Global existence and asymptotic behavior of classical solutions for a 3D two-species Keller--Segel-Stokes system with competitive kinetics

TL;DR

This paper proves the global existence and long-term stability of classical solutions for a complex 3D two-species chemotaxis-fluid system with competitive interactions, addressing challenges posed by chemotaxis, fluid dynamics, and competitive kinetics.

Contribution

It establishes the first results on global solutions for this coupled chemotaxis-fluid system with competitive kinetics in three dimensions.

Findings

Global existence of classical solutions in 3D

Solutions stabilize over time

Large chemotactic sensitivity parameters ensure stability

Abstract

This paper deals with the two-species Keller--Segel-Stokes system with competitive kinetics $(n_1)_t + u\cdot\nabla n_1 =\Delta n_1 - \chi_1\nabla\cdot(n_1\nabla c)+ \mu_1n_1(1- n_1 - a_1n_2)$, $(n_2)_t + u\cdot\nabla n_2 =\Delta n_2 - \chi_2\nabla\cdot(n_2\nabla c) + \mu_2n_2(1- a_2n_1 - n_2), c_t + u\cdot\nabla c =\Delta c - c + \alpha n_1 +\beta n_2$, $u_t= \Delta u + \nabla P+ (\gamma n_1 + \delta n_2)\nabla\phi$, $ \nabla\cdot u = 0$ under homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset \mathbb{R}^3$ with smooth boundary. Many mathematicians study chemotaxis-fluid systems and two-species chemotaxis systems with competitive kinetics. However, there are not many results on coupled two-species chemotaxis-fluid systems which have difficulties of the chemotaxis effect, the competitive kinetics and the fluid influence. Recently, in the two-species chemotaxis-Stokes system, where $-c+\alpha n_1+\beta n_2$ is replaced with $-(\alpha n_1+\beta n_2)c$ in the above system, global existence and asymptotic behavior of classical solutions were obtained in the 3-dimensional case under the condition that $\mu_1,\mu_2$ are sufficiently large. Nevertheless, the above system has not been studied yet; we cannot apply the same argument as in the previous works because of lacking the $L^\infty$-information of $c$. The main purpose of this paper is to obtain global existence and stabilization of classical solutions to the above system in the 3-dimensional case under the largeness conditions for $\mu_1,\mu_2$.

Equations

$$\{\begin{array}{ll}(n_1{)}_t+u\cdot \nabla n_1=\Delta n_1-{\chi }_1\nabla \cdot (n_1\nabla c)+{\mu }_1{n}_1(1-n_1-a_1{n}_2),& x\in \Omega ,t>0,\\ (n_2{)}_t+u\cdot \nabla n_2=\Delta n_2-{\chi }_2\nabla \cdot (n_2\nabla c)+{\mu }_2{n}_2(1-a_2{n}_1-n_2),& x\in \Omega ,t>0,\\ c_t+u\cdot \nabla c=\Delta c-c+\alpha n_1+\beta n_2,& x\in \Omega ,t>0,\\ u_t=\Delta u+\nabla P+(\gamma n_1+\delta n_2)\nabla \varphi ,\nabla \cdot u=0,& x\in \Omega ,t>0\end{array}$$

$$n_t=\Delta n-\nabla \cdot (n\nabla c),c_t=\Delta c-c+n.$$

$$n_t=\Delta n-\chi \nabla \cdot (n\nabla c)+rn-\mu n^2,c_t=\Delta c-c+n,$$

$$(n_1{)}_t=\Delta n_1-{\chi }_1\nabla \cdot (n_1\nabla c)+{\mu }_1{n}_1(1-n_1-a_1{n}_2),$$

$$(n_2{)}_t=\Delta n_2-{\chi }_2\nabla \cdot (n_2\nabla c)+{\mu }_2{n}_2(1-a_2{n}_1-n_2),$$

$$c_t=\Delta c-c+\alpha n_1+\beta n_2,$$

$$n_t+u\cdot \nabla n=\Delta n-\chi \nabla \cdot (n\nabla c)+rn-\mu n^2,$$

$$c_t+u\cdot \nabla c=\Delta c+g(n,c),$$

$$u_t+\kappa (u\cdot \nabla )u=\Delta u+\nabla P+n\nabla \varphi ,\nabla \cdot u=0,$$

$$\{\begin{array}{ll}(n_1{)}_t+u\cdot \nabla n_1=\Delta n_1-{\chi }_1\nabla \cdot (n_1\nabla c)+{\mu }_1{n}_1(1-n_1-a_1{n}_2),& x\in \Omega ,t>0,\\ (n_2{)}_t+u\cdot \nabla n_2=\Delta n_2-{\chi }_2\nabla \cdot (n_2\nabla c)+{\mu }_2{n}_2(1-a_2{n}_1-n_2),& x\in \Omega ,t>0,\\ c_t+u\cdot \nabla c=\Delta c-c+\alpha n_1+\beta n_2,& x\in \Omega ,t>0,\\ u_t=\Delta u+\nabla P+(\gamma n_1+\delta n_2)\nabla \varphi ,\nabla \cdot u=0,& x\in \Omega ,t>0,\\ {\partial }_{\nu }{n}_1={\partial }_{\nu }{n}_2={\partial }_{\nu }c=0,u=0,& x\in \partial \Omega ,t>0,\\ n_i(x,0)=n_{i,0}(x),c(x,0)=c_0(x),u(x,0)=u_0(x),& x\in \Omega ,i=1,2,\end{array}$$

$$\frac{5-\frac{3}{p+1}}{1-\frac{3}{(p+1)\theta }}\cdot (p+1)\cdot \frac{\frac{1}{r}-\frac{3}{(p+1)\theta }}{\frac{1}{2}-\frac{3}{r}}<2$$

$$0<n_{1,0},n_{2,0}\in C^0(\overline{\Omega }),0<c_0\in W^{1,q}(\Omega ),u_0\in D(A^{\vartheta }),$$

$$\varphi \in C^{1+\eta }(\overline{\Omega })$$

$$\chi :=\max \{{\chi }_1,{\chi }_2\}\text{mbox}and\mu :=\min \{{\mu }_1,{\mu }_2\}$$

$$n_1,n_2\in C^0(\overline{\Omega }\times [0,\infty ))\cap C^{2,1}(\overline{\Omega }\times (0,\infty )),$$

$$c\in C^0(\overline{\Omega }\times [0,\infty ))\cap C^{2,1}(\overline{\Omega }\times (0,\infty ))\cap L_{\mathrm{loc}}^{\infty }([0,\infty );W^{1,q}(\Omega )),$$

$$u\in C^0(\overline{\Omega }\times [0,\infty ))\cap C^{2,1}(\overline{\Omega }\times (0,\infty ))\cap L_{\mathrm{loc}}^{\infty }([0,\infty );D(A^{\vartheta })),$$

$$P\in C^{1,0}(\overline{\Omega }\times (0,\infty )).$$

$$\underset{t>0}{\sup }(\Vert n_1(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert n_2(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert c(\cdot ,t){\Vert }_{W^{1,q}(\Omega )}+\Vert u(\cdot ,t){\Vert }_{D(A^{\vartheta })})<\infty .$$

$$4{\delta }_1-(1+{\delta }_1{)}^2{a}_1{a}_2>0\text{mbox}and$$

$$\frac{{\chi }_1^2(1-a_1)}{4a_1{\mu }_1(1-a_1{a}_2)}+\frac{{\delta }_1{\chi }_2^2(1-a_2)}{4a_2{\mu }_2(1-a_1{a}_2)}<\frac{4{\delta }_1-(1+\delta {)}^2{a}_1{a}_2}{a_1{\alpha }^2\delta +a_2{\beta }^2-a_1{a}_2\alpha \beta (1+{\delta }_1)},$$

$$n_1(\cdot ,t)\to N_1,n_2(\cdot ,t)\to N_2,c(\cdot ,t)\to C^{\ast },u(\cdot ,t)\to 0\text{mbox}in{L}^{\infty }(\Omega )\text{mbox}ast\to \infty ,$$

$$N_1:=\frac{1-a_1}{1-a_1{a}_2},N_2:=\frac{1-a_2}{1-a_1{a}_2},C^{\ast }:=\alpha N_1+\beta N_2.$$

$$4{\delta }_1^{\prime }-a_1^{\prime }{a}_2(1+{\delta }_1^{\prime }{)}^2>0\text{mbox}and$$

$${\mu }_2>\frac{{\chi }_2^2{\delta }_1^{\prime }({\alpha }^2{a}_1^{\prime }{\delta }_1^{\prime }+{\beta }^2{a}_2-\alpha \beta a_1^{\prime }{a}_2(1+{\delta }_1^{\prime }))}{4a_2(4{\delta }_1^{\prime }-a_1^{\prime }{a}_2(1+{\delta }_1^{\prime }{)}^2)},$$

$$n_1(\cdot ,t)\to 0,n_2(\cdot ,t)\to 1,c(\cdot ,t)\to \beta ,u(\cdot ,t)\to 0\text{mbox}in{L}^{\infty }(\Omega )\text{mbox}ast\to \infty .$$

$$\displaystyle\int_{\Omega}n_{1}^{p}+\int_{\Omega}n_{2}^{p}\leq C+C\int_{s_{0}}^{t}e^{-(p+1)(t-s)}\int_{\Omega}|\Delta c|^{p+1}-C\int_{s_{0}}^{t}e^{-(p+1)(t-s)}\Bigl{(}\int_{\Omega}n_{1}^{p+1}+\int_{\Omega}n_{2}^{p+1}\Bigr{)}$$

$${\int }_0^{\infty }{\int }_{\Omega }(n_1-N_1{)}^2+{\int }_0^{\infty }{\int }_{\Omega }(n_2-N_2{)}^2+{\int }_0^{\infty }{\int }_{\Omega }(c-C^{\ast }{)}^2\le C$$

$$\frac{d}{dt}E(t)\le -\epsilon {\int }_{\Omega }\left[(n_1-N_1{)}^2+(n_2-N_2{)}^2+(c-C^{\ast }{)}^2\right]$$

$$n_1,n_2\in C^0(\overline{\Omega }\times [0,T_{\max }))\cap C^{2,1}(\overline{\Omega }\times (0,T_{\max })),$$

$$c\in C^0(\overline{\Omega }\times [0,T_{\max }))\cap C^{2,1}(\overline{\Omega }\times (0,T_{\max }))\cap L_{\mathrm{loc}}^{\infty }([0,T_{\max });W^{1,q}(\Omega )),$$

$$u\in C^0(\overline{\Omega }\times [0,T_{\max }))\cap C^{2,1}(\overline{\Omega }\times (0,T_{\max }))\cap L_{\mathrm{loc}}^{\infty }([0,T_{\max });D(A^{\vartheta })),$$

$$P\in C^{1,0}(\overline{\Omega }\times (0,T_{\max })),$$

$$n_1,n_2>0,c>0\text{mbox}in\Omega \times (0,T_{\max }).$$

$$\Vert n_1(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert n_2(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert c(\cdot ,t){\Vert }_{W^{1,q}(\Omega )}+\Vert A^{\vartheta }u(\cdot ,t){\Vert }_{L^2(\Omega )}\to \infty \text{mbox}ast\nearrow T_{\max }.$$

$$c(\cdot ,s_0)\in C^2(\overline{\Omega })\text{mbox}with{\partial }_{\nu }c(\cdot ,s_0)=0\text{mbox}on\partial \Omega .$$

$$\Vert c(\cdot ,s_0){\Vert }_{W^{2,\infty }(\Omega )}\le M$$

Full text

0002010Mathematics Subject Classification. Primary: 35K45; Secondary: 92C17; 35Q35. 000Key words and phrases: Keller–Segel-Stokes; global existence; asymptotic stability.

**Global existence and asymptotic behavior of classical solutions for a 3D two-species Keller–Segel-Stokes system with competitive kinetics **

Xinru Cao

Institut für Mathematik, Universität Paderborn

Warburger Str.100, 33098 Paderborn, Germany

Shunsuke Kurima

Department of Mathematics, Tokyo University of Science

1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

Masaaki Mizukami***Corresponding author

Department of Mathematics, Tokyo University of Science

1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

masaaki.mizukami.math@gmail.com

• Abstract. This paper deals with the two-species Keller–Segel-Stokes system with competitive kinetics

[TABLE]

under homogeneous Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^{3}$ with smooth boundary. Many mathematicians study chemotaxis-fluid systems and two-species chemotaxis systems with competitive kinetics. However, there are not many results on coupled two-species chemotaxis-fluid systems which have difficulties of the chemotaxis effect, the competitive kinetics and the fluid influence. Recently, in the two-species chemotaxis-Stokes system, where $-c+\alpha n_{1}+\beta n_{2}$ is replaced with $-(\alpha n_{1}+\beta n_{2})c$ in the above system, global existence and asymptotic behavior of classical solutions were obtained in the 3-dimensional case under the condition that $\mu_{1},\mu_{2}$ are sufficiently large ([5]). Nevertheless, the above system has not been studied yet; we cannot apply the same argument as in the previous works because of lacking the $L^{\infty}$-information of $c$. The main purpose of this paper is to obtain global existence and stabilization of classical solutions to the above system in the 3-dimensional case under the largeness conditions for $\mu_{1},\mu_{2}$.

1 Introduction and results

It is found in many biological experiments that cells have the ability to adapt their migration in response to a chemical signal in their neighbourhood; especially, cells move towards higher concentration of a chemical substance which is produced by themselves. This mechanism plays a central role in mathematical biology, and has many applications in its variants and extensions (see [2]).

Let $n$ denote the density of the cells and $c$ present the concentration of the chemical signal. The mathematical model describing the above mechanism which was first proposed by Keller–Segel [11] reads as

[TABLE]

In the above system which is called Keller–Segel system, it is known that the size of initial data determine whether classical solutions of the above system exist globally or not ([4, 10, 19, 26]). In the 2-dimensional setting global classical solutions of the above system exist when the mass of an initial data $n_{0}$ is sufficiently small ([19]). On the other hand, there exist initial data such that the mass of the initial data is large enough and a solution blows up in finite time in a 2-dimensional bounded domain ([10]). In the higher dimensional case it is known that, if an initial data $(n_{0},c_{0})$ is sufficiently small with respect to suitable Lebesgue norm, then global existence and boundedness of classical solutions hold ([4, 26]). Then we expect that existence of blow-up solutions hold under some largeness condition for initial data also in the higher-dimensional case; however, Winkler [26] showed that for all $m>0$ there exist initial data $n_{0}$ such that $\|n_{0}\|_{L^{1}(\Omega)}=m$ and a solution blows up in finite time in the case that the domain is a ball in $\mathbb{R}^{N}$ with $N\geq 3$.

As we mentioned above, we can see that whether solutions of the Keller–Segel system blow up or not depends on initial data. On the other hand, it is known that the logistic term can prevent solutions from blowing up even though a initial data is large enough ([13, 20, 27]). Osaki–Tsujikawa–Yagi–Mimura [20] obtained that, in the 2-dimensional case, the chemotaxis system with logistic source

[TABLE]

where $\chi,r,\mu>0$, possesses a unique global solution for all $\chi,r,\mu>0$ and all initial data. In the higher dimensional case, Winkler [27, 29] established existence of global classical solutions under the condition that $\mu>0$ is sufficiently large. Moreover, asymptotic behavior of the solutions was obtained: $n(\cdot,t)\to\frac{r}{\mu}$, $c(\cdot,t)\to\frac{r}{\mu}$ in $L^{\infty}(\Omega)$ as $t\to\infty$ ([7]). Recently, Lankeit [13] obtained global existence of weak solutions to the chemotaxis system with logistic source for all $r\in\mathbb{R}$ and any $\mu>0$ and the eventual smoothness of the solutions was derived if $r\in\mathbb{R}$ is small. More related works which deal with the Keller–Segel system and the chemotaxis system with logistic source can be found in [2, Section 3].

The chemotaxis system with logistic source is a single-species case, and has a classical solution which converges to the constant steady state $(\frac{r}{\mu},\frac{r}{\mu})$. On the other hand, the two-species chemotaxis-competition system

[TABLE]

where $\chi_{1},\chi_{2},\mu_{1},\mu_{2},a_{1},a_{2},\alpha,\beta>0$, which describes the evolution of two competing species which react on a single chemoattractant, has different dynamics depending on $a_{1}$ and $a_{2}$ ([1, 3, 15, 16, 17, 22, 25]). In the 2-dimensional case Bai–Winkler [1] obtained global existence of the above system for all parameters. In the higher-dimensional case global existence and boundedness in the above system were established in [15, 16]. Moreover, it was shown that the solutions of the above system have the same asymptotic behavior as the solutions of Lotka–Volterra competition model: $n_{1}(\cdot,t)\to\frac{1-a_{1}}{1-a_{1}a_{2}}$, $n_{2}(\cdot,t)\to\frac{1-a_{2}}{1-a_{1}a_{2}}$, $c(\cdot,t)\to\frac{\alpha(1-a_{1})+\beta(1-a_{2})}{1-a_{1}a_{2}}$ in $L^{\infty}(\Omega)$ as $t\to\infty$ in the case that $a_{1},a_{2}\in(0,1)$, and $n_{1}(\cdot,t)\to 0$, $n_{2}(\cdot,t)\to 1$, $c(\cdot,t)\to\beta$ in $L^{\infty}(\Omega)$ as $t\to\infty$ in the case that $a_{1}\geq 1>a_{2}>0$ ([1, 16, 17]). More related works can be found in [3, 15, 18, 22, 25].

Recently, the chemotaxis-fluid system

[TABLE]

where $\chi>0$, $r,\mu\geq 0$, $\kappa=0$ (the Stokes case) or $\kappa=1$ (the Navier–Stokes case), which is the chemotaxis system with the fluid influence according to the Navier–Stokes equation, was intensively studied ([14, 23, 24, 28]). In the case that $g(n,c)=-nc$ and $r=\mu=0$ (chemotaxis-Navier–Stokes system) Winkler [28] first overcame the difficulties of the chemotactic effect and the fluid influence, and showed existence of global classical solutions in the 2-dimensional case and global existence of weak solutions to the above system with $\kappa=0$ in the 3-dimensional setting; even though in the 3-dimensional chemotaxis-Stokes case, classical solutions were not found. In the case that $g(n,c)=-c+n$ and $\kappa=0$ (Keller–Segel-Stokes system) Li–Xiao [14] showed global existence and boundedness under the smallness condition for the mass of initial data only in the 2-dimensional case. On the other hand, also in the chemotaxis-fluid system, the logistic source can be helpful for obtaining classical bounded solutions; global classical bounded solutions were established in the 2-dimensional Keller–Segel-Navier–Stokes system ([24]) and in the 3-dimensional Keller–Segel-Stokes system under the condition that $\mu$ is large enough ([23]). For more related works we refer to [2, Section 4].

As we discussed previously, the chemotaxis-competition system and the chemotaxis-fluid system were intensively studied. However, there are not many results on a coupled two-species chemotaxis-fluid system which have difficulties of the chemotaxis effect, the competitive kinetics and the fluid influence. Recently, the problem which is a combination of the chemotaxis-Navier–Stokes system and the chemotaxis-competition system was studied in the 2-dimensional case and the 3-dimensional case ([5, 9]); in the 2-dimensional case global existence and asymptotic behavior of classical bounded solutions to the two-species chemotaxis-Navier–Stokes system were established ([9]), and in the 3-dimensional case existence and stabilization of global classical bounded solutions to the two-species chemotaxis-Stokes system hold under the condition that $\mu_{1},\mu_{2}$ are sufficiently large ([5]). However, the two-species Keller–Segel-Stokes system with competitive kinetics

[TABLE]

where $\Omega$ is a bounded domain, $\chi_{1},\chi_{2},a_{1},a_{2}\geq 0$ and $\mu_{1},\mu_{2},\alpha,\beta,\gamma,\delta>0$ are constants, $n_{1,0},n_{2,0},c_{0},u_{0},\phi$ are known functions, has not been studied yet; we cannot apply the same argument as in [5] because of lacking the $L^{\infty}$-information and only having $L^{1}$-estimate for $c$. in this case. Indeed, we cannot pick $r\in(1,3)$ such that

[TABLE]

holds at [5, (3.9)] (in the case that we use $\|\nabla c\|_{L^{(p+1)\theta^{\prime}}(\Omega)}\leq C\|\Delta c\|_{L^{p+1}(\Omega)}^{a}\|c\|_{L^{1}(\Omega)}^{1-a}$ instead of [5, (3.8)]). The purpose of the present paper is to obtain global existence and asymptotic behavior in (1.1) in a 3-dimensional domain. In order to attain this purpose, we assume throughout this paper that the known functions $n_{1,0},n_{2,0},c_{0},u_{0},\phi$ satisfy

[TABLE]

for some $q>3$, $\vartheta\in\left(\frac{3}{4},1\right)$, $\eta>0$ and $A$ is the Stokes operator (see [21]). Then the main results read as follows. The first theorem gives global existence and boundedness in (1.1).

Theorem 1.1**.**

Let $\Omega\subset\mathbb{R}^{3}$ be a bounded domain with smooth boundary and let $\chi_{1},\chi_{2}>0$, $a_{1},a_{2}\geq 0$, $\mu_{1},\mu_{2},\alpha,\beta,\gamma,\delta>0$. Assume that (1.2) and (1.3) are satisfied. Then there exists a constant $\xi_{0}>0$ such that whenever

[TABLE]

satisfy $\frac{\chi}{\mu}<\xi_{0}$, there is a classical solution $(n_{1},n_{2},c,u,P)$ of the problem (1.1) such that

[TABLE]

Also, the solution is unique in the sense that it allows up to addition of spatially constants to the pressure $P$. Moreover, the above solution is bounded in the following sense:**

[TABLE]

The second theorem asserts asymptotic behavior of solutions to (1.1).

Theorem 1.2**.**

Assume that the assumption of Theorem 1.1 is satisfied. Then the following properties hold:**

1. (i)

In the case that $a_{1},a_{2}\in(0,1)$, under the condition that there exists $\delta_{1}>0$ such that

[TABLE]

the solution of the problem (1.1) converges to a constant stationary solution of (1.1) as follows:**

[TABLE]

where

[TABLE] 2. (ii)

In the case that $a_{1}\geq 1>a_{2}$, under the condition that there exist $\delta_{1}^{\prime}>0$ and $a_{1}^{\prime}\in[1,a_{1}]$ such that

[TABLE]

the solution of the problem (1.1) converges to a constant stationary solution of (1.1) as follows:**

[TABLE]

The strategy for the proof of Theorem 1.1 is to confirm the $L^{p}$-estimates for $n_{1}$ and $n_{2}$ with $p>\frac{3}{2}$. By using the differential inequality we can obtain

[TABLE]

with some $C>0$ and $s_{0}>0$. We will control $\int_{s_{0}}^{t}e^{-(p+1)(t-s)}\int_{\Omega}|\Delta c|^{p+1}$ by applying the variation of the maximal Sobolev regularity (Lemma 2.2) for the third equation in (1.1). Then we can obtain the $L^{p}$-estimates for $n_{1}$ and $n_{2}$. Here the keys of this strategy are the $L^{2}$-estimate for $\nabla c$ (Lemma 3.2) and the maximal Sobolev regularity for the Stokes equation (Lemma 2.3); these enable us to overcome difficulties of applying an argument similar to that in [5]. On the other hand, the strategy for the proof of Theorem 1.2 is to confirm the following inequality:

[TABLE]

with some $C>0$, where $(N_{1},N_{2},C^{*},0)$ is a constant stationary solution of (1.1). In order to obtain this estimate we will find a nonnegative function $E$ satisfying

[TABLE]

with some $\varepsilon>0$. The above inequality and the positivity of $E(t)$ enable us to attain the desired estimate (1.4).

The plan of this paper is as follows. In Section 2 we collect basic facts which will be used later. Section 3 is devoted to proving global existence and boundedness (Theorem 1.1). In Section 4 we show asymptotic stability (Theorem 1.2).

2 Preliminaries

In this section we will give some results which will be used later. We can prove the following lemma which gives local existence of classical solutions to (1.1) by a straightforward adaptation of the reasoning in [28, Lemma 2.1].

Lemma 2.1**.**

Let $\Omega\subset\mathbb{R}^{3}$ be a bounded domain with smooth boundary. Assume that (1.2) and (1.3) are satisfied. Then there exists $T_{\rm max}\in(0,\infty]$ such that the problem (1.1) possesses a classical solution $(n_{1},n_{2},c,u,P)$ satisfying

[TABLE]

Also, the above solution is unique up to addition of spatially constants to the pressure $P$. Moreover, either $T_{\rm max}=\infty$ or

[TABLE]

Given $s_{0}\in(0,T_{\rm max})$, we can derive from the regularity properties that

[TABLE]

In particular, there exists $M=M(s_{0})>0$ such that

[TABLE]

(see e.g., [31]).

The following two lemmas provided as variations of the maximal Sobolev regularity hold keys for global existence and boundedness of solutions to (1.1).

Lemma 2.2**.**

Let $s_{0}\in(0,T_{\rm max})$. Then for all $p>1$ there exists a constant $C=C(p)>0$ such that

[TABLE]

holds for all $t\in(s_{0},T_{\rm max})$.

Proof.

The proof is similar to that of [5, Lemma 2.2]. Let $s_{0}\in(0,T_{\rm max})$ and $t\in(s_{0},T_{\rm max})$. By using the transformation $\widetilde{c}(\cdot,s)=e^{s}c(\cdot,s)$, $s\in(s_{0},t)$, and the maximal Sobolev regularity [8, Theorem 3.1] for $\widetilde{c}$ we obtain this lemma. ∎

Lemma 2.3**.**

Let $s_{0}\in(0,T_{\rm max})$. Then for all $p>1$ there exists a constant $C=C(p,s_{0})>0$ such that

[TABLE]

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

Proof.

Letting $s_{0}\in(0,T_{\rm max})$ and $t\in(s_{0},T_{\rm max})$ and putting $\widetilde{u}(\cdot,s):=e^{s}u(\cdot,s)$, $s\in(s_{0},t)$, we obtain from the forth equation in (1.1) that

[TABLE]

which derives

[TABLE]

where ${\cal P}$ denotes the Helmholtz projection mapping $L^{2}(\Omega)$ onto its subspace $L^{2}_{\sigma}(\Omega)$ of all solenoidal vector field. Thus we derive from [6, Theorem 2.7] that there exist positive constants $C_{1}=C_{1}(p,s_{0})$ and $C_{2}=C_{2}(p,s_{0})$ such that

[TABLE]

for all $t\in(s_{0},T_{\rm max})$. Hence we can prove this lemma. ∎

3 Boundedness. Proof of Theorem 1.1

We will prove Theorem 1.1 by preparing a series of lemmas in this section.

Lemma 3.1**.**

There exists a constant $C>0$ such that for $i=1,2$,

[TABLE]

and

[TABLE]

where $\tau:=\min\{1,\frac{1}{2}T_{\rm max}\}$.

Proof.

The above lemma can be proved by the same argument as in the proof of [9, Lemma 3.1]. ∎

Lemma 3.2**.**

There exists a positive constant $C$ such that

[TABLE]

Moreover, for all $p\in[1,6)$ there exists a positive constant $C(p)$ such that

[TABLE]

Proof.

Integrating the third equation in (1.1) over $\Omega$ together with the $L^{1}$-estimates for $n_{1}$ and $n_{2}$ provided by Lemma 3.1 implies that there exists $C_{1}>0$ such that

[TABLE]

for all $t\in(0,T_{\rm max})$. We next see from an argument similar to that in the proofs of [23, Lemmas 2.5 and 2.6] that there is $C_{2}>0$ such that

[TABLE]

Thanks to (3.1) and (3.2), we have from the Gagliardo–Nirenberg inequality that

[TABLE]

for all $t\in(0,T_{\rm max})$ with some positive constant $C_{3}$, where $a=\frac{6}{5}(1-\frac{1}{p})\in(0,1)$. Thus we can prove this lemma. ∎

Lemma 3.3**.**

For $r\in(1,3)$ there exists a constant $C=C(r)>0$ such that

[TABLE]

Proof.

The $L^{r}$-boundedness of $u$ for $r\in(1,3)$ can be obtained from the well-known Neumann heat semigroup estimates together with Lemma 3.1 (for more details, see [30, Corollary 3.4]). ∎

Now we fix $s_{0}\in(0,T_{\rm max})\cap(0,1]$. We will obtain the $L^{p}$-estimates for $n_{1}$ and $n_{2}$ by preparing a series of lemmas.

Lemma 3.4**.**

For all $p>1$, $\varepsilon>0$ and $\ell>0$ there exists a constant $C=C(p)>0$ such that

[TABLE]

on $(s_{0},T_{\rm max})$, where $\mu=\min\{\mu_{1},\mu_{2}\}$, $\chi=\max\{\chi_{1},\chi_{2}\}$.

Proof.

We can prove this lemma by using the same argument as in the proof of [5, Lemma 3.4]. ∎

Lemma 3.5**.**

For all $p\in(\frac{3}{2},2)$ there exists a constant $C=C(p)>0$ such that

[TABLE]

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

Proof.

Fix $\theta\in(\frac{3}{2},2)$ and put $\theta^{\prime}=\frac{\theta}{\theta-1}\in(2,3)$. From Lemma 2.2 we have

[TABLE]

with some positive constant $C_{1}=C_{1}(p)$. It follows from Lemma 3.2 and the Hölder inequality that there exists a positive constant $C_{2}=C_{2}(p)$ such that

[TABLE]

Here the Gagliardo–Nirenberg inequality and Lemma 3.2 imply

[TABLE]

with some constants $C_{3}=C_{3}(p)>0$ and $C_{4}=C_{4}(p)>0$, where

[TABLE]

We derive from (3.3), (3.4), (3.5) and the Young inequality that there exists a positive constant $C_{5}=C_{5}(p)$ such that

[TABLE]

Hence we can obtain that

[TABLE]

Here we can choose $r\in(1,3)$ such that

[TABLE]

holds. Now it follows from the Gagliardo–Nirenberg inequality, Lemma 3.3 and the Young inequality that there exists a constant $C_{6}=C_{6}(p)>0$ such that

[TABLE]

with

[TABLE]

because

[TABLE]

holds from (3.7). We moreover have from the fact $\frac{1}{1-a}>1$, the Young inequality and (3.8) that

[TABLE]

for all $s\in(s_{0},T_{\rm max})$ with some constant $C_{7}=C_{7}(p)>0$. Thus, combining (3.6), (3.8) and (3.9), we see that there exists a constant $C_{8}=C_{8}(p)>0$ such that

[TABLE]

for all $t\in(s_{0},T_{\rm max})$, which concludes the proof of this lemma. ∎

We give the following lemma to control $\int_{s_{0}}^{t}e^{(p+1)s}\int_{\Omega}|\Delta c|^{p+1}$.

Lemma 3.6**.**

For all $p\in(\frac{3}{2},2)$ there exists a constant $C=C(p)>0$ such that

[TABLE]

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

Proof.

A combination of Lemmas 2.3 and 3.3 implies this lemma. ∎

The following lemma is concerned with the $L^{p}$-estimates for $n_{1}$ and $n_{2}$.

Lemma 3.7**.**

For all $p\in(\frac{3}{2},2)$ there exists $\xi_{0}>0$ such that if $\frac{\chi}{\mu}<\xi_{0}$, then there exists a constant $C>0$ such that

[TABLE]

Proof.

From Lemmas 3.4, 3.5 and 3.6 we have that there exists a constant $K(p)>0$ such that

[TABLE]

for all $t\in(s_{0},T_{\rm max})$, where $\varepsilon>0$ and $\ell>0$. Here there exists a constant $\xi_{0}=\xi_{0}(p)>0$ such that

[TABLE]

If $\frac{\chi}{\mu}<\xi_{0}$, then we see from (3.11) that

[TABLE]

and hence there exists a constant $\ell>0$ such that

[TABLE]

Therefore, under the condition that $\frac{\chi}{\mu}<\xi_{0}$, we can find $\varepsilon>0$ satisfying

[TABLE]

which enables us to obtain from $\eqref{good}$ that

[TABLE]

holds on $(s_{0},T_{\rm max})$. ∎

Lemma 3.8**.**

Assume $\frac{\chi}{\mu}<\xi_{0}$. Then there exists a constant $C>0$ such that

[TABLE]

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

Proof.

Thanks to Lemma 3.7, we can show this lemma by the same argument as in the proof of [5, Lemma 3.9]. ∎

Lemma 3.9**.**

Assume $\frac{\chi}{\mu}<\xi_{0}$. Then there exist $r\in(3,6)\cap(1,q]$ and $C=C(r)>0$ such that

[TABLE]

Proof.

This proof is based on that of [5, Lemma 3.10]. We first note from Lemma 3.7 that for all $p\in(\frac{3}{2},2)$,

[TABLE]

with some constant $C_{1}=C_{1}(p)>0$ and by choosing $\theta\in(r,6)$ we see from Lemmas 3.2 and 3.8 that

[TABLE]

with some constant $C_{2}>0$. Therefore an argument similar to that in the proof of [5, Lemma 3.10] implies this lemma. ∎

Lemma 3.10**.**

Assume $\frac{\chi}{\mu}<\xi_{0}$. Then there exists a constant $C>0$ such that

[TABLE]

Proof.

We can prove this lemma in the same way as in the proof of [5, Lemma 3.11]. ∎

Proof of Theorem 1.1**.**

Lemmas 2.1, 3.8, 3.9 and 3.10 directly drive Theorem 1.1. ∎

4 Asymptotic behavior. Proof of Theorem 1.2

We first recall the following lemma which will give stabilization in (1.1).

Lemma 4.1** ([9, Lemma 4.6]).**

Let $n\in C^{0}(\overline{\Omega}\times[0,\infty))$ satisfy that there exist constants $C>0$ and $\alpha_{0}\in(0,1)$ such that

[TABLE]

Assume that

[TABLE]

with some constant $N^{*}$. Then

[TABLE]

4.1 Case 1: $a_{1},a_{2}\in(0,1)$

Now we assume that $\frac{\chi}{\mu}<\xi_{0}$ and will prove asymptotic behavior of solutions to (1.1) in the case $a_{1},a_{2}\in(0,1)$. In this case we also suppose that there exists $\delta_{1}>0$ such that

[TABLE]

The following lemma asserts that the assumption of Lemma 4.1 is satisfied in the case that $a_{1},a_{2}<1$.

Lemma 4.2**.**

Let $(n_{1},n_{2},c,u,P)$ be a solution to (1.1). If $a_{1},a_{2}\in(0,1)$, then there exist constants $C>0$ and $\alpha_{0}\in(0,1)$ such that

[TABLE]

for all $t\geq 1$. Moreover, $n_{1}$, $n_{2}$ and $c$ satisfy

[TABLE]

where

[TABLE]

Proof.

We can first obtain from Lemmas 3.8, 3.9, 3.10 and [12] that (4.3) holds. Next we will confirm (4.4). We put

[TABLE]

where $\delta_{1}>0$ is a constant defined as in (4.1)–(4.2) and $\delta_{2}>0$ is a constant satisfying

[TABLE]

Then noting from $\nabla\cdot u=0$ that

[TABLE]

for all $i=1,2$ and

[TABLE]

we see from an argument similar to that in the proof of [17, Lemma 2.2] that there exists a constant $\varepsilon_{1}>0$ such that

[TABLE]

Thus we have from the nonnegativity of $E_{1}$ that

[TABLE]

which leads to (4.4). ∎

4.2 Case 2: $a_{1}\geq 1>a_{2}$

In this section we assume that $\frac{\chi}{\mu}<\xi_{0}$ and will obtain stabilization in (1.1) in the case $a_{1}\geq 1>a_{2}$. In this case we also suppose that there exist constants $\delta_{1}^{\prime}>0$ and $a_{1}^{\prime}\in[1,a_{1}]$ such that

[TABLE]

We shall show the following lemma to verify that the assumption of Lemma 4.1 is satisfied in the case that $a_{1}\geq 1>a_{2}$.

Lemma 4.3**.**

Let $(n_{1},n_{2},c,u,P)$ be a solution to (1.1). If $a_{1}\geq 1>a_{2}$, then there exist constants $C>0$ and $\alpha_{0}\in(0,1)$ such that

[TABLE]

for all $t\geq 1$. Moreover, we have

[TABLE]

Proof.

We first see from Lemmas 3.8, 3.9, 3.10 and [12] that (4.7) holds. Next we will show (4.8). We put

[TABLE]

where $\delta_{1}^{\prime}>0$ and $a_{1}^{\prime}\in[1,a_{1}]$ are constants defined as in (4.5)–(4.6) and $\delta_{2}^{\prime}>0$ is a constant satisfying

[TABLE]

Then noting from $\nabla\cdot u=0$ that

[TABLE]

for $i=1,2$ and

[TABLE]

we derive from an argument similar to that in the proof of [16, Lemma 4.1] that there exists a constant $\varepsilon_{2}>0$ such that

[TABLE]

Hence we obtain from the nonnegativity of $E_{1}$ that

[TABLE]

which means that the desired estimate (4.8) holds. ∎

4.3 Convergence for $u$

Finally we provide the following lemma with respect to the decay properties of $u$.

Lemma 4.4**.**

Under the assumptions of Theorems 1.1 and 1.2, the solution of (1.1) has the following property:**

[TABLE]

Proof.

From Lemmas 4.2 and 4.3 the same argument as in the proof of [5, Lemma 4.6] implies this lemma. ∎

4.4 Proof of Theorem 1.2

Proof of Theorem 1.2**.**

A combination of Lemmas 4.1, 4.2, 4.3 and 4.4 directly leads to Theorem 1.2. ∎

Acknowledgement

M.M. is supported by JSPS Research Fellowships for Young Scientists (No. 17J00101). A major part of this work was written while S.K. and M.M. visited Universität Paderborn under the support from Tokyo University of Science.