A new result for the global existence (and boundedness), regularity and stabilization of a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization

TL;DR

This paper establishes the global existence, boundedness, regularity, and stabilization of solutions for a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization, under certain conditions on the rotational effect.

Contribution

It proves the existence of global weak solutions and their boundedness, as well as their convergence to equilibrium, for a complex chemotaxis-fluid system with rotational effects.

Findings

Global weak solutions exist for the system.

Solutions are uniformly bounded in relevant norms.

Solutions tend to equilibrium as time approaches infinity.

Abstract

This paper deals with the following quasilinear Keller-Segel-Navier-Stokes system modeling coral fertilization $(*)$: $$\left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\nabla c)-nm,\quad x\in \Omega, t>0, c_t+u\cdot\nabla c=\Delta c-c+m,\quad x\in \Omega, t>0, m_t+u\cdot\nabla m=\Delta m-nm,\quad x\in \Omega, t>0,\\ u_t+\kappa(u \cdot \nabla)u+\nabla P=\Delta u+(n+m)\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0,\quad x\in \Omega, t>0 \end{array}\right.$$ under no-flux boundary conditions in a bounded domain $\Omega\subset \mathbb{R}^3$ with smooth boundary, where $\phi\in W^{2,\infty} (\Omega)$. Here the matrix-valued function $S(x,n,c)$ denotes the rotational effect which satisfies $|S(x,n,c)|\leq S_0 (c)(1 + n)^{-\alpha}$ with $\alpha\geq0$ and some nonnegative nondecreasing function $S_0$. Based on this inequality and some carefully analysis, if $\alpha>0$, then for any $\kappa\in\mathbb{R},$ system $(*)$ possesses a global weak solution for which there exists $T > 0$ such that $(n,c,m , u)$ is smooth in $\Omega\times( T ,\infty)$. Furthermore, for any $p>1,$ this solution is uniformly bounded in with respect to the norm in $L^p(\Omega)\times L^\infty(\Omega) \times L^\infty(\Omega)\times L^2 (\Omega; \mathbb{R}^3)$. Building on this boundedness property and some other analysis, it can finally even be proved that in the large time limit, any such solution approaches the spatially homogeneous equilibrium $(\hat{n},\hat{m},\hat{m},0)$ in an appropriate sense, where $\hat{n}=\frac{1}{|\Omega|}\{\int_{\Omega}n_0-\int_{\Omega}m_0\}_{+}$ and $\hat{m}=\frac{1}{|\Omega|}\{\int_{\Omega}m_0 -\int_{\Omega}n_0\}_{+}$.

Equations

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

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

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

$$|S(x,n,c)|\le (1+n{)}^{-\alpha }S(c)\text{mbox}forall(x,n,c)\in \Omega \times [0,\infty {)}^2\text{mbox}withS(c)\text{mbox}nondecreasingon[0,\infty )$$

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

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

$$|S(x,n,c)|\le (1+n{)}^{-\alpha }\text{mbox}forall(x,n,c)\in \Omega \times [0,\infty {)}^2$$

$$\left\{\begin{array}[]{ll}\rho_{t}+u\cdot\nabla\rho=\Delta\rho-\chi\nabla\cdot(\rho\nabla c)-\varepsilon\rho^{q},\\ \displaystyle{0=\Delta c+\rho,}\\ \end{array}\right.$$

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

$$\Vert n(\cdot ,t){\Vert }_{L^2(\Omega )}^2+\Vert \nabla c(\cdot ,t){\Vert }_{L^2(\Omega )}^2+\Vert u(\cdot ,t){\Vert }_{W^{1,2}(\Omega )}^2,$$

$$\left\{\begin{array}[]{rl}\displaystyle{\int_{\Omega}n_{\varepsilon}^{4\alpha+\frac{2}{3}}+\int_{\Omega}|\nabla c_{\varepsilon}|^{2}+\int_{\Omega}|{u_{\varepsilon}}|^{2}~{}~{}~{}\mbox{if}~{}~{}\alpha\neq\frac{1}{12},}\\ \displaystyle{\int_{\Omega}n_{\varepsilon}\ln n_{\varepsilon}+\int_{\Omega}|\nabla c_{\varepsilon}|^{2}+\int_{\Omega}|{u_{\varepsilon}}|^{2}~{}~{}~{}~{}\mbox{if}~{}~{}\alpha=\frac{1}{12},}\end{array}\right.$$

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

$$\left\{\begin{array}[]{ll}\displaystyle{n_{0}\in C(\bar{\Omega})~{}~{}~{}~{}\mbox{with}~{}~{}n_{0}\geq 0~{}~{}\mbox{and}~{}~{}n_{0}\not\equiv 0},\\ \displaystyle{c_{0}\in W^{1,\infty}(\Omega)~{}~{}\mbox{with}~{}~{}c_{0}\geq 0~{}~{}\mbox{in}~{}~{}\bar{\Omega},}\\ \displaystyle{m_{0}\in C(\bar{\Omega})~{}~{}~{}~{}\mbox{with}~{}~{}m_{0}\geq 0~{}~{}\mbox{and}~{}~{}m_{0}\not\equiv 0},\\ \displaystyle{u_{0}\in D(A^{\gamma}_{r})~{}~{}\mbox{for~{}~{} some}~{}~{}\gamma\in(\frac{3}{4},1)~{}~{}\mbox{and any}~{}~{}{r}\in(1,\infty),}\\ \end{array}\right.$$

$$\alpha >0,$$

$$\left\{\begin{array}[]{ll}n\in L^{\infty}_{loc}([0,\infty),L^{p}(\Omega))\cap L^{2}_{loc}([0,\infty),W^{1,2}(\Omega))~{}~{}~{}\mbox{for any}~{}~{}p>1,\\ c\in L^{\infty}(\Omega\times(0,\infty))\cap L^{2}_{loc}([0,\infty),W^{2,2}(\Omega))\cap L^{4}_{loc}([0,\infty),W^{1,4}(\Omega)),\\ m\in L^{\infty}(\Omega\times(0,\infty))\cap L^{2}_{loc}([0,\infty),W^{2,2}(\Omega))\cap L^{4}_{loc}([0,\infty),W^{1,4}(\Omega)),\\ u\in L^{2}_{loc}([0,\infty),W^{1,2}_{0,\sigma}(\Omega))\cap L^{\infty}_{loc}([0,\infty),L^{2}(\Omega)),\\ \end{array}\right.$$

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

$$n,c,m\in C^{2+\iota ,1+\frac{\iota }{2}}(\bar{\Omega }\times [T,\infty )),u\in C^{2+\iota ,1+\frac{\iota }{2}}(\bar{\Omega }\times [T,\infty );{\mathbb{R}}^3).$$

$$n(\cdot ,t)\to \hat{n},c(\cdot ,t)\to \hat{m}\text{mbox}aswellasm(\cdot ,t)\to \hat{m}\text{mbox}andu(\cdot ,t)\to 0\text{mbox}in{L}^{\infty }(\Omega ),$$

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

$$\begin{array}[]{ll}S_{\varepsilon}(x,n,c)=\rho_{\varepsilon}(x)\chi_{\varepsilon}(n)S(x,n,c),~{}~{}x\in\bar{\Omega},~{}~{}n\geq 0,~{}~{}c\geq 0\end{array}$$

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

$$0\le {\rho }_{\epsilon }\le 1\text{mbox}in\Omega ,{\rho }_{\epsilon }\nearrow 1\text{mbox}in\Omega \text{mbox}as\epsilon \searrow 0$$

$$0\le {\chi }_{\epsilon }\le 1\text{mbox}in[0,\infty ),{\chi }_{\epsilon }\nearrow 1\text{mbox}in[0,\infty )\text{mbox}as\epsilon \searrow 0.$$

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

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

$$\Vert e^{-tA}\mathcal{P}\phi {\Vert }_{L^p(\Omega )}\le {\kappa }_1(\Omega )t^{-\gamma }\Vert \phi {\Vert }_{L^q(\Omega )}\text{mbox}forallt>0\text{mbox}andany\phi \in L^q(\Omega )$$

$$\Vert e^{-tA}\mathcal{P}\nabla \cdot \phi {\Vert }_{L^p(\Omega )}\le {\kappa }_2(\Omega )t^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q}-\frac{1}{p})}\Vert \phi {\Vert }_{L^q(\Omega )}\text{mbox}forallt>0\text{mbox}andany\phi \in L^q(\Omega )$$

$$\Vert e^{-tA}\mathcal{P}\phi {\Vert }_{L^p(\Omega )}\le {\kappa }_3(\Omega )\Vert \phi {\Vert }_{L^p(\Omega )}\text{mbox}forallt>0\text{mbox}andany\phi \in L^p(\Omega ).$$

$$\frac{d}{dt}{\int }_{\Omega }{c}_{\epsilon }(\cdot ,t)\le 0,\text{mbox}and\frac{d}{dt}{\int }_{\Omega }{m}_{\epsilon }(\cdot ,t)\le 0\text{mbox}forallt\in (0,T_{max,\epsilon }),$$

$$\Vert c_{\epsilon }(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert m_{\epsilon }(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}\le \lambda \text{mbox}forallt\in (0,T_{max,\epsilon }),$$

Full text

[TABLE]

under no-flux boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^{3}$ with smooth boundary, where $\phi\in W^{2,\infty}(\Omega)$. Here $S(x,n,c)$ denotes the rotational effect which satisfies $|S(x,n,c)|\leq S_{0}(c)(1+n)^{-\alpha}$ with $\alpha\geq 0$ and some nonnegative nondecreasing function $S_{0}$. Based on a new weighted estimate and some carefully analysis, if $\alpha>0$, then for any $\kappa\in\mathbb{R},$ system $(*)$ possesses a global weak solution for which there exists $T>0$ such that $(n,c,m,u)$ is smooth in $\Omega\times(T,\infty)$. Furthermore, for any $p>1,$ this solution is uniformly bounded in with respect to the norm in $L^{p}(\Omega)\times L^{\infty}(\Omega)\times L^{\infty}(\Omega)\times L^{2}(\Omega;\mathbb{R}^{3})$. Building on this boundedness property and some other analysis, it can finally even be proved that in the large time limit, any such solution approaches the spatially homogeneous equilibrium $(\hat{n},\hat{m},\hat{m},0)$ in an appropriate sense, where $\hat{n}=\frac{1}{|\Omega|}\{\int_{\Omega}n_{0}-\int_{\Omega}m_{0}\}_{+}$ and $\hat{m}=\frac{1}{|\Omega|}\{\int_{\Omega}m_{0}-\int_{\Omega}n_{0}\}_{+}$.

Key words: Navier-Stokes system; Keller-Segel model; Global existence; Large time behavior; Tensor-valued sensitivity

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

1 Introduction

This work is concerned with the following chemotaxis-fluid system modelling coral fertilization:

[TABLE]

where $\Omega\subset\mathbb{R}^{3}$ is a bounded domain with smooth boundary $\partial\Omega$ and the matrix-valued function $S(x,n,c)$ indicates the rotational effect which satisfies

[TABLE]

and

[TABLE]

and $\alpha\geq 0.$ As described in [19, 18, 9, 10], problems of this type arise in the modeling of the phenomenon of coral broadcast spawning, where the sperm $n$ 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 (see also [22]). Here $\kappa,u,P$ and $\phi$ denote, respectively, the strength of nonlinear fluid convection, the velocity field, the associated pressure of the fluid and the potential of the gravitational field. We further note that the sensitivity tensor $S(x,n,c)$ may take values that are matrices possibly containing nontrivial off-diagonal entries, 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 [56, 55] for the detailed model derivation).

Chemotaxis is the directed movement of the cells as a response to gradients of the concentration of the chemical signal substance in their environment, where the chemical signal substance may be produced or consumed by cells themselves (see e.g. Hillen and Painter [12] and [2]). The classical chemotaxis system was introduced in 1970 by Keller and Segel ([17]), which is called Keller-Segel system. Since then, the Keller-Segel model has attracted more and more attention, and also has been constantly modified by various authors to characterize more biological phenomena (see Cieślak and Stinner [5], Cieślak and Winkler [7], Ishida et al. [15], Painter and Hillen [27], Hillen and Painter [12], Wang et al. [34, 35], Winkler et al. [7, 13, 54, 43, 45, 44, 47], Zheng [58] and references therein for detailed results). For related works in this direction, we mention that a corresponding quasilinear version ( see e.g. [31, 54, 60, 58, 61]), the logistic damping or the signal consumed by the cells, has been deeply investigated by Cieślak and Stinner [5, 6], Tao and Winkler [31, 42, 54], and Zheng et al. [58, 66, 60, 67].

In various situations, however, the interaction of chemotactic movement of the gametes and the surrounding fluid is not negligible (see Tuval et al. [33]). In 2005, Tuval et al. ([33]) proposed the following prototypical signal consuming model (with tensor-valued sensitivity):

[TABLE]

where $f(c)$ denotes the consumption rate of the oxygen by the cells. Here $S$ is a tensor-valued function or a scalar function which is the same as (2.1). The model (1.4) describes the interaction of oxygen-taxis bacteria with a surrounding incompressible viscous fluid in which the oxygen is dissolved. After this, assume that the chemotactic sensitivity $S(x,n,c):=S(c)$ is a scalar function. This kind of models have been studied by many researchers by making use of energy-type functionals (see e.g. Chae et. al. [4], Duan et. al. [8], Liu and Lorz [25, 26], Tao and Winkler [32, 46, 48, 50], Zhang and Zheng [57] and references therein). In fact, if $S(x,n,c):=S(c)$, Winkler ([46] and [48]) proved that in two-dimensional space (1.4) admits a unique global classical solution which stabilizes to the spatially homogeneous equilibrium $(\frac{1}{|\Omega|}\int_{\Omega}n_{0},0,0)$ in the large time limit. While in three-dimensional setting, he (see [50]) also showed that there exists a globally defined weak solution to (1.4).

Experiment [56] show that the chemotactic movement could be not directly along the signal gradient, but with a rotation, so that, the the corresponding chemotaxis-fluid system with tensor-valued sensitivity loses entropy-like functional structure, which gives rise to considerable mathematical difficulties during the process of analysis. The global solvability of corresponding initial value problem for chemotaxis-fluid system with tensor-valued sensitivity have been deeply investigated by Cao, Lankeit [3], Ishida [14], Wang et al. [36, 39] and Winkler [49].

If $-nf(c)$ in the $c$-equation is replaced by $-c+n$, and the $u$-equation is a (Navier-)Stokes equation, then (1.4) becomes the following chemotaxis-(Navier-)Stokes system in the context of signal produced other than consumed by cells (see [52, 38, 39, 40, 64, 16])

[TABLE]

Due to the presence of the tensor-valued sensitivity $S(x,n,c)$ as well as the strongly nonlinear term $(u\cdot\nabla)u$ and lower regularity for $n$, the analysis of (1.5) with tensor-valued sensitivity began to flourish (see [52, 38, 39, 40, 64, 16]). In fact, the global boundedness of classical solutions to the Stokes-version ($\kappa=0$ in the third equation of system (1.5)) of system (1.5) with the tensor-valued $S$ satisfying $|S(x,n,c)|\leq C_{S}(1+n)^{-\alpha}$ with some $C_{S}>0$ and $\alpha>0$ which implies that the effect of chemotaxis is weakened when the cell density increases has been proved for any $\alpha>0$ in two dimensions (see Wang and Xiang [39]) and for $\alpha>\frac{1}{2}$ in three dimensions (see Wang and Xiang [40]). Then Wang-Winkler-Xiang ([38]) further shows that when $\alpha>0$ and $\Omega\subset R^{2}$ is a bounded convex domain with smooth boundary, system (1.5) possesses a global-in-time classical and bounded solution. Recently, Zheng ([62]) extends the results of [38] to the general bounded domain by some new entropy-energy estimates. More recently, by using new entropy-energy estimates, Zheng and Ke ([16]) presented the existence of global and weak solutions for the system (1.5) under the assumption that $S$ satisfies (1.2) and

[TABLE]

with $\alpha>\frac{1}{3}$, which, in light of the known results for the fluid-free system (see Horstmann and Winkler [13] and Bellomo et al. [2] ), is an optimal restriction on $\alpha$. For more works about the chemotaxis-(Navier-)Stokes models (1.5), we mention that a corresponding quasilinear version or the logistic damping has been deeply investigated by Zheng [59], Wang and Liu [24], Tao and Winkler [32], Wang et. al. [39, 40].

Other variants of the model (1.5) has been used in the mathematical study of coral broad- cast spawning. In fact, Kiselev and Ryzhik ([19] and [18]) introduced the following Keller-Segel type system to model coral fertilization:

[TABLE]

where $\rho,u,\chi$ and $-\varepsilon\rho^{q}$, respectively, denote the density of egg (sperm) gametes, the smooth divergence free sea fluid velocity as well as the positive chemotactic sensitivity constant and the reaction (fertilization) phenomenon. In fact, under suitable conditions, the global-in-time existence of the solution to (1.6) is presented by Kiselev and Ryzhik in [19]. Moreover, they proved that the total mass $m_{0}(t)=\int_{R^{2}}\rho(x,t)dx$ approaches a positive constant whose lower bound is $C(\chi,\rho_{0},u)$ as $t\rightarrow\infty$ when $q>2$. In the critical case of $N=q=2$, a corresponding weaker but yet relevant effect within finite time intervals is detected (see [18]).

In order to analyze a further refinement of the model (1.6) which explicitly distinguishes between sperms and eggs, Espejo and Winkler ([10]) have recently considered the Navier-Stokes version of (2.1):

[TABLE]

in a bounded domain $\Omega\subset\mathbb{R}^{2}$. If $N=2,$ Espejo and Winkler ([10]) established the global existence of classical solutions to the associated initial-boundary value problem (1.7), which tend towards a spatially homogeneous equilibrium in the large time limit. Furthermore, if $S(x,n,c)$ satisfying (1.2) and (1.3) with $\alpha\geq\frac{1}{3}$ or $\alpha\geq 0$ and the initial data satisfy a certain smallness condition, Li-Pang-Wang ([22]) proved the same result for the three-dimensional Stokes ($\kappa=0$ in the fourth equation of (1.1)) version of system (1.1). From [22], we know that $\alpha\geq\frac{1}{3}$ is enough to warrant the boundedness of solutions to system (2.1) for any large data (see Li-Pang-Wang [22]). We should point that the core step of [22] is to establish the estimates of the functional

[TABLE]

which strongly relies on $\alpha\geq\frac{1}{3}$ and $\kappa=0$ (see the proof of Lemma 3.1 of [22]). To the best of our knowledge, it is yet unclear whether for $\alpha<\frac{1}{3}$ or $\kappa\neq 0$, the solutions of (2.1) exist (or even bounded) or not. Recently, relying on the functional

[TABLE]

we ([63]) presented the existence of global weak solutions for the system (1.1) under the assumption that $S$ satisfies (1.2) and (1.3) with $\alpha>0$. However, the existence of global (stronger than the result of [63]) weak solutions is still open. In this paper, by using a new weighted estimate (see Lemma 3.2), we try to obtain enough regularity and compactness properties (see Lemmas 3.2, 3.3, and 3.5), then show that system (1.1) possesses a globally defined weak solution, which improves the result of [63]. Therefore, collecting the above results, it is meaningful to analyze the following question:

Whether or not the assumption of $\alpha$ is optimal? Can we further relax the restriction on $\alpha$, say, to $\alpha>0$? Moreover, can we consider the regularity of global solution for system (1.1)?

Inspired by the above works, the first result of paper is to prove the existence of global (and bounded) solution for any $\alpha>0.$ Moreover, we also show that the corresponding solutions converge to a spatially homogeneous equilibrium exponentially as $t\rightarrow\infty$ as well.

Throughout this paper, we assume that

[TABLE]

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

[TABLE]

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 ${r}\in(1,\infty)$ ([29]).

In the context of these assumptions, the first of our main results can be read as follows.

Theorem 1.1**.**

Let $\Omega\subset\mathbb{R}^{3}$ be a bounded domain with smooth boundary. Suppose that the assumptions (1.2)–(1.3) and (1.8)–(1.9) hold. If

[TABLE]

then for any $\kappa\in\mathbb{R}$, there exist

[TABLE]

such that $(n,c,m,u)$ is a global weak solution of the problem (2.1) in the natural sense as specified in [63]. Moreover, if $\kappa=0$, the problem (2.1) possesses at least one global classical solution $(n,c,m,u,P)$. Moreover, this solution is bounded in $\Omega\times(0,\infty)$ in the sense that

[TABLE]

Remark 1.1**.**

(i) Theorem 1.1 indicates that $\alpha>0$ and $\kappa=0$ is enough to ensure the global existence and uniform boundedness of solution of the three-dimensional Keller-Segel-Stokes system (1.1), which improves the result obtained in [22], therein $\alpha\geq\frac{1}{3}$ is required.

(ii)This result also improves the result of our recent paper ([63]), where the more weak solution than our result was obtained by using different method (see [63]).

We can secondly prove that in fact any such weak solution $(n,c,m,u)$ becomes smooth ultimately, and that it approaches the unique spatially homogeneous steady state $(\hat{n},\hat{m},\hat{m},0)$, where $\hat{n}=\frac{1}{|\Omega|}\{\int_{\Omega}n_{0}-\int_{\Omega}m_{0}\}_{+}$ and $\hat{m}=\frac{1}{|\Omega|}\{\int_{\Omega}m_{0}-\int_{\Omega}n_{0}\}_{+}$.

Theorem 1.2**.**

Under the assumptions of Theorem 1.1, then there are $T>0$ and $\iota\in(0,1)$ such that the solution $(n,c,m,u)$ given by Theorem 1.1 satisfies

[TABLE]

Moreover,

[TABLE]

where $\hat{n}=\frac{1}{|\Omega|}\{\int_{\Omega}n_{0}-\int_{\Omega}m_{0}\}_{+}$ and $\hat{m}=\frac{1}{|\Omega|}\{\int_{\Omega}m_{0}-\int_{\Omega}n_{0}\}_{+}$.

Remark 1.2**.**

Theorem 1.1 indicates that if $\alpha>0$, then for arbitrarily large initial data and for any $\kappa\in\mathbb{R}$, this problem admits at least one global weak solution for which there exists $T>0$ such that $(n,c,m,u)$ is smooth in $\Omega\times(T,\infty)$. Moreover, it is asserted that such solutions are shown to approach a spatially homogeneous equilibrium in the large time limit, which improves the result obtained in [22], therein $\kappa=0$ is required.

Mathematical challenges for the regularity and stabilization of the solution for system (1.1). System (1.1) incorporates fluid and rotational flux, which involves more complex cross-diffusion mechanisms and brings about many considerable mathematical difficulties. Firstly, even when posed without any external influence, that is, $n=c=m\equiv 0,$ the corresponding Navier-Stokes system (1.1) does not admit a satisfactory solution theory up to now (see Leray [21] and Sohr [29], Wiegner [41]). As far as we know that the question of global solvability in classes of suitably regular functions yet remains open except in cases when the initial data are appropriately small (see e.g. Wiegner [41]). Moreover, the tensor-valued sensitivity functions result in new mathematical difficulties, mainly linked to the fact that a chemotaxis system with such rotational fluxes thereby loses an energy-like structure (see e.g. [56]). In [22] and [10], relying on globally bounded for the solution, Espejo-Winkler ([10]) Li-Pang-Wang ([22]) proved that all these solutions of problem (1.1) are shown to approach a spatially homogeneous equilibrium in the large time limit when $N=2$ or $N=3$ and $\kappa=0,$ respectively. As already mentioned in the above, in the case $N=3$, it is not only unknown whether the incompressible Navier-Stokes equations possess global smooth solutions for arbitrarily large smooth initial data (see e.g. Wiegner [41] and Sohr [29]). Therefore, when $\kappa\not=0$ and $N=3$, we can not use the idea of [22] and [10] to discuss the large time behavior to problem (1.1), since, the globally bounded for the solutions are needed in [22] and [10].

In order to derive these theorems, in Section 2, we introduce the regularized system of (1.1), establish some basic estimates of the solutions and recall a local existence result. In Section 3, a key step of the proof of our main results is to establish a bound for $n_{\varepsilon}(\cdot,t)$ in $L^{{p}}(\Omega)$ for any $p>1$. The approach is based on the weighted estimate of $\int_{\Omega}n_{\varepsilon}^{{p}}g(c_{\varepsilon})$ with some weight function $g(c_{\varepsilon})$ which is uniformly bounded both from above and below by positive constants. Here $n_{\varepsilon}$ and $c_{\varepsilon}$ are components of the solutions to (2.1) below. On the basis of the previously established estimates and the compactness properties thereby implied, we shall pass to the limit along an adequate sequence of numbers $\varepsilon=\varepsilon_{j}\searrow 0$ and thereby verify Theorem 1.1. Using the basic relaxation properties expressed in (2.13) and (2.14), Section 4 is devoted to showing the large time behavior of global solutions to (1.1) obtained in the above section. To this end, thanks to the decay property of $m_{\varepsilon}(\cdot,t)-\hat{m}+n_{\varepsilon}(\cdot,t)-\hat{n}$ formulated in Lemmas 4.6 and 4.8, this actually entails a certain eventual regularity and decay of $u_{\varepsilon}$ also in the present situation, where $\hat{n}=\frac{1}{|\Omega|}\{\int_{\Omega}n_{0}-\int_{\Omega}m_{0}\}_{+}$ and $\hat{m}=\frac{1}{|\Omega|}\{\int_{\Omega}m_{0}-\int_{\Omega}n_{0}\}_{+}$. Using these bounds (see Lemmas 4.6–4.10), based on maximal Sobolev regularity in the Stokes evolution system as well as inhomogeneous linear heat equations and the standard Schauder theory, we then prove eventual Hölder regularity and smoothness of solution $(n_{\varepsilon},c_{\varepsilon},m_{\varepsilon},u_{\varepsilon})$ (see Lemmas 4.11–4.12). For convergence as $t\rightarrow\infty$, we draw upon uniform Hölder bounds and smoothness for solution $(n,c,m,u)$ (see Lemmas 4.13–4.14). Finally, applying an Ehrling-type lemma, we can prove any such solution approaches the spatially homogeneous equilibrium by using the above convergence properties (see Lemma 4.15).

2 Preliminaries

As mentioned in the introduction, the chemotactic sensitivity $S$ in the first equation in (1.1) and the nonlinear convective term $\kappa(u\cdot\nabla)u$ in the Navier-Stokes subsystem of (1.1) bring about a great challenge to the study of system (1.1). To deal with these difficulties, according to the ideas in [50] (see also [53, 16, 51]), we first consider the approximate problems given by

[TABLE]

where

[TABLE]

and

[TABLE]

is the standard Yosida approximation. Here $(\rho_{\varepsilon})_{\varepsilon\in(0,1)}\in C^{\infty}_{0}(\Omega)$ and $(\chi_{\varepsilon})_{\varepsilon\in(0,1)}\in C^{\infty}_{0}([0,\infty))$ are a family of functions which satisfy

[TABLE]

and

[TABLE]

Without essential difficulty, the local existence of approximate solutions to (2.1) can be easily proved according to the corresponding procedure in Lemma 2.1 of [50] (see also [49] and Lemma 2.1 of [27]). Therefore, we give the following lemma without proof.

Lemma 2.1**.**

Assume that $\varepsilon\in(0,1).$ Then there exist $T_{max,\varepsilon}\in(0,\infty]$ and a classical solution $(n_{\varepsilon},c_{\varepsilon},m_{\varepsilon},u_{\varepsilon},P_{\varepsilon})$ of (2.1) in $\Omega\times(0,T_{max,\varepsilon})$ such that

[TABLE]

classically solving (2.1) in $\Omega\times[0,T_{max,\varepsilon})$. Moreover, $n_{\varepsilon},c_{\varepsilon}$ and $m_{\varepsilon}$ are nonnegative in $\Omega\times(0,T_{max,\varepsilon})$, and

[TABLE]

where $\gamma$ is given by (1.9).

Lemma 2.2**.**

([13, 43, 65]) The Stokes operator $A$ denotes the realization of the Stokes operator under homogeneous Dirichlet boundary conditions in the solenoidal subspace $L^{2}_{\sigma}(\Omega)$ of $L^{2}(\Omega)$. Let $\mathcal{P}:L^{p}(\Omega)\rightarrow L^{p}_{\sigma}(\Omega)$ stand for the Helmholtz projection in $L^{p}(\Omega).$ Then there exist positive constants $\kappa_{i}(i=1,\ldots,3)$ such that

[TABLE]

as well as

[TABLE]

and

[TABLE]

Invoking the divergence free of the fluid and the homogeneous Neumann boundary conditions on $n_{\varepsilon},m_{\varepsilon}$ and $c_{\varepsilon}$, we can establish the following basic estimates by using the maximum principle to the second and third equations. The proof of this lemma is very similar to that of Lemmas 2.2 and 2.6 of [32] (see also Lemma 3.2 of [37]), so we omit its proof here

Lemma 2.3**.**

There exists $\lambda>0$ such that the solution of (2.1) satisfies

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

as well as

[TABLE]

and

[TABLE]

For simplicity, here and hereafter, we take the notations

[TABLE]

by using (2.10) and (1.3).

3 A-priori estimates

In this section we want to ensure that the time-local solutions obtained in Lemma 2.1 are in fact global solutions. To this end, for any $p>1,$ we firstly obtain boundedness of $n_{\varepsilon}$ in $L^{p}(\Omega)$ under the assumption that $\alpha>0.$ Inspired by the weighted estimate argument developed in [44] (see also [48, 51]), we shall invoke a weight function $g(c_{\varepsilon})$ which is uniformly bounded from above and below by positive constants. Before deriving the uniform of $L^{p}$ norm of $n_{\varepsilon}$, let us first recalling the well-known facts for $g(c_{\varepsilon})$.

Lemma 3.1**.**

Let

[TABLE]

where

[TABLE]

and $C_{S}$ is given by (2.16). Then for any $s\in(0,\|c_{0}\|_{L^{\infty}(\Omega)}]$,

[TABLE]

*and *

[TABLE]

Proof.

Obviously, (3.2) holds. On the other hand, a direct computation shows

[TABLE]

This combined with the fact that $\beta=\frac{1}{8\|c_{0}\|_{L^{\infty}(\Omega)}^{2}}$ implies (3.3). ∎

Lemma 3.2**.**

Let $\alpha>0$. Then for any $p>1$, there exists $C>0$ such that the solution of (2.1) satisfies

[TABLE]

Proof.

Firstly, we define a functional

[TABLE]

where $p>\max\{1,2\alpha\}$, $g(c_{\varepsilon})=e^{\beta c^{2}}$ and $\beta$ is the same as (3.1). Using the first two equations in (2.1), we find:

[TABLE]

In the following, we will estimate the right-hand sides of (3.6) one by one. To this end, firstly, applying the elementary calculus identity

[TABLE]

and the fact that

[TABLE]

we once more integrate by parts to find that

[TABLE]

by using $u_{\varepsilon}=0,x\in\partial\Omega,t>0$. Next, we derive from the non-negativity of $g^{\prime},g$ (see (3.4) and (3.2)), $c_{\varepsilon},m_{\varepsilon}$ and $n_{\varepsilon}$ that

[TABLE]

which combined with (3.6) and (3.7) yields

[TABLE]

Now we proceed to estimate the fourth term on the right-hand side herein by using (2.10) and (3.3) to find that

[TABLE]

where $\mu_{1}$ is the same as (3.3).

Now we estimate the term $\int_{\Omega}n_{\varepsilon}^{p-1}g(c_{\varepsilon})\Delta n_{\varepsilon}$ and $\frac{1}{p}\int_{\Omega}n_{\varepsilon}^{p}g^{\prime}(c_{\varepsilon})\Delta c_{\varepsilon}$ in the right hand side of (3.9). In fact, we once more integrate by parts to see that

[TABLE]

by using the Young inequality. Next, recall (1.3), we can estimate second term on the right-hand side of (3.9) as follows:

[TABLE]

where in the last inequality, we have used the Young inequality. Now, collecting (3.9)–(3.12), we may have

[TABLE]

whence returning to the definition of $g(c_{\varepsilon})$ we conclude that

[TABLE]

by using $p>\max\{1,2\alpha\}.$ In view of $\beta=\frac{1}{8\|c_{0}\|_{L^{\infty}(\Omega)}^{2}}$, thus, (3.14) implies that

[TABLE]

On the other hand, due to $\alpha>0,$ we may have

[TABLE]

so that, there exists $\eta_{0}>0$, such that for any $s>\eta_{0}$,

[TABLE]

Therefore, by some basic calculation, we derive from (2.10) that

[TABLE]

with

[TABLE]

by using (2.10) and $p>\max\{1,2\alpha\}$. Substituting (3.16) into (3.15), we have

[TABLE]

Now, according to (2.10), we therefore obtain on using the Gagliardo-Nirenberg inequality that

[TABLE]

for some positive constants $C_{1}$ and $C_{2}$, where in the last inequality, we have used (3.2). Collecting (3.17) and (3.18), we have

[TABLE]

by using the Young inequality. Therefore,

[TABLE]

To track the time evolution of $c_{\varepsilon}$, testing the second equation in (2.1) by $c_{\varepsilon}$ and using $\nabla\cdot u_{\varepsilon}=0$ and (2.10) yields that for some positive constant $C_{4}$ such that

[TABLE]

wereafter integrating the above inequality in time yields

[TABLE]

for some $C_{5}>0$ by an integration. This yields to

[TABLE]

with

[TABLE]

Finally, (3.23) in conjunction with Lemma 2.3 of [38] (see also [62]) and (3.20) establish (3.25).

∎

In a straightforward manner, the estimates gained above can be seen to imply the following $\varepsilon$-independent estimates, which plays an important role in proving Theorem 1.1.

Lemma 3.3**.**

Let $\alpha>0$. Then there exists $C>0$ such that the solution of (2.1) satisfies

[TABLE]

Moreover, for $t\in(0,T_{max,\varepsilon}-\tau)$, it holds that

[TABLE]

*where $\tau$ is the same as (3.24). *

Proof.

We multiply the second equation in (2.1) by $-\Delta c_{\varepsilon}$ and integrate by parts to see that

[TABLE]

where we have used the fact that

[TABLE]

On the other hand, by the Young inequality and (2.14),

[TABLE]

In the last summand in (3.27), we use the Cauchy-Schwarz inequality to obtain

[TABLE]

Now thanks to (2.10) and in view of the Gagliardo-Nirenberg inequality, we can find $C_{1}>0$ and $C_{2}>0$ fulfilling integrate by parts to find that

[TABLE]

This, together with the Young inequality, yields

[TABLE]

Inserting (3.28) and (3.31) into (3.27), we have

[TABLE]

Now, multiplying the third equation of (2.1) by $u_{\varepsilon}$, integrating by parts and using $\nabla\cdot u_{\varepsilon}=0$

[TABLE]

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

[TABLE]

by using (3.25) and (2.10). Inserting (3.34) into (3.33) and integrating in time to see that

[TABLE]

and

[TABLE]

where we use that once more employing the Gagliardo-Nirenberg inequality, the Hölder inequality and the Young inequality we can find $C_{8}>0$ and $C_{9}>0$ satisfying

[TABLE]

Next, combining (3.32), (3.35) and rearranging shows that

[TABLE]

and

[TABLE]

by using (3.30).

Testing the third equation in (2.1) by $-\Delta m_{\varepsilon}$ and integrating by parts and using (3.25) and (3.35), one can finally derive

[TABLE]

and

[TABLE]

∎

With Lemmas 2.3 and 3.2–3.3 at hand, we can proceed to show that our approximate solutions are actually global in time.

Lemma 3.4**.**

For any $\varepsilon>0,$ then one can find $C>0$ such that the solutions of (2.1) fulfill

Proof.

Firstly, under the assumption that $T_{max,\varepsilon}<\infty$, for any $\varepsilon>0$, Lemmas 2.3–3.3 would provide us with $C_{1}>0$ such that

[TABLE]

[TABLE]

as well as

[TABLE]

and

[TABLE]

Then, aided by the $L^{2}$-estimate for $\nabla u_{\varepsilon}$ (from a testing argument), we can obtain that

[TABLE]

with some $C_{2}=C_{2}(\varepsilon)>0$. Thus, the continuous embedding $D(A^{\gamma})\hookrightarrow L^{\infty}(\Omega)$ implies the $L^{\infty}$ -estimate for $u_{\varepsilon}.$ Therefore, employing the same arguments as in the proof of Lemma 3.2 in [63] (see also [16, 61, 49, 50]), and taking advantage of (3.42)–(3.46), we conclude the estimates

[TABLE]

as well as

[TABLE]

and

[TABLE]

and some positive constant $C_{3}.$ In view of (3.46)–(3.49), we apply Lemma 2.1 to reach a contradiction.

∎

3.1 Further a-priori estimates

With the help of Lemma 3.2 and the Gagliardo–Nirenberg inequality, one can derive the following Lemma:

Lemma 3.5**.**

Let $\alpha>0$. Then for each $T\in(0,T_{max,\varepsilon})$, there exists $C>0$ independent of $\varepsilon$ such that the solution of (2.1) satisfies

[TABLE]

Proof.

Multiply the first equation in $(\ref{1.1fghyuisda})$ by $n_{\varepsilon}$ and using $\nabla\cdot u_{\varepsilon}=0$, we derive

[TABLE]

Recalling (1.3) and using $\alpha\geq 0$, from Young inequality again, we derive from Lemma 3.2 that

[TABLE]

which combined with (3.51) implies that

[TABLE]

so that, gather (3.26) and (3.53), one can get (3.50). ∎

4 Passing to the limit: The proof of Theorem 1.1

With the help of a priori estimates, in this subsection, by means of a standard extraction procedure we can now derive the following lemma which actually contains our main existence result (Theorem 1.1) already.

Lemma 4.1**.**

Assume that $\alpha>0$. Then for any $\kappa\in\mathbb{R}$, there exists $(\varepsilon_{j})_{j\in\mathbb{N}}\subset(0,1)$ such that $\varepsilon_{j}\rightarrow 0$ as $j\rightarrow\infty$ and that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

as well as

[TABLE]

*and *

[TABLE]

*some quadruple $(n,c,m,u)$ which is a global weak solution of (1.1) in the natural sense as specified in [63]. *

Proof.

Firstly, applying the discussion in Section 3, under the assumptions of Theorem 1.1, for each $T>0$, we can find $\varepsilon$-independent constant $C(T)$ such that

[TABLE]

as well as

[TABLE]

and

[TABLE]

Now, choosing $\varphi\in W^{1,2}(\Omega)$ as a test function in the first equation in (2.1) and using (4.12), we have

[TABLE]

for all $t>0$. Along with (4.12) and (4.13), further implies that

[TABLE]

where $C_{1}$ and $C_{2}$ are positive constants independent of $\varepsilon$. Now, due to the Hölder inequality, we have

[TABLE]

by using (4.12) and (4.13).

Inserting (4.17) into (4.16) and applying (4.12) and (4.13), we can obtain for some positive constant $C_{1}(T)$ such that

[TABLE]

by using (4.13) and Lemma 3.4.

In a similar way, one can derive

[TABLE]

and

[TABLE]

with some $C_{2}(T)>0.$

Next, for any given $\varphi\in C^{\infty}_{0,\sigma}(\Omega;\mathbb{R}^{3})$, we infer from the fourth equation in (2.1) that

[TABLE]

Now, by virtue of (4.14), Lemma 3.2 and Lemma 2.3, we thus infer that there exist positive constants $C_{3},C_{4}$ and $C_{5}$ such that

[TABLE]

Here we have used the fact that

[TABLE]

Combining estimates (4.12)–(4.21), we conclude from Aubin-Lions lemma (see e.g. [28]) that $(u_{\varepsilon})_{\varepsilon\in(0,1)}$ is relatively compact in $L^{2}_{loc}(\Omega\times[0,\infty);\mathbb{R}^{3})$ and $(n_{\varepsilon})_{\varepsilon\in(0,1)}$, $(c_{\varepsilon})_{\varepsilon\in(0,1)}$, $(m_{\varepsilon})_{\varepsilon\in(0,1)}$ are relatively compact in $L^{2}_{loc}(\Omega\times[0,\infty)).$ Therefore, in conjunction with (4.12)–(4.14) and standard compactness arguments, we can thus find a sequence $(\varepsilon_{j})_{j\in N}\subset(0,1)$ such that $\varepsilon_{j}\searrow 0$ as $j\rightarrow\infty$, and such that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

as well as

[TABLE]

and

[TABLE]

for some limit function $(n,c,m,u).$ On the other hand, according to the bounds provided by Lemma 2.3 and Lemmas 3.2–3.3, this readily yields that, for any $\varepsilon\in(0,1)$,

[TABLE]

where we have used the fact that

[TABLE]

by using Lemma 3.3. Therefore, in light of (4.33), regularity estimates for the second equation of (2.1) (see e.g. [20]) ensure that $(c_{\varepsilon})_{\varepsilon\in(0,1)}$ is bounded in $L^{\frac{5}{3}}((0,T);W^{2,\frac{5}{3}}(\Omega))$. Hence, by virtue of (4.20), we derive form the Aubin–Lions lemma that $(c_{\varepsilon})_{\varepsilon\in(0,1)}$ relatively compacts in $L^{\frac{5}{3}}((0,T);W^{1,\frac{5}{3}}(\Omega))$. Thus, we can choose an appropriate subsequence that is still written as $(\varepsilon_{j})_{j\in\mathbb{N}}$ such that $\nabla c_{\varepsilon_{j}}\rightarrow z_{1}$ in $L^{\frac{5}{3}}(\Omega\times(0,T))$ for all $T\in(0,\infty)$ and some $z_{1}\in L^{\frac{5}{3}}(\Omega\times(0,T))$ as $j\rightarrow\infty$. Therefore, by (4.26), we can also derive that $\nabla c_{\varepsilon_{j}}\rightarrow z_{1}$ a.e. in $\Omega\times(0,\infty)$ as $j\rightarrow\infty$. In view of (4.26) and the Egorov theorem, we conclude that $z_{1}=\nabla c$ and hence

[TABLE]

This combined with (4.22), (4.23) as well as (4.26) and (1.2) implies that

[TABLE]

by using the Egorov theorem. Next we shall prove that $(n,c,m,u)$ is a weak solution of problem (2.1). To this end, testing the first equation in (2.1) by $\varphi\in C^{\infty}_{0}(\Omega\times[0,\infty))$, we obtain

[TABLE]

for all $\varepsilon\in(0,1)$. Then (4.24)–(4.35) and the dominated convergence theorem enables us to conclude

[TABLE]

by a limit procedure. Next, multiplying the second equation and the third equation in (2.1) by $\varphi\in C^{\infty}_{0}(\Omega\times[0,\infty))$, we derive from a limit procedure that

[TABLE]

and

[TABLE]

in a completed similar manner (see [63] for details). Then testing the fourth equation of (2.1) by $\varphi\in C_{0}^{\infty}(\bar{\Omega}\times[0,T);\mathbb{R}^{3})$, we obtain

[TABLE]

by using Lemma 3.4 and a limit procedure (see [63] for details). This means that $(n,c,m,u)$ is a weak solution of (2.1), in the natural sense as specified in [63].

∎

Moreover, if in addition we assume that $\kappa\in\mathbb{R}$, then our solutions will actually be bounded and smooth and hence classical. In fact, by applying the standard parabolic regularity and the classical Schauder estimates for the Stokes evolution, we will show that it is sufficiently regular so as to be a classical solution.

Lemma 4.2**.**

Let $(n,c,m,u)$ be a weak solution of (1.1). Assume that $\alpha>0$ and $\kappa=0$. Then $(n,c,m,u)$ solves (1.1) in the classical sense in $\Omega\times(0,\infty).$ Moreover, this solution is bounded in $\Omega\times(0,\infty)$ in the sense that

[TABLE]

Proof.

In what follows, let $C,C_{i}$ denote some different constants, and if no special explanation, they depend at most on $\Omega,\phi,m,n_{0},c_{0}$ and $u_{0}$.

Step 1. The boundedness of $\|A^{\gamma}u(\cdot,t)\|_{L^{2}(\Omega)}$ and $\|u(\cdot,t)\|_{L^{\infty}(\Omega)}$ for all $t\in(0,T_{max,\varepsilon})$

On the basis of the variation-of-constants formula for the projected version of the third equation in (1.1), we derive that

[TABLE]

On the other hand, in view of Lemma 3.2 as well as (1.8) and (2.10),

[TABLE]

with $h:=\mathcal{P}((n(\cdot,\tau)+m(\cdot,\tau)\nabla\phi)$. Therefore, according to standard smoothing properties of the Stokes semigroup we see that there exist $C_{1},C_{2}>0$ and $\lambda_{1}>0$ such that

[TABLE]

with $\gamma\in(\frac{3}{4},1),$ where in the last inequality, we have used the fact that

[TABLE]

by using $-\gamma-\frac{3}{2}(\frac{1}{{4+2\alpha}}-\frac{1}{2})>-1.$ (4.42) implies to

[TABLE]

by using the fact that $D(A^{\gamma})$ is continuously embedded into $L^{\infty}(\Omega)$ (by $\gamma>\frac{3}{4}$).

Step 2. The boundedness of $\|\nabla c(\cdot,t)\|_{L^{4}(\Omega)}$ for all $t\in(0,T_{max,\varepsilon})$

Now, multiply the second equation in $(\ref{334451.1fghyuisda})$ by $-\Delta c$, in view of (3.25), (2.10) and (4.43), we derive from (3.22) and the Young inequality that

[TABLE]

Considering the fact that $\nabla c\cdot\nabla\Delta c=\frac{1}{2}\Delta|\nabla c|^{2}-|D^{2}c|^{2}$, by a straightforward computation using the second equation in (1.1) and several integrations by parts, we find that

[TABLE]

for all $t\in(0,T_{max,\varepsilon})$. On the other hand, since Lemma 2.2 of [67], we derive from (2.10) and the Young inequality that

[TABLE]

where $\sigma_{0}$ is the same as (4.43) and $\kappa_{0}$ and $\kappa_{1}$ are some positive constants. Thanks to the pointwise inequality $|\Delta c|\leq\sqrt{3}|D^{2}c|$, along with (1.8) as well as (2.10) and (2.10) this implies that

[TABLE]

and

[TABLE]

by using (3.25) and the Young inequality. Now, inserting (4.46) into (4.48), this shows that

[TABLE]

by using (4.43). Again, from the Young inequality, (1.8) as well as (2.10) and (3.25), we have

[TABLE]

and

[TABLE]

Given the the boundedness of $\|\nabla c\|_{L^{2}(\Omega)}^{2}$ (see (4.44)), it is well-known that (cf. [15, 32, 59]) the boundary trace embedding implies that

[TABLE]

Now, together with (4.45), (4.47)–(4.52), we can derive that, for some positive constant $C_{9}$,

[TABLE]

which combined with (4.46) yields to

[TABLE]

This implies

[TABLE]

by integration.

Step 3. The boundedness of $\|\nabla m(\cdot,t)\|_{L^{4}(\Omega)}$ for all $t\in(0,T_{max,\varepsilon})$

An application of the variation of constants formula for $c$ leads to

[TABLE]

To estimate the terms on the right of (4.56), in light of (2.10) and (3.25), applying the $L^{p}$-$L^{q}$ estimates associated heat semigroup, for some positive constant $\lambda_{1}$ such that

[TABLE]

as well as

[TABLE]

and

[TABLE]

where $\iota=\frac{13}{28},\tilde{\kappa}=\frac{1}{56}$. Inserting (4.57)–(4.59) into (4.56), one has

[TABLE]

Step 4. The boundedness of $\|c(\cdot,t)\|_{W^{1,\infty}(\Omega)}$ and $\|m(\cdot,t)\|_{W^{1,\infty}(\Omega)}$ for all $t\in(\tau,T_{max,\varepsilon})$ with $\tau\in(0,T_{max,\varepsilon})$

Choosing $\theta\in(\frac{1}{2}+\frac{3}{8},1),$ then the domain of the fractional power $D((-\Delta+1)^{\theta})\hookrightarrow W^{1,\infty}(\Omega)$ (see e.g. [13, 43]). Thus, in light of $\alpha>0$, using the Hölder inequality and the $L^{p}$-$L^{q}$ estimates associated heat semigroup,

[TABLE]

and

[TABLE]

with $\tau\in(0,T_{max,\varepsilon})$, where we have used (2.10), (4.55), (4.43), (4.60) as well as the Hölder inequality and

[TABLE]

Step 5. The boundedness of $\|c(\cdot,t)\|_{W^{1,\infty}(\Omega)}$ and $\|n(\cdot,t)\|_{L^{\infty}(\Omega)}$ for all $t\in(0,T_{max,\varepsilon})$

Recalling Lemma 2.1, (4.61) and (4.62), we infer that

[TABLE]

and

[TABLE]

Step 6. The boundedness of $\|n(\cdot,t)\|_{L^{\infty}(\Omega)}$ for all $t\in(\tau,T_{max,\varepsilon})$ with $\tau\in(0,T_{max,\varepsilon})$

Fix $T\in(0,T_{max,\varepsilon})$. Let $M(T):=\sup_{t\in(0,T)}\|n(\cdot,t)\|_{L^{\infty}(\Omega)}$ and $\tilde{h}:=nS(x,n,c)\nabla c+u$. Then by (3.25), (1.3) and (4.55), there exists $C_{28}>0$ such that

[TABLE]

where we have used (2.5) and the boundedness of $\|c(\cdot,t)\|_{W^{1,\infty}(\Omega)}$ for all $t\in(\tau,T_{max,\varepsilon})$ with $\tau\in(0,T_{max,\varepsilon})$. Hence, due to the fact that $\nabla\cdot u=0$, again, by means of an associate variation-of-constants formula for $n$, we can derive

[TABLE]

where $t_{0}:=(t-1)_{+}$. As the last summand in (4.67) is non-positive by the maximum principle, we can thus estimate

[TABLE]

If $t\in(0,1]$, by virtue of the maximum principle, we derive that

[TABLE]

while if $t>1$ then with the help of the $L^{p}$-$L^{q}$ estimates for the Neumann heat semigroup and (2.10), we conclude that

[TABLE]

Finally, we fix an arbitrary $\frac{7}{2}\in(3,4)$ and then once more invoke known smoothing properties of the Stokes semigroup and the Hölder inequality to find $C_{4}>0$ such that

[TABLE]

In combination with (4.67)–(4.70) and using the definition of $M(T)$ we obtain $C_{35}>0$ such that

[TABLE]

Hence, with some basic calculation, in light of $T\in(0,T_{max,\varepsilon})$ was arbitrary, one can get

[TABLE]

Finally, by virtue of Lemma 2.1 and (4.42), (4.63)–(4.64), (4.72), the local solution can be extend to the global-in-time solutions.

Employing almost exactly the same arguments as in the proof of Lemma 3.1 in [23] (see also [62]), and taking advantage of (4.41), we conclude the regularity theories for the Stokes semigroup and the Hölder estimate for local solutions of parabolic equations, we can obtain weak solution $(n,c,m,u)$ is a classical solution. ∎

The most important consequence of Lemmas 4.1–4.2 is the following:

Proof of Theorem 1.1: The theorem 1.1 is part of the statement proven by Lemmas 4.1–4.2.

4.1 Eventual smoothness and asymptotics

Given the preliminary lemma collected in the above, in this subsection, we now establish the claimed asymptotic behavior of the solutions to (2.1) under $\alpha>0$. Before going further, we list the following lemma, which will be used to derive the convergence properties of solution with respect to the norm in $L^{2}(\Omega)$.

Lemma 4.3**.**

*(Lemma 4.6 of [10]) Let $\lambda>0,C>0$, and suppose that $y\in C^{1}([0,\infty))$ and $h\in C^{0}([0,\infty))$ are nonnegative functions satisfying $y^{\prime}(t)+\lambda y(t)\leq h(t)$ for some $\lambda>0$ and all $t>0$. Then if $\int_{0}^{\infty}h(s)ds\leq C,$ we have $\lim_{t\rightarrow\infty}y(t)=0$. *

To begin with, let us collect some basic solution properties which essentially have already been used in [10].

Lemma 4.4**.**

The global solution of (2.1) satisfies

[TABLE]

Proof.

These properties are immediate consequences of (2.15) and (2.12). ∎

As an immediate consequence, we obtain the following which will firstly serve as a fundament for our proof of stabilization in the first and third solution components.

Lemma 4.5**.**

Under the assumptions of Lemma 4.1, for any $\eta>0$, there are $T>0$ and $\varepsilon_{0}>0$ such that for any $t>T$ and such that for any $\varepsilon\in(0,\varepsilon_{0})$

[TABLE]

and

[TABLE]

where

[TABLE]

and

[TABLE]

Proof.

Pursuing a strategy demonstrated in lemma 4.2 of [48], we start by noting that as a first consequence of Lemma 4.4 we know that

[TABLE]

Next, in view of (3.25), by using the Hölder inequality and the Poincaré inequality, for some positive constant $K$,

[TABLE]

Inserting (4.78) into (4.79), we obtain

[TABLE]

Now if $\int_{\Omega}n_{0}-\int_{\Omega}m_{0}\geq 0,$ (2.11) warrants that $\int_{\Omega}n_{\varepsilon}-\int_{\Omega}m_{\varepsilon}\geq 0$, which along with (4.80) implies that

[TABLE]

Noticing that $\int_{\Omega}m_{\varepsilon}(s)\geq\int_{\Omega}m_{\varepsilon}(t)$ for all $t\geq s,$ we have

[TABLE]

where we invoke (2.11) to obtain

[TABLE]

By very similar argument, one can see that $\int_{\Omega}n_{\varepsilon}\rightarrow 0$ and $\int_{\Omega}m_{\varepsilon}\rightarrow\int_{\Omega}m_{0}-\int_{\Omega}n_{0}$ as $t\rightarrow\infty$ in the case of $\int_{\Omega}n_{0}-\int_{\Omega}m_{0}<0$. This readily establishes (4.74) and (4.75). ∎

Lemma 4.6**.**

Under the assumptions of Lemma 4.1, for any $\eta>0$, there are $T>0$ and $\varepsilon_{0}>0$ such that for any $t>T$ and such that for any $\varepsilon\in(0,\varepsilon_{0})$

[TABLE]

and

[TABLE]

*where $\hat{m}$ is give by (4.76). *

Proof.

Firstly, since Lemma 3.4 asserts the existence of $\kappa_{1}$ such that

[TABLE]

and since (2.11) implies that

[TABLE]

thus, by (4.74) we infer from the interpolation inequality and the Hölder inequality that

[TABLE]

which immediately implies (4.82). Here we have used the fact that

[TABLE]

by using (4.74). Next, for any $p>1$, in view of Lemma 2.3, we derive from the the interpolation and the Hölder inequality that

[TABLE]

which yields (4.83) directly. ∎

Lemma 4.7**.**

Under the assumptions of Lemma 4.1, for any $\eta>0$, there are $T>0$ and $\varepsilon_{0}>0$ such that for any $t>T$ and such that for any $\varepsilon\in(0,\varepsilon_{0})$

[TABLE]

and

[TABLE]

*where $\hat{m}$ is give by (4.76). *

Proof.

Firstly, by means of the testing procedure, we may derive from the Young inequality that

[TABLE]

where we have used the fact that $\nabla\cdot u_{\varepsilon}=0$ and $u_{\varepsilon}|_{\partial\Omega}=0$. On the other hand, the bounds from 4.6 entails

[TABLE]

This together with (4.90) and Lemma 4.3 imply (4.88) and (4.89). ∎

Lemma 4.8**.**

Under the assumptions of Lemma 4.1, for any $p>1$ and $\eta>0$, there are $T>0$ and $\varepsilon_{0}>0$ such that for any $t>T$ and such that for any $\varepsilon\in(0,\varepsilon_{0})$

[TABLE]

where $\hat{n}$ is given by (4.77).

Proof.

Firstly, for any $p>1$, by Lemma 3.2, there exist positive constants $\alpha_{1}$ and $q>p$ such that

[TABLE]

By the interpolation and the Hölder inequality, we have

[TABLE]

by using (4.75). From (4.94) we readily derive (4.92) and thereby completes the proof.

∎

The stabilization property implied by Lemmas 4.6 and 4.8 can now be turned into a preliminary statement on decay of $u_{\varepsilon}$ by making use of Lemma 4.3 and the standard testing procedures.

Lemma 4.9**.**

Under the assumptions of Lemma 4.1, for any $\eta>0$, there are $T>0$ and $\varepsilon_{0}>0$ such that for any $t>T$ and such that for any $\varepsilon\in(0,\varepsilon_{0})$

[TABLE]

[TABLE]

as well as

[TABLE]

and

[TABLE]

*for any $q\in[1,6)$. *

Proof.

From the fourth equation in (2.1) we obtain the associated Navier-Stokes energy inequality in the form

[TABLE]

where we have used the fact that $\hat{n}=-\hat{m}$ as well as $\nabla\cdot u_{\varepsilon}=0$ and $u_{\varepsilon}|_{\partial\Omega}=0$. Due to the Poincaré inequality again, we have

[TABLE]

therefore, collecting (4.92) and (4.82), we derive from (4.99) that

[TABLE]

and

[TABLE]

and thereby proves (4.95)–(4.96). Finally, we make use of the embedding $W^{1,2}(\Omega)\hookrightarrow L^{q}(\Omega)$ (for any $q\in[1,6)$) and the Young inequality to find that (4.97) and (4.98) hold. ∎

Using thge decay property of $m_{\varepsilon}(\cdot,t)-\hat{m}+n_{\varepsilon}(\cdot,t)-\hat{n}$ (see Lemmas 4.6 and 4.8), by means of a contraction mapping argument, we may derive a certain eventual regularity and decay of $u_{\varepsilon}$ in $L^{p}(\Omega)$ with some $p\geq 6$.

Lemma 4.10**.**

For any $p\in[6,\infty)$ and $\eta>0$, there are $T>0$ and $\varepsilon_{0}>0$ such that for any $t>T$ and such that for any $\varepsilon\in(0,\varepsilon_{0})$

[TABLE]

Proof.

We let $p\geq 6$ and choose $q\in(3,6)$ which is close to $6$ (e.g. $q=\frac{6p}{p+2}$) such that

[TABLE]

Now, we define $\gamma_{0}=\frac{3}{2}(\frac{1}{q}-\frac{1}{p})$. Next, in view of (4.103) and using $p\geq 6$, $q\in(3,6)$, we have

[TABLE]

Therefore,

[TABLE]

where $\kappa_{2}$ is give by Lemma 2.2. Thus for any $\eta>0$, we any choose $\eta_{0}\in(0,\eta)$ small enough such that

[TABLE]

On the other hand, by (4.98), Lemmas 4.6–4.9, we then pick $T_{0}$ and $\varepsilon_{0}>0$ such that for any $t>T$ and such that for any $\varepsilon\in(0,\varepsilon_{0})$

[TABLE]

where $\kappa_{1}$ and $\kappa_{3}$ are given by Lemma 2.2. In view of (4.106), for any $t_{1}>T_{0}$ and $\varepsilon\in(0,\varepsilon_{t_{1}})$, one can find $\tilde{t}_{0}\in(t_{1},t_{1}+1)$ such that

[TABLE]

Now, we define

[TABLE]

In the following, we will prove that (4.102) holds for any $t>T.$ To this end, for the above $\tilde{t}_{0},p,\gamma_{0}$ and $\eta_{0}$, we let

[TABLE]

Then we consider the mapping $\varphi:X\rightarrow\mathbb{R}$ defined by

[TABLE]

Now, we will show that $\varphi$ is a contraction on $X$. In fact, in view of Lemma 2.2, for any $s>1$ and for any such $v$ we may derive from the Hölder inequality that

[TABLE]

Therefore, in light of (4.104), (4.107) as well as (4.108) and (4.106), we see that for every $t\in(\tilde{t}_{0},\tilde{t}_{0}+3)$ and every $v\in X$

[TABLE]

from which it readily follows that $\varphi(X)\subset X$. Likewise, for $v\in X$ and $w\in X$ we can use Lemma 2.2 to find that

[TABLE]

On the other hand, (4.105) implies that

[TABLE]

whence (4.111) shows that $\varphi$ acts as a contraction on $X$ and hence possesses a unique fixed point on $X$, which, in view of the definition of $\varphi$, must coincide with the unique weak solution $u_{\varepsilon}$ of fourth equation of (2.1) on $(\tilde{t}_{0},\tilde{t}_{0}+3)$ (see e.g. Thm. V.2.5.1 of [29]). Now, by (4.110), we also derive that

[TABLE]

from $(\tilde{t}_{0}+1,\tilde{t}_{0}+3)\supset(t_{1}+2,t_{1}+3)$ we readily derive (4.102).

∎

In the following lemmas, we next plan to prove Hölder regularity of the components of a solution on intervals of the form $(T_{0},T_{0}+1)$ for $T_{0}>0$ by using the maximal Sobolev regularity. To this end, we introduce the following cut-off functions, which will play a key role in deriving higher order regularity for solution of problem (2.1).

Definition 4.1**.**

Let $\xi:\mathbb{R}\rightarrow[0,1]$ be a smooth, monotone function, satisfying $\xi\equiv 0$ on $(-\infty,0]$ and $\xi\equiv 0$ on $(1,\infty)$ and for any $t_{0}\in\mathbb{R}$ we let $\xi_{t0}:=\xi(t-t_{0})$.

Due to the above cut-off function, it follows from maximal Sobolev regularity that the solution $(n_{\varepsilon},c_{\varepsilon},m_{\varepsilon},u_{\varepsilon})$ even satisfies estimates in appropriate Hölder spaces:

Lemma 4.11**.**

Let $\alpha>0$. Then one can find $\mu\in(0,1)$ and $T,\varepsilon_{0},C>0$ such that for all $\varepsilon\in(0,\varepsilon_{0})$

[TABLE]

[TABLE]

as well as

[TABLE]

*and *

[TABLE]

Proof.

Firstly, for any $T_{0}>0$ let $\xi:=\xi_{T_{0}}$ and $v:=\xi u_{\varepsilon}.$

[TABLE]

where

[TABLE]

To estimate the inhomogeneity $h$ herein, we first note that the known maximal Sobolev regularity estimate for the Stokes semigroup ([11]) yields a constant $k_{1}>0$ such that

[TABLE]

From the boundedness of the Helmholtz projection in $L^{s}$-spaces and the Hölder inequality we derive from Lemma 4.10 that there exist positive constant $k_{2}$ and $k_{3}$ such that for any $T_{0}>T$

[TABLE]

where $l>2s$ and $\frac{1}{l}+\frac{1}{l^{\prime}}=1.$ Thanks to the Gagliardo-Nirenberg inequality, from Lemma 4.10 again, we can estimate the right of (4.118) by following:

[TABLE]

where $a\in(0,1)$ satisfies

[TABLE]

Therefore, inserting (4.119) into (4.118) and applying the Young inequality, we find $k_{6}>0$ such that for all $t_{0}>T$

[TABLE]

Moreover, we derive from Definition 4.1 and Lemmas 4.6, 4.9 and 4.10, there is $k_{7}>0$ such that

[TABLE]

so that invoking (4.120) and (4.117) we can estimate

[TABLE]

Therefore, by the definition of $\xi$, for any $s>1$, there exist positive constants $C$ and $T$ such that for any $t>T$ there is $\varepsilon_{0}>0$ satisfying that for any $\varepsilon\in(0,\varepsilon_{0})$

[TABLE]

which in view of a known embedding result ([1]) implies that for all $t_{0}>0$, we can find $\theta_{1}\in(0,1)$ and $C_{10}$ such that

[TABLE]

Likewise, again using the maximal Sobolev regularity estimates and the Gagliardo-Nirenberg inequality, we can claim that (4.114)–(4.116) hold by applying Lemmas 2.3, 3.2 and 4.10. ∎

Straightforward applications of standard Schauder estimates for the Stokes evolution equation and the heat equation, respectively, finally yield eventual smoothness of the solution $(n_{\varepsilon},c_{\varepsilon},m_{\varepsilon},u_{\varepsilon})$.

Lemma 4.12**.**

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

[TABLE]

[TABLE]

as well as

[TABLE]

*and *

[TABLE]

Proof.

We first combine Lemma 4.11 to infer the existence of $\alpha_{1}\in(0,1)$, $T_{1}>0$ and $C_{1}>0$ such that for all $t>T_{1}$,

[TABLE]

Standard parabolic Schauder estimates applied to the second and third equation in (2.1) ([20]) thus provide $C_{2}>0$ fulfilling

[TABLE]

According to Lemma 4.11, it is possible to fix $\alpha_{2}\in(0,1)$, $T_{2}>0$ and $C_{3}>0$ such that

[TABLE]

We next set $T:=T_{2}+1$ and let $t_{0}>T$ be given. Then with $\xi_{t_{0}}$ taken from Definition 4.1, we again use that $v(\cdot,t):=\xi_{t_{0}}u_{\varepsilon}(\cdot,t),(x\in\Omega,t>t_{0}-1)$, is a solution of

[TABLE]

where

[TABLE]

Now from (4.131) and the smoothness of $\xi$ we readily obtain $\alpha_{3}\in(0,1)$ and $C_{4}>0$ fulfilling

[TABLE]

so that regularity estimates from Schauder theory for the Stokes evolution equation ([30]) ensure that (4.132) possesses a classical solution $\bar{v}\in{C^{2+\alpha_{3},1+\frac{\alpha_{3}}{2}}(\bar{\Omega}\times[t_{0}-1,t_{0}+1])}$ satisfying

[TABLE]

with some $C_{5}>0$ which is independent of $t_{0}$. This combined with the uniqueness property of (4.132), one can prove

[TABLE]

Again relying on Lemma 4.11, this in turn warrants that for some $\alpha_{4}\in(0,1),T_{4}>0$ and $C_{7}>0$ such that for all $t>T_{4}$

[TABLE]

which along with the Schauder theory says establishes

[TABLE]

Finally, choose $T=\max\{T_{1},T_{1},T_{2},T_{3},T_{4}\}$ and $\mu=\min\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\}$, then (4.130), (4.135), (4.137) imply (4.125)–(4.128). ∎

Having found uniform Hölder bounds on $n_{\varepsilon},c_{\varepsilon},m_{\varepsilon}$ and $u_{\varepsilon}$ for $\varepsilon>0$ in the previous three lemmas (see Lemmas 4.11 and 4.12), also $n,c,m$ and $u$ share this regularity and these bounds.

Lemma 4.13**.**

Assume that $\alpha>0$. There exist $\theta\in(0,1)$ as well as $T_{0}>0$, $(\varepsilon_{j})_{j\in\mathbb{N}}\subset(0,1)$ of the sequence from Lemma 4.1 such that for any $t>T_{0}$

[TABLE]

that $\varepsilon_{j}\searrow 0$ as $j\rightarrow\infty$ and

[TABLE]

as $\varepsilon=\varepsilon_{j}\searrow 0$. Moreover, there is $C>0$ such that

[TABLE]

as well as

[TABLE]

Proof.

In conjunction with Lemmas 4.12 and 4.1 and the standard compactness arguments (see [28]), we can thus find a sequence $(\varepsilon_{j})_{j\in\mathbb{N}}\subset(0,1)$ such that $\varepsilon_{j}\searrow 0$ as $j\rightarrow\infty$, and such that (4.138)–(4.141) hold. The proof of Lemma 4.13 is completed. ∎

Lemma 4.14**.**

Let $\alpha>0$. Then one can find $\theta\in(0,1)$ and $T>0$ such that

[TABLE]

*as well as *

[TABLE]

and

[TABLE]

Proof.

Let $g:=-\xi c+m\xi-\xi u\cdot\nabla c+c\xi^{\prime}$, where $\xi:=\xi_{T_{0}}$ is given by Definition 4.1 and $T_{0}$ is same as the previous lemmas. Then we consider the following problem

[TABLE]

In view of Lemma 4.13 and Definition 4.1, we drive that

[TABLE]

so that, regularity estimates from Schauder theory for the parabolic equation (see e.g. III.5.1 of [20]) ensure that problem (4.145) admits a unique solution $\tilde{c}\in C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[T_{0}+1,\infty)).$ This combined with the property of $\xi$ implies that

[TABLE]

Applying the same argument one can derive the third equation of (1.1) that

[TABLE]

Finally, employing almost exactly the same arguments as in the proof of Lemma 4.12 (the minor necessary changes are left as an easy exercise to the reader), and taking advantage of (4.141), we conclude that

[TABLE]

and

[TABLE]

whence combining the result of (4.146) with (4.147) completes the proof. ∎

On the basis of the eventual uniform continuity properties implied by the estimates in this section (see Lemma 4.14), by using the interpolation inequality, we can now turn the weak stabilization properties of $n,c,m$ and $u$ from Lemmas 4.6–4.9 into convergence with regard to the norm in $L^{\infty}(\Omega)$.

Lemma 4.15**.**

Let $\alpha>0$. The solution $(n,c,m,u)$ of (2.1) constructed in Lemma 4.1 satisfies

[TABLE]

*where $\hat{n}=\frac{1}{|\Omega|}\{\int_{\Omega}n_{0}-\int_{\Omega}m_{0}\}_{+}$ and $\hat{m}=\frac{1}{|\Omega|}\{\int_{\Omega}m_{0}-\int_{\Omega}n_{0}\}_{+}$. *

Proof.

Firsly, due to Lemmas 4.6–4.9, we derive from Lemma 4.14 that

[TABLE]

where $\hat{m}$ and $\hat{n}$ are given by (4.76) and (4.77), respectively. Next, due to Lemma 4.13, one can obtain there exist positive constants $\kappa_{1}$ and $T$ such that for all $t>T$

[TABLE]

Therefore, for any $\eta>0$, we may use the compactness of the first of the embeddings $C^{2+\theta}(\bar{\Omega})\hookrightarrow\hookrightarrow L^{\infty}(\Omega)\hookrightarrow L^{2}(\Omega)$ to fix, through an associated Ehrling lemma, a constant $\kappa_{2}>0$ such that

[TABLE]

[TABLE]

as well as

[TABLE]

and

[TABLE]

Now due to (4.150), we may choose $t_{0}>\max\{1,T\}$ large enough such that for all $t>t_{0}$,

[TABLE]

Combined with (4.152)–(4.156), this shows that in fact

[TABLE]

[TABLE]

as well as

[TABLE]

and

[TABLE]

which together with the fact that $\eta>0$ was arbitrary implies the claimed estimates.

∎

In order to prove Theorem 1.2, we now only have to collect the results prepared during this section:

Proof of Theorem 1.2.

Proof.

Combining Lemmas 4.13–4.15 this convergence statement results immediately. ∎

Acknowledgement: This work is partially supported by the National Natural Science Foundation of China (No. 11601215), Shandong Provincial Science Foundation for Outstanding Youth (No. ZR2018JL005) and Project funded by China Postdoctoral Science Foundation (No. 2019M650927, 2019T120168).