (1.4) if the following conditions are satisfied

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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,

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and

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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

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

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for any $\varphi\in C_{0}^{\infty}(\bar{\Omega}\times[0,T);\mathbb{R}^{3})$ fulfilling $\nabla\varphi\equiv 0$ in $\Omega\times(0,T)$. If $\Omega\times(0,\infty)\longrightarrow\mathbb{R}^{5}$ is a weak solution of (1.4) in $\Omega\times(0,T)$ for all $T>0$, then we call $(n,c,u)$ a global weak solution of (1.4).

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 $A_{r}$ denotes the Stokes operator with domain $D(A_{r}):=W^{2,{r}}(\Omega)\cap W^{1,{r}}_{0}(\Omega)\cap L^{r}_{\sigma}(\Omega)$, and $L^{r}_{\sigma}(\Omega):=\{\varphi\in L^{r}(\Omega)|\nabla\cdot\varphi=0\}$ for ${r}\in(1,\infty)$ ([28]).

Theorem 2.1**.**

Let (2.6) hold, and suppose that

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Then for any choice of $n_{0},c_{0}$ and $u_{0}$ fulfilling (2.7), the problem (1.4) possesses at least one global weak solution $(n,c,u,P)$ in the sense of Definition 2.1.

Remark 2.1**.**

From Theorem 2.1, we conclude that if the exponent $m$ of nonlinear diffusion is large than $2$, then model (1.4) exists a global solution, which implies the nonlinear diffusion term benefits the global of solutions, which seems partly extends the results of Tao and Winkler [32], who proved the possibility of boundedness, in the case that $m=1$, the coefficient of logistic source suitably large and the strength of nonlinear fluid convection $\kappa=0$.

Our intention is to construct a global weak solution of (1.4) as the limit of smooth solutions of appropriately regularized problems. To this end, in order to deal with the strongly nonlinear term $(u\cdot\nabla)u$ and $\Delta n^{m}$, we need to introduce the following approximating equation of (1.4):

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where

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is the standard Yosida approximation. In light of the well-established fixed point arguments (see [41], Lemma 2.1 of [25] and Lemma 2.1 of [42]), we can prove that (2.9) is locally solvable in classical sense, which is stated as the following lemma.

Lemma 2.1**.**

Assume that $\varepsilon\in(0,1).$ Then there exist $T_{max,\varepsilon}\in(0,\infty]$ and a classical solution $(n_{\varepsilon},c_{\varepsilon},u_{\varepsilon},P_{\varepsilon})$ of (2.9) in $\Omega\times(0,T_{max,\varepsilon})$ such that

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classically solving (2.9) in $\Omega\times[0,T_{max,\varepsilon})$. Moreover, $n_{\varepsilon}$ and $c_{\varepsilon}$ are nonnegative in $\Omega\times(0,T_{max,\varepsilon})$, and

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where $\gamma$ is given by (2.7).

3 A priori estimates

In this section, we are going to establish an iteration step to develop the main ingredient of our result. The iteration depends on a series of a priori estimate. The proof of this lemma is very similar to that of Lemmata 2.2 and 2.6 of [32], so we omit its proof here.

Lemma 3.1**.**

There exists $\lambda>0$ independent of $\varepsilon$ such that the solution of (2.9) satisfies

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Lemma 3.2**.**

Let $m>2$. Then there exists $C>0$ independent of $\varepsilon$ such that the solution of (2.9) satisfies

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

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Proof.

Taking ${c_{\varepsilon}}$ as the test function for the second equation of (2.9) and using $\nabla\cdot u_{\varepsilon}=0$ and the Young inequality yields that

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On the other hand, due to the Gagliardo–Nirenberg inequality, (3.1), in light of the Young inequality and $m>2$, we obtain that for any $\delta_{1}>0,$

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with some positive constants $C_{1},C_{2}$ and $C_{3}$ independent of $\varepsilon$. Hence, in light of (3.4) and (3.5), we derive that

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and some positive constant $C_{3}$ independent of $\varepsilon.$ Next, multiply the first equation in $(\ref{1.1fghyuisda})$ by $({n_{\varepsilon}}+\varepsilon)^{m-2}$ and combining with the second equation, using $\nabla\cdot u_{\varepsilon}=0$ and the Young inequality implies that

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Now, multiplying the third equation of (2.9) by $u_{\varepsilon}$, integrating by parts and using $\nabla\cdot u_{\varepsilon}=0$, we derive that

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Here we use the Hölder inequality and (2.6) and the continuity of the embedding $W^{1,2}(\Omega)\hookrightarrow L^{6}(\Omega)$ and to find $C_{4}>0$ and $C_{5}>0$ such that

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which together with (3.5) implies that for any $\delta_{2}$

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where $C_{6}$ is a positive constant independent of $\varepsilon.$ Inserting (3.10) into (3.9) and using the Young inequality and $m>2$, we conclude that there exists a positive constant $C_{7}$ such that

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Take an evident linear combination of the inequalities provided by (3.6), (3.7) and (3.11), we conclude

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where $C_{8}$ and $L$ are positive constants. Now, choosing $L=\frac{2m}{(m-2)}$ and $\delta_{1}=\delta_{2}=\frac{L}{4}\frac{m(m-2)}{(m-1)^{2}}$ in (3.12), we can conclude that (3.2) and (3.3). ∎

With the help of Lemma 3.2, in light of the Gagliardo–Nirenberg inequality and an application of well-known arguments from parabolic regularity theory, we can derive the following Lemma:

Lemma 3.3**.**

Let $m>2$. Then there exists $C>0$ independent of $\varepsilon$ such that the solution of (2.9) satisfies

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

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Proof.

Firstly, due to (3.2) and (3.3), in light of the Gagliardo–Nirenberg inequality, for some $C_{1}$ and $C_{2}>0$ which are independent of $\varepsilon$, we derive that

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Next, taking ${c_{\varepsilon}^{\frac{8m-11}{3}}}$ as the test function for the second equation of (2.9) and using $\nabla\cdot u_{\varepsilon}=0$ and the Young inequality yields that

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with some positive constant $C_{3}.$ Hence, due to (3.15) and (3.16), we can find $C_{4}>0$ such that

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and

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Now, due to (3.17) and (3.18), in light of the Gagliardo–Nirenberg inequality, we derive that there exist positive constants $C_{5}$ and $C_{6}$ such that

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Finally, collecting (3.15), and (3.17)–(3.19), we can get the results. ∎

Lemma 3.4**.**

There exists a positive constant $C:=C(\varepsilon)$ depends on $\varepsilon$ such that

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and

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Proof.

Firstly, due to $D(1+\varepsilon A):=W^{2,2}(\Omega)\cap W_{0,\sigma}^{1,2}(\Omega)\hookrightarrow L^{\infty}(\Omega),$ by (3.2), we derive that for some $C_{1}>0$ and $C_{2}>0$,

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Next, let $A=-\mathscr{2}\Delta$; testing the third equation by $Au_{\varepsilon}$ implies

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On the other hand, in light of the Gagliardo–Nirenberg inequality, the Young inequality and (3.22), there exists a positive constant $C_{3}$ such that

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Here we have the well-known fact that $\|A(\cdot)\|_{L^{2}(\Omega)}$ defines a norm equivalent to $\|\cdot\|_{W^{2,2}(\Omega)}$ on $D(A)$ (see Theorem 2.1.1 of [28]). Therefore, recalling that $A=-\mathscr{2}\Delta$ and hence

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inserting the above equation and (3.24) into (3.23), we can conclude that

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with some positive constant $C_{4}.$ Collecting (3.15) and (3.25) and applying the Young inequality, we can get the results. ∎

Lemma 3.5**.**

There exists $C:=C(\varepsilon)>0$ depends on $\varepsilon$ such that

[TABLE]

and

[TABLE]

Proof.

Firstly, testing the second equation in (2.9) against $-\Delta c_{\varepsilon}$ and employing the Young inequality yields

[TABLE]

for all $t\in(0,T_{max,\varepsilon})$. Now, employing (3.2) and (3.21), the Gagliardo–Nirenberg inequality and the Young inequality, we derive there exist positive constants $C_{1},C_{2}$ and $C_{3}$ such that

[TABLE]

for all $t\in(0,T_{max,\varepsilon})$. Now, in view of the Gagliardo-Nirenberg inequality and the well-known fact that $\|A(\cdot)\|_{L^{2}(\Omega)}$ defines a norm equivalent to $\|\cdot\|_{W^{2,2}(\Omega)}$ on $W^{2,2}(\Omega)\cap W_{0}^{1,2}(\Omega)$ (see p. 129, Theorem e of [28]), we have

[TABLE]

where $C_{4}$ is a positive constant. Hence, in together with (3.30) and (3.21), we conclude there exists a positive constant $C_{5}$ such that for all $T\in(0,T_{max,\varepsilon})$,

[TABLE]

Inserting (3.30) and (3.29) into (3.28) and using (3.15) and (3.31), we can derive (3.26) and (3.27). This completes the proof of Lemma 3.5. ∎

With Lemmata 3.2–3.5 at hand, we are now in the position to prove the solution of approximate problem (2.9) is actually global in time.

Lemma 3.6**.**

Let $m>2$. Then for all $\varepsilon\in(0,1),$ the solution of (2.9) is global in time.

Proof.

Assuming that $T_{max,\varepsilon}$ be finite for some $\varepsilon\in(0,1)$. Next, applying almost exactly the same arguments as in the proof of Lemma 3.4 in [48], we may derive the following estimate: the solution of (2.9) satisfies that for all $\beta>1$

[TABLE]

where $C_{1}$ is a positive constant. On the other hand, due to (3.20), we derive that there exists a positive constant $C_{2}$ such that

[TABLE]

Hence , in light of the Hölder inequality and the Gagliardo–Nirenberg inequality, (3.26) and the Young inequality, we conclude that

[TABLE]

with some positive constants $C_{3}$ and $C_{4}.$ Now, inserting (3.34) into (3.32), we derive that there exists a positive constant $C_{5}$ such that

[TABLE]

Next, with the help of the Young inequality, we derive that there exists a positive constant $C_{6}$ such that

[TABLE]

Now, choosing $\beta=\frac{4}{3}$, in (3.35) and (3.36), we conclude that

[TABLE]

Here we have use the fact that $4(2\beta-2)=2\beta.$ Hence, in light of (3.15) and $m>2$, by (3.37), we derive that there exists a positive constant $C_{7}$ such that

[TABLE]

Now, employing almost exactly the same arguments as in the proof of Lemma 3.3 in [48], we conclude that the solution of (2.9) satisfies that for all $p>1$,

[TABLE]

for all $t\in(0,T_{max,\varepsilon})$ and some positive constant $C_{7}.$ By the Hölder inequality and (3.38) and using $m>2$ and the Gagliardo–Nirenberg inequality, we derive there exist positive constants $C_{9},C_{10}$ and $C_{11}$ such that

[TABLE]

where

[TABLE]

Since, $m>2$ yields to $\frac{12p-12m+9}{6p+6m-8}<2$, in light of (3.40) and the Young inequality, we derive that there exists a positive constant $C_{12}$ such that

[TABLE]

Hence, inserting (3.41) into (3.39), we derive that

[TABLE]

Now, with some basic analysis, we may derive that for all $p>1,$ there exists a positive constant $C_{13}$ such that

[TABLE]

Let $h_{\varepsilon}(x,t)=\mathcal{P}[-\kappa(Y_{\varepsilon}u_{\varepsilon}\cdot\nabla)u_{\varepsilon}+n_{\varepsilon}\nabla\phi]$. Then along with (3.2) and (3.43), there exists a positive constant $C_{13}$ such that $\|h_{\varepsilon}(\cdot,t)\|_{L^{2}(\Omega)}\leq C_{14}$ for all $t\in(0,T_{max,\varepsilon})$. Hence, we pick an arbitrary $\gamma\in(\frac{3}{4},1),$ then in light of the smoothing properties of the Stokes semigroup ([10]), we derive that for some $C_{15}>0$, we have

[TABLE]

Observe that $\gamma>\frac{3}{4},$ $D(A^{\gamma})$ is continuously embedded into $L^{\infty}(\Omega)$, therefore, due to (3.44), we derive that there exists a positive constant $C_{16}$ such that

[TABLE]

Now, for any $\beta>1$, choosing $p>0$ large enough such that $p>2\beta$, then due to (3.43) and (3.35), in light of the Young inequality, we derive that there exists a positive constant $C_{17}$ such that

[TABLE]

Now, integrating the above inequality in time, we derive that there exists a positive constant $C_{18}$ such that

[TABLE]

In order to get the boundedness of $\|\nabla c_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}$, we rewrite the variation-of-constants formula for $c_{\varepsilon}$ in the form

[TABLE]

Now, we choose $\theta\in(\frac{7}{8},1),$ then the domain of the fractional power $D((-\Delta+1)^{\theta})\hookrightarrow W^{1,\infty}(\Omega)$ ([44]). Hence, in view of $L^{p}$-$L^{q}$ estimates associated heat semigroup, (2.7), (3.43), (3.45) and (3.47), we derive that there exist positive constants $C_{19},C_{20}$ and $C_{21}$ such that

[TABLE]

with $\tau\in(0,T_{max,\varepsilon})$. Next, using the outcome of (3.39) with suitably large $p$ as a starting point, we may employ a Moser-type iteration (see e.g. Lemma A.1 of [30]) applied to the first equation of (2.9) to get that

[TABLE]

and some positive constant $C_{22}$. In view of (3.45), (3.48) and (3.49), we apply Lemma 2.1 to reach a contradiction. ∎

4 Regularity properties of time derivatives

In this subsection, we provide some time-derivatives uniform estimates of solutions to the system (2.9). The estimate is used in this Section to construct the weak solution of the equation (1.4). This will be the purpose of the following three lemma:

Lemma 4.1**.**

Let $m>2$, (2.6) and (2.7) hold. Then for any $T>0,$ one can find $C>0$ independent if $\varepsilon$ such that

[TABLE]

as well as

[TABLE]

and

[TABLE]

Proof.

Firstly, due to (3.2), (3.3) and (3.15), employing the Hölder inequality (with two exponents $\frac{4m-1}{4(m-1)}$ and $\frac{4(m-1)}{3}$) and the Gagliardo-Nirenberg inequality, we conclude that there exist positive constants $C_{1},C_{2},C_{3}$ and $C_{4}$ such that

[TABLE]

and

[TABLE]

Next, testing the first equation of (2.9) by certain $({m-1})n_{\varepsilon}^{m-2}\varphi\in C^{\infty}(\bar{\Omega})$, we have

[TABLE]

for all $t>0$. Hence, observe that the embedding $W^{2,q}(\Omega)\hookrightarrow W^{1,\infty}(\Omega)(q>3)$, due to (3.15), (3.3) and (4.5), applying $m>2$ and the Young inequlity, we deduce $C_{1},C_{2}$ and $C_{3}$ such that

[TABLE]

which implies (4.1).

Likewise, given any $\varphi\in C^{\infty}(\bar{\Omega})$, we may test the second equation in (2.9) against $\varphi$ to conclude that

[TABLE]

Thus, due to (3.3), (3.14)–(3.15) and (4.5), in light of $m>2$ and the Young inequality, we derive that there exist positive constant $C_{8}$ and $C_{9}$ such that

[TABLE]

Hence, (4.2) holds.

Finally, for any given $\varphi\in C^{\infty}_{0,\sigma}(\Omega;\mathbb{R}^{3})$, we infer from the third equation in (2.9) that

[TABLE]

Now, by (3.3), (3.14) and (3.22), we also get that there exist positive constants $C_{10},C_{11}$ and $C_{12}$ such that

[TABLE]

Hence, (4.3) is hold. ∎

In order to prove the limit functions $n$ and $c$ gained below, we will rely on an additional regularity estimate for $u_{\varepsilon}\cdot\nabla c_{\varepsilon}$, $n_{\varepsilon}\nabla c_{\varepsilon}$ and $n_{\varepsilon}u_{\varepsilon}$.

Lemma 4.2**.**

Let $m>2$, (2.6) and (2.7) hold. Then for any $T>0,$ one can find $C>0$ independent of $\varepsilon$ such that

[TABLE]

and

[TABLE]

Proof.

In light of (3.3), (3.15), (4.5) and the Young inequality, we derive that there exist positive constants $C_{1}$ and $C_{2}$ such that

[TABLE]

and

[TABLE]

These readily establish (4.12) and (4.13).

∎

5 Passing to the limit. Proof of Theorem 2.1

With the above compactness properties at hand, by means of a standard extraction procedure we can now derive the following lemma which actually contains our main existence result already.

The proof of Theorem 2.1 Firstly, in light of Lemmata 3.2–3.3 and 4.1, we conclude that there exists a positive constant $C_{1}$ such that

[TABLE]

as well as

[TABLE]

and

[TABLE]

Hence, collecting (5.2)–(5.3) and employing the the Aubin-Lions lemma (see e.g. [27]), we conclude that

[TABLE]

and

[TABLE]

Therefore, there exists a subsequence $\varepsilon=\varepsilon_{j}\subset(0,1)_{j\in\mathbb{N}}$ and the limit functions $n$ and $c$ such that

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Next, in view of (5.1), an Aubin–Lions lemma (see e.g. [27]) applies to yield strong precompactness of $(n_{\varepsilon}^{m-1})_{\varepsilon\in(0,1)}$ in $L^{2}(\Omega\times(0,T)),$ whence along a suitable subsequence we may derive that $n_{\varepsilon}^{m-1}\rightarrow z^{m-1}_{1}$ and hence $n_{\varepsilon}\rightarrow z_{1}$ a.e. in $\Omega\times(0,\infty)$ for some nonnegative measurable $z_{1}:\Omega\times(0,\infty)\rightarrow\mathbb{R}$. Now, with the help of the Egorov theorem, we conclude that necessarily $z_{1}=n,$ thus

[TABLE]

Therefore, observing that ${\frac{8(m-1)}{4m-1}}>1,{\frac{8(m-1)}{3}}>1,$ due to (4.4)–(4.5), (3.15), there exists a subsequence $\varepsilon=\varepsilon_{j}\subset(0,1)_{j\in\mathbb{N}}$ such that $\varepsilon_{j}\searrow 0$ as $j\rightarrow\infty$

[TABLE]

as well as

[TABLE]

and

[TABLE]

Next, let $g_{\varepsilon}(x,t):=-c_{\varepsilon}+n_{\varepsilon}-u_{\varepsilon}\cdot\nabla c_{\varepsilon}.$ Therefore, recalling (3.15), (3.3) and (4.13), we conclude that $c_{\varepsilon t}-c_{\varepsilon}=g_{\varepsilon}$ is bounded in $L^{\frac{5}{4}}(\Omega\times(0,T))$ for any $\varepsilon\in(0,1)$, we may invoke the standard parabolic regularity theory to infer that $(c_{\varepsilon})_{\varepsilon\in(0,1)}$ is bounded in $L^{\frac{5}{4}}((0,T);W^{2,\frac{5}{4}}(\Omega))$. Thus, by (4.2) and the Aubin–Lions lemma we derive that the relative compactness of $(c_{\varepsilon})_{\varepsilon\in(0,1)}$ in $L^{\frac{5}{4}}((0,T);W^{1,\frac{5}{4}}(\Omega))$. We can pick an appropriate subsequence which is still written as $(\varepsilon_{j})_{j\in\mathbb{N}}$ such that $\nabla c_{\varepsilon_{j}}\rightarrow z_{2}$ in $L^{\frac{5}{4}}(\Omega\times(0,T))$ for all $T\in(0,\infty)$ and some $z_{2}\in L^{\frac{5}{4}}(\Omega\times(0,T))$ as $j\rightarrow\infty$, hence $\nabla c_{\varepsilon_{j}}\rightarrow z_{2}$ a.e. in $\Omega\times(0,\infty)$ as $j\rightarrow\infty$. In view of (5.8) and the Egorov theorem we conclude that $z_{2}=\nabla c,$ and whence

[TABLE]

In the following, we shall prove $(n,c,u)$ is a weak solution of problem (1.4) in Definition 2.1. In fact, with the help of (5.6)–(5.9), (5.13), we can derive (2.1). Now, by the nonnegativity of $n_{\varepsilon}$ and $c_{\varepsilon}$, we derive $n\geq 0$ and $c\geq 0$. Next, due to (5.9) and $\nabla\cdot u_{\varepsilon}=0$, we conclude that $\nabla\cdot u=0$ a.e. in $\Omega\times(0,\infty)$. On the other hand, in view of (3.3) and (3.15), we can infer from (4.12) that

[TABLE]

Next, due to (5.6), (5.10) and (5.14), we derive that

[TABLE]

Therefore, by the Egorov theorem, we can get $z_{3}=n\nabla c,$ and hence

[TABLE]

Next, due to $\frac{3}{8(m-1)}+\frac{3}{10}<\frac{3}{4}$, in view of (5.12) and (5.13), we also infer that for each $T\in(0,\infty)$

[TABLE]

and moreover, (5.7) and (5.10) imply that

[TABLE]

which combined with the Egorov theorem implies that

[TABLE]

for each $T\in(0,\infty).$ As a straightforward consequence of (5.6) and (5.7), it holds that

[TABLE]

Next, by (5.7) and using the fact that $\|Y_{\varepsilon}\varphi\|_{L^{2}(\Omega)}\leq\|\varphi\|_{L^{2}(\Omega)}(\varphi\in L^{2}_{\sigma}(\Omega))$and $Y_{\varepsilon}\varphi\rightarrow\varphi$ in $L^{2}(\Omega)$ as $\varepsilon\searrow 0$, we derive that there exists a positive constant $C_{2}$ such that

[TABLE]

and

[TABLE]

Now, thus, by (5.7), (5.20) and (5.21) and the dominated convergence theorem, we derive that

[TABLE]

which implies that

[TABLE]

Now, collecting (5.7) and (5.23), we derive

[TABLE]

Therefore, by (5.16)–(5.19) and (5.24) we conclude that the integrability of $n\nabla c,nu$ and $cu,u\otimes u$ in (2.2). Finally, according to (5.6)–(5.19) and (5.23)–(5.24), we may pass to the limit in the respective weak formulations associated with the the regularized system (2.9) and get the integral identities (2.3)–(2.5).

Acknowledgement: The authors are very grateful to the anonymous reviewers for their carefully reading and valuable suggestions which greatly improved this work. This work is partially supported by the National Natural Science Foundation of China (No. 11601215), the Natural Science Foundation of Shandong Province of China (No. ZR2016AQ17) and the Doctor Start-up Funding of Ludong University (No. LA2016006).