Global classical small-data solutions for a three-dimensional Keller--Segel--Navier--Stokes system modeling coral fertilization

TL;DR

This paper proves the existence of global classical solutions with exponential decay for a three-dimensional Keller--Segel--Navier--Stokes system modeling coral fertilization, under small initial data and bounded sensitivity tensor.

Contribution

It establishes the first global classical solutions with exponential decay for this complex coupled system in 3D with small initial data.

Findings

Global classical solutions exist under small initial data.

Solutions exhibit exponential decay over time.

Bounded sensitivity tensor ensures well-posedness.

Abstract

We are concerned with the Keller--Segel--Navier--Stokes system \begin{equation*} \left\{ \begin{array}{ll} \rho_t+u\cdot\nabla\rho=\Delta\rho-\nabla\cdot(\rho \mathcal{S}(x,\rho,c)\nabla c)-\rho m, &\!\! (x,t)\in \Omega\times (0,T), \\ m_t+u\cdot\nabla m=\Delta m-\rho m, &\!\! (x,t)\in \Omega\times (0,T), \\ c_t+u\cdot\nabla c=\Delta c-c+m, & \!\! (x,t)\in \Omega\times (0,T), \\ u_t+ (u\cdot \nabla) u=\Delta u-\nabla P+(\rho+m)\nabla\phi,\quad \nabla\cdot u=0, &\!\! (x,t)\in \Omega\times (0,T) \end{array}\right. \end{equation*} subject to the boundary condition $(\nabla\rho-\rho \mathcal{S}(x,\rho,c)\nabla c)\cdot \nu\!\!=\!\nabla m\cdot \nu=\nabla c\cdot \nu=0, u=0$ in a bounded smooth domain $\Omega\subset\mathbb R^3$. It is shown that the corresponding problem admits a globally classical solution with exponential decay properties under the hypothesis that $\mathcal{S}\in C^2(\overline\Omega\times [0,\infty)^2)^{3\times 3}$ satisfies $|\mathcal{S}(x,\rho,c)|\leq C_S $ for some $C_S>0$, and the initial data satisfy certain smallness conditions.

Equations

$$\left\{\begin{array}[]{ll}\rho_{t}+u\cdot\nabla\rho=\Delta\rho-\nabla\cdot(\rho\mathcal{S}(x,\rho,c)\nabla c)-\rho m,&\!\!(x,t)\in\Omega\times(0,T),\\ m_{t}+u\cdot\nabla m=\Delta m-\rho m,&\!\!(x,t)\in\Omega\times(0,T),\\ c_{t}+u\cdot\nabla c=\Delta c-c+m,&\!\!(x,t)\in\Omega\times(0,T),\\ u_{t}+(u\cdot\nabla)u=\Delta u-\nabla P+(\rho+m)\nabla\phi,\quad\nabla\cdot u=0,&\!\!(x,t)\in\Omega\times(0,T)\end{array}\right.$$

$$n_t+U\cdot \nabla n=\Delta n+\chi \nabla \cdot (n\nabla (\Delta {)}^{-1}n)-\mu n^{q}in{\mathbb{R}}^N\times (0,T)$$

$${\int }_{{\mathbb{R}}^2}n(x,t)dx\to n_{\infty }(\chi ,n_0,U)ast\to \infty$$

$$\left\{\begin{array}[]{ll}n_{t}+U\cdot\nabla n=\Delta n-\chi\nabla\cdot(n\nabla c)-\mu n^{2},&\quad x\in\Omega,t>0,\\ c_{t}+U\cdot\nabla c=\Delta c-c+n,&\quad x\in\Omega,t>0\end{array}\right.$$

$$\left\{\begin{array}[]{ll}n_{t}+u\cdot\nabla n=\Delta n-\nabla\cdot(n\mathcal{S}(x,n,c)\nabla c)-\mu n^{2},\\ c_{t}+u\cdot\nabla c=\Delta c-c+n,\\ u_{t}+\kappa(u\cdot\nabla)u=\Delta u-\nabla P+n\nabla\phi,\\ \nabla\cdot u=0.\end{array}\right.$$

$$\left\{\begin{array}[]{ll}\rho_{t}+u\cdot\nabla\rho=\Delta\rho-\nabla\cdot(\rho\mathcal{S}(x,\rho,c)\nabla c)-\rho m,&\!\!(x,t)\in\Omega\times(0,T),\\ m_{t}+u\cdot\nabla m=\Delta m-\rho m,&\!\!(x,t)\in\Omega\times(0,T),\\ c_{t}+u\cdot\nabla c=\Delta c-c+m,&\!\!(x,t)\in\Omega\times(0,T),\\ u_{t}+(u\cdot\nabla)u=\Delta u-\nabla P+(\rho+m)\nabla\phi,\quad\nabla\cdot u=0,&\!\!(x,t)\in\Omega\times(0,T),\\ \!\!(\nabla\rho-\rho\mathcal{S}(x,\rho,c)\nabla c)\cdot\nu\!\!=\!\nabla m\cdot\nu=\nabla c\cdot\nu=0,u=0,&\!\!\!\!(x,t)\in\partial\Omega\times(0,T),\\ \!\!\!\rho(x,0)\!=\!\rho_{0}(x),m(x,0)\!=\!m_{0}(x),c(x,0)\!=\!c_{0}(x),u(x,0)\!=\!u_{0}(x),&\!\!x\in\Omega,\end{array}\right.$$

$$|\mathcal{S}(x,\rho ,c)|\le C_{\mathcal{S}}\text{for some}{C}_{\mathcal{S}}>0,$$

$$\left\{\begin{array}[]{ll}\rho_{0}\in C^{0}(\overline{\Omega}),~{}\rho_{0}\geq 0~{}\hbox{and}~{}\rho_{0}\not\equiv 0,\\ m_{0}\in C^{0}(\overline{\Omega}),~{}m_{0}\geq 0~{}\hbox{and}~{}m_{0}\not\equiv 0,\\ c_{0}\in W^{1,\infty}(\Omega),~{}c_{0}\geq 0~{}\hbox{and}~{}c_{0}\not\equiv 0,\\ u_{0}\in D(A^{\beta})~{}\hbox{for all}~{}\beta\in(\frac{3}{4},1),\end{array}\right.$$

$$\Vert {\rho }_0-{\rho }_{\infty }{\Vert }_{L^{p_0}(\Omega )}<\epsilon ,\Vert m_0{\Vert }_{L^{q_0}(\Omega )}<\epsilon ,\Vert c_0{\Vert }_{L^{\infty }(\Omega )}<\epsilon ,\Vert u_0{\Vert }_{L^3(\Omega )}<\epsilon ,$$

$$\Vert m(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}\le$$

$$\Vert \rho (\cdot ,t)-{\rho }_{\infty }{\Vert }_{L^{\infty }(\Omega )}\le$$

$$\Vert c(\cdot ,t){\Vert }_{W^{1,\infty }(\Omega )}\le$$

$$\Vert u(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}\le$$

$$\Vert {\rho }_0{\Vert }_{L^{p_0}(\Omega )}\le \epsilon ,\Vert m_0-m_{\infty }{\Vert }_{L^{q_0}(\Omega )}\le \epsilon ,\Vert \nabla c_0{\Vert }_{L^3(\Omega )}\le \epsilon ,\Vert u_0{\Vert }_{L^3(\Omega )}\le \epsilon ,$$

$$\Vert m(\cdot ,t)-m_{\infty }{\Vert }_{L^{\infty }(\Omega )}\le K_1{e}^{-{\alpha }_{1}t},$$

$$\Vert \rho (\cdot ,t){\Vert }_{L^{\infty }(\Omega )}\le K_2{e}^{-{\alpha }_{1}t},$$

$$\Vert c(\cdot ,t)-m_{\infty }{\Vert }_{W^{1,\infty }(\Omega )}\le K_3{e}^{-{\alpha }_{2}t},$$

$$\Vert u(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}\le K_4{e}^{-{\alpha }_{2}t}.$$

$$\Vert e^{t\Delta }\omega {\Vert }_{L^p(\Omega )}\le C_1\left(1+t^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{p})}\right)e^{-{\lambda }_{1}t}\Vert \omega {\Vert }_{L^q(\Omega )}$$

$$\Vert e^{t\Delta }\omega {\Vert }_{L^p(\Omega )}\le C_1\left(1+t^{-\frac{N}{2}(\frac{1}{q}-\frac{1}{p})}\right)e^{-{\lambda }_{1}t}\Vert \omega {\Vert }_{L^q(\Omega )};$$

$$\Vert \nabla e^{t\Delta }\omega {\Vert }_{L^p(\Omega )}\le C_2\left(1+t^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q}-\frac{1}{p})}\right)e^{-{\lambda }_{1}t}\Vert \omega {\Vert }_{L^q(\Omega )};$$

$$\Vert \nabla e^{t\Delta }\omega {\Vert }_{L^p(\Omega )}\le C_3\left(1+t^{-\frac{N}{2}(\frac{1}{q}-\frac{1}{p})}\right)e^{-{\lambda }_{1}t}\Vert \nabla \omega {\Vert }_{L^q(\Omega )};$$

$$\Vert e^{t\Delta }\nabla \cdot \omega {\Vert }_{L^p(\Omega )}\le C_4\left(1+t^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q}-\frac{1}{p})}\right)e^{-{\lambda }_{1}t}\Vert \omega {\Vert }_{L^q(\Omega )}.$$

$$\Vert A^{\gamma }{e}^{-tA}\varphi {\Vert }_{L^p(\Omega )}\le C_5(p,\gamma )t^{-\gamma }{e}^{-\mu t}\Vert \varphi {\Vert }_{L^p(\Omega )};$$

$$\Vert e^{-tA}\varphi {\Vert }_{L^q(\Omega )}\le C_6(p,q)t^{-\frac{N}{2}\left(\frac{1}{p}-\frac{1}{q}\right)}{e}^{-\mu t}\Vert \varphi {\Vert }_{L^p(\Omega )};$$

$$\Vert \nabla e^{-tA}\varphi {\Vert }_{L^q(\Omega )}\le C_7(p,q)t^{-\frac{1}{2}-\frac{N}{2}\left(\frac{1}{p}-\frac{1}{q}\right)}{e}^{-\mu t}\Vert \varphi {\Vert }_{L^p(\Omega )};$$

$$\Vert \varphi {\Vert }_{W^{1,p}(\Omega )}\le C_8(\gamma ,p,q)\Vert A^{\gamma }\varphi {\Vert }_{L^q(\Omega )}.$$

$${\int }_0^t(1+s^{-\alpha })(1+(t-s{)}^{-\beta })e^{-\gamma s}{e}^{-\delta (t-s)}ds\le C_{10}(\alpha ,\beta ,\gamma ,\delta )\left(1+t^{\min \{0,1-\alpha -\beta \}}\right)e^{-\min \{\gamma ,\delta \}t}.$$

$$\mathcal{S}(x,\rho ,c)=0,(x,\rho ,c)\in \partial \Omega \times [0,\infty )\times [0,\infty )$$

$$\Vert \rho (\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert m(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert c(\cdot ,t){\Vert }_{W^{1,\infty }(\Omega )}+\Vert A^{\beta }u(\cdot ,t){\Vert }_{L^2(\Omega )}\to \infty ast\to T_{max}.$$

$$\Vert \rho (\cdot ,t){\Vert }_{L^1(\Omega )}\le \Vert {\rho }_0{\Vert }_{L^1(\Omega )},\Vert m(\cdot ,t){\Vert }_{L^1(\Omega )}\le \Vert m_0{\Vert }_{L^1(\Omega )},$$

$${\int }_0^t\Vert \rho (\cdot ,s)m(\cdot ,s){\Vert }_{L^1(\Omega )}ds\le \min \{\Vert {\rho }_0{\Vert }_{L^1(\Omega )},\Vert m_0{\Vert }_{L^1(\Omega )}\},$$

$$\Vert \rho (\cdot ,t){\Vert }_{L^1(\Omega )}-\Vert m(\cdot ,t){\Vert }_{L^1(\Omega )}=\Vert {\rho }_0{\Vert }_{L^1(\Omega )}-\Vert m_0{\Vert }_{L^1(\Omega )},$$

$$\Vert m(\cdot ,t){\Vert }_{L^2(\Omega )}^2+2{\int }_0^t\Vert \nabla m(\cdot ,s){\Vert }_{L^2(\Omega )}^{2}ds\le \Vert m_0{\Vert }_{L^2(\Omega )}^2,$$

$$\Vert m(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}\le \Vert m_0{\Vert }_{L^{\infty }(\Omega )},$$

Full text

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subject to the boundary condition $(\nabla\rho-\rho\mathcal{S}(x,\rho,c)\nabla c)\cdot\nu\!\!=\!\nabla m\cdot\nu=\nabla c\cdot\nu=0,u=0$ in a bounded smooth domain $\Omega\subset\mathbb{R}^{3}$. It is shown that the corresponding problem admits a globally classical solution with exponential decay properties under the hypothesis that $\mathcal{S}\in C^{2}(\overline{\Omega}\times[0,\infty)^{2})^{3\times 3}$ satisfies $|\mathcal{S}(x,\rho,c)|\leq C_{S}$ for some $C_{S}>0$, and the initial data satisfy certain smallness conditions.

Keywords: Keller–Segel system; Navier–Stokes; tensor–value sensitivity; decay estimates.

AMS Subject Classification: 35B65; 35B40; 35K55; 92C17; 35Q92.

1 Introduction

Chemotaxis, the biased movement of individuals in response to gradients of certain chemicals, has a significant effect on pattern formation in numerous biological contexts (see [2, 12, 23]). In particular, the chemotaxis plays an important role in the reproduction of some invertebrates such as corals, anemones and sea urchins. Indeed, there is experimental evidence that eggs can release a chemical which attracts sperms during the process of coral fertilization ([5, 6, 21, 24, 25]).

The important effect of chemotaxis on the efficiency of coral fertilization is investigated by Kiselev and Ryzhik ([14, 15]) via the following chemotaxis system (the densities of egg and sperm gametes are assumed to be identical)

[TABLE]

where $n$ represents the density of egg (sperm) gametes, $U$ is a prescribed solenoidal sea fluid velocity, and $\chi>0$ denotes the chemotactic sensitivity constant, $\epsilon n^{q}$ ($q\geq 2$) denotes the fertilization phenomenon. For the Cauchy problem in $\mathbb{R}^{2}$ with initial datum $n(\cdot,0)=n_{0}$, the global-in-time existence of solutions to (1.1) ($N=2,3$) is proved under the suitable conditions on initial data. In addition, they showed that the total mass

[TABLE]

with $n_{\infty}(\chi,n_{0},U)>0$ satisfying $n_{\infty}(\chi,n_{0},U)\rightarrow 0$ as $\chi\rightarrow\infty$ in the case $q>2$ of supercritical reaction ([15]), whereas in the critical case $q=2$, the decay rate of $\int_{\mathbb{R}^{2}}n(x,t)dx$ is faster than that of $1/\log t$ as $t\rightarrow\infty$, and a weaker effect of chemotaxis is observed within finite time intervals ([14]). Recently, the total mass behavior of solution to (1.1) is investigated in [1, 3, 13] when the chemical concentration is governed by a parabolic equation. In particular, the results of [3, 13] indicate that unlike in the Cauchy problem, the dynamical behavior of solution to (1.1) with $q=2$ in the framework of bounded domains is essentially independent of the effect from chemotactic cross-diffusion. More precisely, it is shown in [3, 13] that whenever $U$ is a bounded and sufficiently regular solenoidal vector field, the component $n$ of any non-trivial classical bounded solution to

[TABLE]

decays to zero in either of the spaces $L^{1}(\Omega)$ and $L^{\infty}(\Omega)$, which can be controlled by appropriate multiples of $1/(t+1)$ from above and below, respectively.

Experiments indicate that in certain of chemotaxis motion in a liquid environment, the interaction between cells and the surrounding fluid may substantially affect the behavior thereof ([16, 20]). In the style of [7, 28], we hence suppose that this interaction occurs not only through transport but possibly also through a buoyancy-driven feedback of sperm (egg) gametes to the fluid velocity. Accordingly, it leads to a refinement of (1.2) in the framework of chemotaxis–(Navier–)Stokes system

[TABLE]

for the unknown density of sperm (egg) gametes $n$, the signal concentration $c$, the fluid velocity $u$ and the associated pressure $P$ in the physical domain $\Omega\subset\mathbb{R}^{3}$. Here the evolution of velocity $u$ is governed by the incompressible (Navier)-Stokes equations, in addition, it is driven by gametes through buoyant forces within a gravitational potential $\phi$, $\phi\in W^{2,\infty}(\Omega)$ and the chemotactic sensitivity tensor $\mathcal{S}(x,n,c)=(s_{ij}(x,n,c))\in C^{2}(\overline{\Omega}\times[0,\infty)^{2})$, $i,j\in\{1,2,3\}$, which reflects that the chemotactic migration may not necessarily be oriented along the gradient of the chemical signal, but may rather involve rotational flux components (see [22, sec. 4.2.1] or [36] for tensor-valued sensitivities in the chemotaxis system).

In view of mathematical analysis, the model (1.3) compounds the known difficulties in the study of the three-dimensional fluid dynamics with the typical intricacies in the study of chemotactic cross-diffusion reinforced by signal production. In fact, three-dimensional Navier–Stokes equations are yet lacking complete existence theory, particularly the global solvability in classes of suitably regular functions is yet left as an open problem except in the cases that the initial data are appropriately small ([30]). In addition, it is observed that when $\mathcal{S}=\mathcal{S}(x,\rho,c)$ is a tensor, the corresponding chemotaxis–fluid system loses the natural energy structure, which plays a key role in the analysis of the scalar-valued case ([34, 32, 35, 33]). Despite these challenges, some comprehensive results on the global-boundedness and large time behavior of solutions are available in the literature (see [4, 17, 19, 26, 29, 35, 37] for example). Indeed, by a continuation argument, authors of [37] established the global classical solutions of (1.3) with $\kappa=1,\mu=0$ decaying to ($\bar{n}_{0},\bar{n}_{0},0$) exponentially with $\bar{n}_{0}=\frac{1}{|\Omega|}\int_{\Omega}n_{0}(x)dx$ if $\|n_{0}\|_{L^{3}(\Omega)},\|\nabla c_{0}\|_{L^{3}(\Omega)}$ and $\|u_{0}\|_{L^{3}(\Omega)}$ are small enough. In particular, for the 3D chemotaxis–Stokes variant of (1.3) with $rn-\mu n^{2}$ instead of $\mu n^{2}$ and $\mathcal{S}=\chi$ in the $n-$equation, the existence of global bounded smooth solutions is proved for appropriately large $\mu>0$ ([26]); while the corresponding two-dimensional Navier–Stokes variant thereof possesses a global bounded classical solution for arbitrary $\mu>0$ ([27]). In addition, the latter two works also provide some results on the asymptotic decay of solutions when $r=0$, which, in the light of results of [3, 13], indeed seems to decay in time like $\frac{1}{t+1}$. Furthermore, in the very recent paper [35], Winkler showed that in the delicate three-dimensional setting, the Keller–Segel–Navier–Stokes system considered in [27] possesses at least one globally generalized solution, and that under an explicit condition on the size of $\mu$ this solution approach a spatially homogeneous equilibrium in their first two components.

From a biological point of view, it is more realistic to distinguish between eggs and sperms, and it thereby becomes possible to take into account that only spermatozoids will be affect by chemotactic attraction, whereas the eggs are governed by random diffusion, fluid transport and degradation upon contact with sperms during the coral fertilization process ([8, 9, 15]). In addition, the interaction of the gametes and the ambient fluid is not negligible. The gametes are assumed to be transported by the fluid, in turn, the motion of the latter is driven by gametes through buoyant forces within a gravitational potential $\phi$.

As an important step toward the comprehensive understanding of the coral fertilization process, we shall consider the large time behavior of the egg–sperm chemotaxis–fluid system. More precisely, this paper is concerned with the following Keller–Segel–Navier–Stokes system in the spatially three–dimensional setting

[TABLE]

where the sperm $\rho$ chemotactically moves toward the higher concentration of the chemical $c$ released by the egg $m$, while the egg $m$ is merely affected by random diffusion, fluid transport and degradation upon contact with the sperm, $\mathcal{S}=\mathcal{S}(x,\rho,c)$ satisfies

[TABLE]

and

[TABLE]

where $A$ denotes the realization of the Stokes operator in $L^{2}(\Omega)$.

In the context of these assumptions, our main result can be stated as follows:

Theorem 1.1**.**

Suppose that (1.5) hold and $\int_{\Omega}\rho_{0}>\int_{\Omega}m_{0}$. Let $p_{0}\in(\frac{3}{2},3)$, $q_{0}\in(3,\frac{3p_{0}}{3-p_{0}})$. There exists $\varepsilon>0$ such that for any initial data $(\rho_{0},m_{0},c_{0},u_{0})$ fulfilling (1.6) as well as

[TABLE]

(1.4) admits a global classical solution $(\rho,m,c,u,P)$. In particular, for any $\alpha_{1}\in(0,\min\{\lambda_{1},\rho_{\infty}\})$, $\alpha_{2}\in(0,\min\{\alpha_{1},\lambda_{1}^{\prime},1\})$, there exist constants $K_{i}$, $i=1,2,3,4$, such that for all $t\geq 1$

[TABLE]

Here $\lambda^{\prime}_{1}$ is the first eigenvalue of $A$, $\lambda_{1}$ is the first nonzero eigenvalue of $-\Delta$ on $\Omega$ under the Neumann boundary condition, and $\rho_{\infty}=\frac{1}{|\Omega|}(\int_{\Omega}\rho_{0}-\int_{\Omega}m_{0})$.

As for the case $\int_{\Omega}\rho_{0}<\int_{\Omega}m_{0}$, i.e., $m_{\infty}=\frac{1}{|\Omega|}(\int_{\Omega}m_{0}-\int_{\Omega}\rho_{0})>0$, we have

Theorem 1.2**.**

Assume that (1.5) and $\int_{\Omega}\rho_{0}<\int_{\Omega}m_{0}$ hold, and let $p_{0}\in(2,3)$, $q_{0}\in(3,\frac{3p_{0}}{2(3-p_{0})})$. Then there exists $\varepsilon>0$ such that for any initial data $(\rho_{0},m_{0},c_{0},u_{0})$ fulfilling (1.6) as well as

[TABLE]

(1.4) admits a global classical solution $(\rho,m,c,u,P)$. Furthermore, for any $\alpha_{1}\!\in\!(0,\min\{\lambda_{1},m_{\infty},1\})$, $\alpha_{2}\!\in\!(0,\min\{\alpha_{1},\lambda_{1}^{\prime}\})$, there exist constants $K_{i}>0$, $i=1,2,3,4$, such that

[TABLE]

Remark 1.1**.**

In our results, we have excluded the case $\int_{\Omega}\rho_{0}=\int_{\Omega}m_{0}$. Indeed, in the light of results of [3, 13], algebraical decay rather than exponential decay of the solutions is expected in this case.

It is noted that a similar result was proved in [18] for the three-dimensional Stoke variant of (1.4). However, as is well-known, the nonlinear convection $(u\cdot\nabla)u$ in the three-dimensional Navier–Stokes equation may enforce the spontaneous emergence of singularities in the sense of blow-up with respect to the norm in $L^{\infty}(\Omega)$, we thereby subject the study of classical solutions of (1.4) to small initial data by an essentially one-step contradiction argument, unlike that in the two-dimensional case ([9]). Moreover, in comparison with the chemotaxis–fluid system considered in [4, 37], due to

[TABLE]

for all $\omega\in L^{q}(\Omega)$ with $\int_{\Omega}\omega=0$, $-\rho m$ in the first equation of (1.4) gives rise to some difficulty in mathematical analysis despite its dissipative feature. Indeed, the core of this argument is to verify that the interval $(0,T)$ on which solutions enjoy some exponential decay properties can be extended to $(0,\infty)$, which accordingly requires an appropriate combination of the mass conservation of $\rho(x,t)-m(x,t)$ with the $L^{p}-L^{q}$ estimates for the Neumann heat semigroup.

The plan of this paper is as follows: In Section 2, we give a local existence result and some useful estimates. In Section 3, in the case of $\mathcal{S}$ vanishing on the boundary, we give the proof of the main results according to either $\int_{\Omega}\rho_{0}>\int_{\Omega}m_{0}$ or $\int_{\Omega}\rho_{0}<\int_{\Omega}m_{0}$. In the last section, on the basis of certain a priori estimates, the proof of our main results for the general $\mathcal{S}$ satisfying (1.5) is realized via an approximation procedure.

2 Preliminaries

In this section, we provide some preliminary results that will be used in the subsequent sections. We begin by recalling the important $L^{p}-L^{q}$ estimates for the Neumann heat semigroup on bounded domains ([31]).

Lemma 2.1**.**

(Lemma 1.3 of [31]) Let $(e^{t\Delta})_{t>0}$ denote the Neumann heat semigroup in the domain $\Omega$ and $\lambda_{1}>0$ denote the first nonzero eigenvalue of $-\Delta$ in $\Omega\subset\mathbb{R}^{N}$ under the Neumann boundary condition. There exists $C_{i}$, $i=1,2,3,4$, such that for all $t>0$,

$(i)$* If $1\leq q\leq p\leq\infty$, then for all $\omega\in L^{q}(\Omega)$ with $\int_{\Omega}\omega=0$,*

[TABLE]

$(ii)$* If $1\leq q\leq p\leq\infty$, then for all $\omega\in L^{q}(\Omega)$,*

[TABLE]

$(iii)$* If $2\leq q\leq p<\infty$, then for all $\omega\in W^{1,q}(\Omega)$,*

[TABLE]

$(iv)$* If $1\leq q\leq p\leq\infty$ or $1<q<\infty$ and $p=\infty$, then for all $\omega\in(L^{q}(\Omega))^{N}$,*

[TABLE]

Next we introduce the Stokes operator and recall estimates for the corresponding semigroup. With $L_{\sigma}^{p}(\Omega):=\{\varphi\in L^{p}(\Omega)|\nabla\cdot\varphi=0\}$ and $\mathcal{P}$ representing the Helmholtz projection of $L^{p}(\Omega)$ onto $L_{\sigma}^{p}(\Omega)$, the Stokes operator on $L_{\sigma}^{p}(\Omega)$ is defined as $A_{p}=-\mathcal{P}\Delta$ with domain $D(A_{p}):=W^{2,p}(\Omega)\cap W^{2,p}_{0}(\Omega)\cap L_{\sigma}^{p}(\Omega)$. Since $A_{p_{1}}$ and $A_{p_{2}}$ coincide on the intersection of their domains for $p_{1}$, $p_{2}\in(1,\infty)$, we will drop the index in the following.

Lemma 2.2**.**

(Lemma 2.3 of [4]) The Stokes operator $A$ generates the analytic semigroup $(e^{-tA})_{t>0}$ in $L_{\sigma}^{r}(\Omega)$. Its spectrum satisfies $\lambda_{1}^{\prime}=\hbox{inf}~{}\hbox{Re}\sigma(A)>0$ and we fix $\mu\in(0,\lambda_{1}^{\prime})$. For any such $\mu$, we have

$(i)$* For any $p\in(1,\infty)$ and $\gamma\geq 0$, there is $C_{5}(p,\gamma)>0$ such that for all $\phi\in L^{p}_{\sigma}(\Omega)$,*

[TABLE]

$(ii)$* For any $p$, $q$ with $1<p\leq q<\infty$, there is $C_{6}(p,q)>0$ such that for all $\phi\in L^{p}_{\sigma}(\Omega)$,*

[TABLE]

$(iii)$* For any $p$, $q$ with $1<p\leq q<\infty$, there is $C_{7}(p,q)>0$ such that for all $\phi\in L^{p}_{\sigma}(\Omega)$,*

[TABLE]

$(iv)$* If $\gamma\geq 0$ and $1<p<q<\infty$ satisfy $2\gamma-\frac{N}{q}\geq 1-\frac{N}{p}$, there is $C_{8}(\gamma,p,q)>0$ such that for all $\phi\in D(A_{q}^{\gamma})$,*

[TABLE]

Lemma 2.3**.**

(Theorem 1 and Theorem 2 of [11]) The Helmholtz projection $\mathcal{P}$ defines a bounded linear operator $\mathcal{P}$: $L^{p}(\Omega)\rightarrow L^{p}_{\sigma}(\Omega)$; in particular, for any $p\in(1,\infty)$, there exists $C_{9}(p)>0$ such that $\|\mathcal{P}\omega\|_{L^{p}(\Omega)}\leq C_{9}(p)\|\omega\|_{L^{p}(\Omega)}$ for every $\omega\in L^{p}(\omega)$.

The following elementary lemma provides some useful information on both the short-time and the large-time behavior of certain integrals, which is used in the proof of the main results.

Lemma 2.4**.**

(Lemma 1.2 of [31]) Let $\alpha\in(0,1),\beta\in(0,1)$, and $\gamma$, $\delta$ be positive constants such that $\gamma\neq\delta$. Then there exists $C_{10}(\alpha,\beta,\gamma,\delta)>0$ such that

[TABLE]

Next we recall the result on the local existence of classical solutions, which can be proved by a straightforward adaptation of well-known fixed point argument (see [32] for example).

Lemma 2.5**.**

Suppose that (1.5), (1.6) and

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hold. Then there exist $T_{max}\in(0,\infty]$ and a classical solution $(\rho,m,c,u,P)$ of (1.4) on $(0,T_{max})$. Moreover, $\rho,m,c$ are nonnegative in $\Omega\times(0,T_{max})$, and if $T_{max}<\infty$, then for $\beta\in(\frac{3}{4},1)$,

[TABLE]

This solution is unique, up to addition of constants to $P$.

The following elementary properties of the solutions in Lemma 2.5 are immediate consequences of the integration of the first and second equations in (1.4), as well as an application of the maximum principle to the second and third equations.

Lemma 2.6**.**

Suppose that (1.5), (1.6) and (2.1) hold. Then for all $t\in(0,T_{max})$, the solution of (1.4) from Lemma 2.5 satisfies

[TABLE]

3 Proof of Theorems for $\mathcal{S}=0$ on $\partial\Omega$

In this section, we give the proofs of Theorem 1.1 and Theorem 1.2 when $\mathcal{S}=0$ on $\partial\Omega$, respectively, i.e. the proof of Proposition 3.1 and Proposition 3.2 below, under which the boundary condition for $\rho$ in (1.4) actually reduces to the homogeneous Neumann condition $\nabla\rho\cdot\nu=0$.

In the case $\int_{\Omega}\rho_{0}>\int_{\Omega}m_{0}$, i.e., $\rho_{\infty}>0$, $m_{\infty}=0$, we have

Proposition 3.1**.**

Suppose that (1.5) hold and $\int_{\Omega}\rho_{0}>\int_{\Omega}m_{0}$. Let $p_{0}\in(\frac{3}{2},3)$, $q_{0}\in(3,\frac{3p_{0}}{3-p_{0}})$. There exists $\varepsilon>0$ such that for any initial data $(\rho_{0},m_{0},c_{0},u_{0})$ fulfilling (1.6) as well as

[TABLE]

(1.4) admits a global classical solution $(\rho,m,c,u,P)$. In particular, for any $\alpha_{1}\in(0,\min\{\lambda_{1},\rho_{\infty}\})$, $\alpha_{2}\in(0,\min\{\alpha_{1},\lambda_{1}^{\prime},1\})$, there exist constants $K_{i}$, $i=1,2,3,4$, such that for all $t\geq 1$

[TABLE]

Proposition 3.1 is the consequence of the following lemmas. In the proof thereof the constants $C_{i}>0$, $i=1,\ldots,10$, refer to those in Lemma 2.1–2.4, respectively. The following verifiable observations will warrant the choice in these lemmas.

Lemma 3.1**.**

Under the assumptions of Proposition 3.1 and $\sigma=\int_{0}^{\infty}\left(1+s^{-\frac{3}{2p_{0}}}\right)e^{-\alpha_{1}s}ds$, there exist $M_{1}>0,M_{2}>0$ and $\varepsilon\in(0,1)$ such that

[TABLE]

Let

[TABLE]

Then $T>0$ is well-defined by Lemma 2.5 and (1.6). Now we claim that $T=T_{max}=\infty$ if $\varepsilon$ is sufficiently small. To this end, by the contradiction argument, it only needs to verify that all of the estimates mentioned in (3.19) also hold with even smaller coefficients on the right-side thereof, which mainly rely on $L^{p}-L^{q}$ estimates for the Neumann heat semigroup and the fact that the classical solution on $(0,T_{max})$ can be written as

[TABLE]

for all $t\in(0,T_{max})$ according to the variation-of-constants formula.

Although the proof of Lemma 3.2 and Lemma 3.3 below is very similar to that of Lemma 3.11 and Lemma 3.12 in [18], respectively, we give their proofs for the convenience of the interested reader.

Lemma 3.2**.**

Under the assumptions of Proposition 3.1, for all $t\in(0,T)$ and $\theta\in[q_{0},\infty]$, there exists constant $M_{5}>0$ such that

[TABLE]

Proof.

Due to $e^{t\Delta}\rho_{\infty}=\rho_{\infty}$ and $\int_{\Omega}(\rho_{0}-m_{0}-\rho_{\infty})=0$, the definition of $T$ and Lemma 2.1(i) show that for all $t\in(0,T)$ and $\theta\in[q_{0},\infty]$,

[TABLE]

where $M_{5}=M_{1}+C_{1}+C_{1}|\Omega|^{\frac{1}{p_{0}}-\frac{1}{q_{0}}}$.

Lemma 3.3**.**

Under the assumptions of Proposition 3.1, for any $k>1$,

[TABLE]

with $\sigma=\int_{0}^{\infty}(1+s^{-\frac{3}{2p_{0}}})e^{-\alpha_{1}s}ds$ and $M_{6}=e^{M_{5}\sigma\varepsilon}$.

Proof.

Testing the first equation in (1.1) with $m^{k-1}$ ($k>1$) and integrating by parts, it holds that

[TABLE]

In view of $-\rho\leq|\rho-m-\rho_{\infty}|-m-\rho_{\infty}\leq-\rho_{\infty}+|\rho-m-\rho_{\infty}|$, Lemma 3.2 yields

[TABLE]

and thus

[TABLE]

from which (3.24) follows immediately.

Lemma 3.4**.**

Under the assumptions of Proposition 3.1, we have

[TABLE]

Proof.

For $\alpha_{2}<\lambda_{1}^{\prime}$, we fix $\mu\in(\alpha_{2},\lambda_{1}^{\prime})$. According to (3.23), Lemma 2.2(ii) and Lemma 2.3, we infer that

[TABLE]

where $\mathcal{P}(\overline{\rho+m}\nabla\phi)=\overline{\rho+m}\mathcal{P}(\nabla\phi)=0$ is used.

Due to $\alpha_{1}<\rho_{\infty}$, the application of Lemma 3.2 and Lemma 3.3 shows that

[TABLE]

with $M_{7}^{\prime}=M_{5}+4e^{M_{5}\sigma\varepsilon}$.

On the other hand, by the Hölder inequality and definition of $T$, we have

[TABLE]

Now, plugging (3.26), (3.27) into (3.25) and applying Lemma 2.4, we end up with

[TABLE]

where (3.7), (3.13) and the fact that $\frac{3}{2}(\frac{1}{p_{0}}-\frac{1}{q_{0}})<1$ are used.

The estimate for the gradient is also preserved.

Lemma 3.5**.**

Under the assumptions of Proposition 3.1, we have

[TABLE]

Proof.

According to (3.23), we have

[TABLE]

Applying Lemma 2.2(iii), Lemma 2.3 and the Hölder inequality, we arrive at

[TABLE]

where $\mathcal{P}(\overline{\rho+m}\nabla\phi)=\overline{\rho+m}\mathcal{P}(\nabla\phi)=0$ is used.

Due to $\alpha_{1}<\rho_{\infty}$, the application of Lemma 3.2 and Lemma 3.3 shows that

[TABLE]

On the other hand, from the Hölder inequality and definition of $T$, it follows that

[TABLE]

Therefore, inserting (3.30), (3.29) into (3.28) and applying Lemma 2.4, we get

[TABLE]

where (3.8), (3.12) and the fact that $q_{0}\in(3,\frac{3p_{0}}{3-p_{0}}),p_{0}\in(\frac{3}{2},3)$ are used.

Lemma 3.6**.**

Under the assumptions of Proposition 3.1, we have

[TABLE]

Proof.

By (3.22) and Lemma 2.1(ii), we have

[TABLE]

Now we estimate the last two integrals on the right-side of the above inequality. From Lemma 2.1(ii), Lemma 2.4, Lemma 3.3 with $k=q_{0}$ and the fact that $q_{0}>3$, it follows that

[TABLE]

On the other hand, by Lemma 2.1(ii), Lemma 2.4, Lemma 3.4 and the definition of $T$, we obtain

[TABLE]

From (3.31)–(3.33), it follows that

[TABLE]

due to the choice of $M_{2},M_{3}$ and $\varepsilon$ in (3.5) and (3.11), and thereby completes the proof.

Lemma 3.7**.**

Under the assumptions of Proposition 3.1, for all $\theta\in[q_{0},\infty]$ and $t\in(0,T)$,

[TABLE]

Proof.

According to (3.20), Lemma 2.1(iv), we have

[TABLE]

Now we need to estimate $I_{1}$ and $I_{2}$. Firstly, from Lemma 3.2 and Lemma 3.3, we obtain

[TABLE]

with $M_{8}=e^{(1+C_{1}+C_{1}|\Omega|^{\frac{1}{p_{0}}-\frac{1}{q_{0}}})\sigma}+\rho_{\infty}|\Omega|^{\frac{1}{q_{0}}}$, which along with Lemma 3.6 and Lemma 2.1 implies that

[TABLE]

where we have used (3.6) and (3.9) and $\frac{1}{p_{0}}-\frac{1}{q_{0}}<\frac{1}{3}$.

On the other hand, from Lemma 3.2 and Lemma 3.4, it follows that

[TABLE]

where we have used (3.10) and $\frac{1}{p_{0}}-\frac{1}{q_{0}}<\frac{1}{3}$. Hence combining the above inequalities leads to our conclusion immediately.

Now we are ready to complete the proof of Theorem 1.1 in the case $\mathcal{S}=0$ on $\partial\Omega$.

Proof of Proposition 3.1.  First from Lemma 3.4–3.7 and Definition (3.19), it follows that $T=T_{max}$. It remains to show that $T_{max}=\infty$ and convergence result asserted in Proposition 3.1. Supposed that $T_{max}<\infty$, we only need to show that for all $t\leq T_{max}$,

[TABLE]

with $\beta\in(\frac{3}{4},1)$ according to the extensibility criterion in Lemma 2.5.

Let $t_{0}:=\min\{1,\frac{T_{max}}{3}\}$. Then from Lemma 3.3, there exists $K_{1}>0$ such that for $t\in(t_{0},T_{max})$,

[TABLE]

Moreover, from Lemma 3.2 and the fact that

[TABLE]

it follows that for all $t\in(t_{0},T_{max})$ and some constant $K_{2}>0$,

[TABLE]

Furthermore, Lemma 3.6 implies that there exists $K_{3}^{\prime}>0$ such that

[TABLE]

Hence it only remains to show that

[TABLE]

for some constant $C>0$. In fact, we will show that

[TABLE]

for $t_{0}<t<T_{max}$ with some constant $C>0$.

By (3.23), we have

[TABLE]

According to Lemma 2.2,

[TABLE]

From Lemma 2.2, 2.3, 3.2 and the Hölder inequality, it follows that there exists $l_{1}>0$ such that

[TABLE]

On the other hand, let $M(t):=e^{-\alpha_{2}t}\|A^{\beta}u(\cdot,t)\|_{L^{2}(\Omega)}$ for $0<t<T_{max}$. By Lemma 2.2(iv) and the Gagliardo–Nirenberg type inequality, one can see that

[TABLE]

for some $l_{2}>0$ with $\vartheta=\frac{1}{q_{0}}/(\frac{1}{q_{0}}-\frac{1}{2}+\frac{2\beta}{3})$, and thereby the application of Lemma 2.2, 2.3, 3.4 and 3.5 gives

[TABLE]

for some $l_{3}>0$.

Hence inserting the above inequalities into (3.41), we arrive at

[TABLE]

which implies that for some $l_{4}>0$ depending on $t_{0}$, we have

[TABLE]

On the other hand, from Lemma 2.5, $\displaystyle\max_{0\leq t\leq t_{0}}M(t)\leq l_{5}.$ Therefore, we get

[TABLE]

Due to $\vartheta<1$, we infer that $M(t)\leq l_{6}$ for all $t\in(0,T_{max})$ for some $l_{6}>0$ independent of $T_{max}$ hence arrive at (3.40).

Furthermore, due to $D(A^{\beta})\hookrightarrow L^{\infty}(\Omega)$ with $\beta\in(\frac{3}{4},1)$ and Lemma 3.4, we get

[TABLE]

Now we turn to show that there exists $K_{3}^{\prime\prime}>0$ such that

[TABLE]

From (3.22), it follows that

[TABLE]

An application of (3.24) with $k=\infty$ yields

[TABLE]

On the other hand, from (3.42) and (3.39), we can see that

[TABLE]

Hence, inserting (3.45), (3.46) into (3.44), we arrive at the conclusion (3.43). Therefore we have $T_{max}=\infty$, and the decay estimates in (3.1)–(3.4) follow from (3.32)–(3.35) and (3.38), respectively.

As for the case $\int_{\Omega}\rho_{0}<\int_{\Omega}m_{0}$, i.e., $m_{\infty}>0$, $\rho_{\infty}=0$, we also have

Proposition 3.2**.**

Assume that (1.5) and $\int_{\Omega}\rho_{0}<\int_{\Omega}m_{0}$ hold, and let $p_{0}\in(2,3)$, $q_{0}\in(3,\frac{3p_{0}}{2(3-p_{0})})$. Then there exists $\varepsilon>0$ such that for any initial data $(\rho_{0},m_{0},c_{0},u_{0})$ fulfilling (1.6) as well as

[TABLE]

(1.4) admits a global classical solution $(\rho,m,c,u,P)$. Furthermore, for any $\alpha_{1}\!\in\!(0,\min\{\lambda_{1},m_{\infty},1\})$, $\alpha_{2}\!\in\!(0,\min\{\alpha_{1},\lambda_{1}^{\prime}\})$, there exist constants $K_{i}>0$, $i=1,2,3,4$, such that

[TABLE]

The basic strategy in the proof of Proposition 3.2 parallels that in the proof of Proposition 3.1 to a certain extent. However, due to differences in the properties of $\rho$ and $m$, there are significant differences in the details of their proofs. Thus for the convenience of the reader, we will sketch the proof of Proposition 3.2.

The following elementary observations can be also verified easily:

Lemma 3.8**.**

Under the assumptions of Proposition 3.2, it is possible to choose $M_{1}>0,M_{2}>0$ and $\varepsilon>0$ such that

[TABLE]

Define

[TABLE]

By Lemma 2.5 and (1.6), $T>0$ is well-defined. As in the previous subsection, we first show $T=T_{max}$, and then $T_{max}=\infty$. To this end, we will show that all of the estimates mentioned in (3.67) are valid with even smaller coefficients on the right hand side than that in (3.67). The derivation of these estimates will mainly rely on $L^{p}-L^{q}$ estimates for the Neumann heat semigroup and the corresponding semigroup for Stokes operator, and the fact that the classical solution of (1.1) on $(0,T)$ can be represented as

[TABLE]

The proofs of the following two lemmas are same as that of [18], so we omit it here.

Lemma 3.9**.**

(Lemma 3.17 in [18]) Under the assumptions of Proposition 3.2,

[TABLE]

for all $t\in(0,T)$ and $\theta\in[q_{0},\infty]$ with $M_{5}=1+C_{1}+C_{1}|\Omega|^{\frac{1}{p_{0}}-\frac{1}{q_{0}}}$.

Lemma 3.10**.**

(Lemma 3.18 in [18]) Under the assumptions of Proposition 3.2,

[TABLE]

Lemma 3.11**.**

Under the assumptions of Proposition 3.2, we have

[TABLE]

Proof.

For any given $\alpha_{2}<\lambda_{1}^{\prime}$, we can fix $\mu\in(\alpha_{2},\lambda_{1}^{\prime})$. By (3.71), Lemma 2.2, Lemma 2.3 and $\mathcal{P}(\nabla\phi)=0$, we obtain that

[TABLE]

By Lemma 3.10 and the definition of $T$, we get

[TABLE]

Inserting (3.73) into (3.72), by the definition of $T$ and noting that $\frac{3}{2}(\frac{1}{p_{0}}-\frac{1}{q_{0}})<1$, we have

[TABLE]

where we have used (3.52) and (3.55).

Lemma 3.12**.**

Under the assumptions of Proposition 3.2, we have

[TABLE]

Proof.

According to (3.71), and applying Lemma 2.2(iii) and Lemma 2.3, we arrive at

[TABLE]

where $\mathcal{P}(m_{\infty}\nabla\phi)=m_{\infty}\mathcal{P}(\nabla\phi)=0$ is used.

From (3.73), it follows that

[TABLE]

In addition, an application of the Hölder inequality and definition of $T$ shows that

[TABLE]

Therefore, inserting (3.76), (3.75) into (3.74) and applying Lemma 2.4, we get

[TABLE]

where (3.53), (3.57) are used.

Lemma 3.13**.**

Under the assumptions of Proposition 3.2, we have

[TABLE]

Proof.

From (3.70) and the standard regularization properties of the Neumann heat semigroup $(e^{\tau\Delta})_{\tau>0}$ in [31], one can conclude that

[TABLE]

In the second inequality, we have used $\nabla e^{(t-s)(\Delta-1)}m_{\infty}=0$.

From Lemma 2.1(ii), Lemma 3.10 and Lemma 2.4, it follows that

[TABLE]

On the other hand, by Lemma 2.1(ii), Lemma 2.4 and the definition of $T$, we obtain

[TABLE]

Hence combining above inequalities and applying (3.51) and (3.54), we arrive at the conclusion.

Lemma 3.14**.**

Under the assumptions of Proposition 3.2, we have

[TABLE]

Proof.

From (3.69), we have

[TABLE]

By Lemma 2.1, the result in Section 2 of [31] and $\alpha_{1}<\min\{\lambda_{1},m_{\infty}\}$, we obtain

[TABLE]

According to the definition of $T$, Lemma 3.13 and Lemma 2.4, this shows that

[TABLE]

Similarly, we can also get

[TABLE]

and

[TABLE]

where the fact that $q_{0}\in(3,\frac{3p_{0}}{2(3-p_{0})})$ warrants $-\frac{3}{p_{0}}+\frac{3}{2q_{0}}>-1$ is used. Hence the combination of the above inequalities yields $\|\rho(\cdot,t)\|_{L^{\theta}(\Omega)}\leq\frac{M_{1}}{2}\varepsilon(1+t^{-\frac{3}{2}(\frac{1}{p_{0}}-\frac{1}{\theta})})e^{-\alpha_{1}t}$, thanks to (3.60), (3.59) and (3.56).

Lemma 3.15**.**

Under the assumptions of Proposition 3.2, we have

[TABLE]

Proof.

From (3.68) and Lemma 2.1(iv), it follows that

[TABLE]

From the definition of $T$ and (3.58), we have

[TABLE]

From Lemma 3.9, Lemma 3.11 and (3.61), it follows that

[TABLE]

Combining the above inequalities, we arrive at

[TABLE]

and thus complete the proof of this lemma.

By the above lemmas, one can see that $T=T_{max}$, and the further estimates of solutions are needed to ensure $T_{max}=\infty$.

Lemma 3.16**.**

Under the assumptions of Proposition 3.2, for all $\beta\in(\frac{3}{4},\min\{\frac{5}{4}-\frac{3}{2q_{0}},1\})$ there exists $M_{6}>0$ such that

[TABLE]

Proof.

The proof is similar to that of (3.40), and thus is omitted here.

Lemma 3.17**.**

Under the assumptions of Proposition 3.2, there exists $M_{7}>0$ such that $\|c(\cdot,t)-m_{\infty}\|_{L^{\infty}(\Omega)}\leq M_{7}e^{-\alpha_{2}t}$ for all $(t_{0},T_{max})$ with $t_{0}=\min\{\frac{T_{max}}{6},1\}$.

Proof.

We refer the readers to the proof of Lemma 3.24 in [18].

At this position, we can show the proof of Theorem 1.2 in the case $\mathcal{S}=0$ on $\partial\Omega$.

Proof of Proposition 3.2.  We first show that the solution is global, i.e. $T_{max}=\infty$. To this end, according to the extensibility criterion in Lemma 2.5, it suffices to show that there exists $C>0$ such that for all $t_{0}<t<T_{max}$

[TABLE]

From Lemma 3.10, Lemma 3.14 and Lemma 3.16, there exists $K_{i}>0$, $i=1,2,3,4$, such that

[TABLE]

[TABLE]

for $t\in(t_{0},T_{max})$. Furthermore, Lemma 3.17 implies that $\|c(\cdot,t)-m_{\infty}\|_{W^{1,\infty}(\Omega)}\leq K_{3}^{\prime}e^{-\alpha_{2}t}$ with some $K_{3}^{\prime}>0$ for all $t\in(t_{0},T_{max})$. Since $D(A^{\beta})\hookrightarrow L^{\infty}(\Omega)$ with $\beta\in(\frac{3}{4},1)$, it follows from Lemma 3.16 that $\|u(\cdot,t)\|_{L^{\infty}(\Omega)}\leq K_{4}e^{-\alpha_{2}t}$ for some $K_{4}>0$ for all $t\in(t_{0},T_{max})$. This completes the proof of Proposition 3.2.

4 Proof of main results for general $\mathcal{S}$

In this section, we give the proof of our results for the general matrix-valued $\mathcal{S}$ by a rather standard argument, which is accomplished by an approximation procedure (see [4] for example). In order to make the previous results applicable, we introduce a family of smooth functions $\rho_{\eta}\in C_{0}^{\infty}(\Omega)$ and $0\leq\rho_{\eta}(x)\leq 1$ for $\eta\in(0,1),$ $\lim_{\eta\to 0}\rho_{\eta}(x)=1$ and let $\mathcal{S}_{\eta}(x,\rho,c)=\rho_{\eta}(x)\mathcal{S}(x,\rho,c).$ Using this definition, we regularize (1.4) as follows

[TABLE]

with the initial data

[TABLE]

It is observed that $\mathcal{S}_{\eta}$ satisfies the additional condition $\mathcal{S}=0$ on $\partial\Omega$. Therefore based on the discussion in Section 3, under the assumptions of Theorem 1.1 and Theorem 1.2, problem (4.1)–(4.2) admits a global classical solution $(\rho_{\eta},m_{\eta},c_{\eta},u_{\eta},P_{\eta})$ that satisfies

[TABLE]

for some constants $K_{i}$, $i=1,2,3,4$, and all $t\geq 0$. Applying a standard procedure such as in Lemma 5.2 and Lemma 5.6 of [4], one can obtain a subsequence of $\{\eta_{j}\}_{j\in\mathbb{N}}$ with $\eta_{j}\to 0$ as $j\to\infty$ such that $\rho_{\eta_{j}}\rightarrow\rho,~{}m_{\eta_{j}}\rightarrow m,~{}c_{\eta_{j}}\rightarrow c,u_{\eta_{j}}\rightarrow u\quad\hbox{in}~{}C_{loc}^{\nu,\frac{\nu}{2}}(\overline{\Omega}\times(0,\infty))$ as $j\rightarrow\infty$ for some $\nu\in(0,1)$. Moreover, by the arguments as in Lemma 5.7, Lemma 5.8 of [4], one can also show that $(\rho,m,c,u,P)$ is a classical solution of (1.4) with the decay properties asserted in Theorem 1.1 and Theorem 1.2, respectively. The proof of our main results is thus complete.