2D Stochastic Chemotaxis-Navier-Stokes System

TL;DR

This paper proves the existence and uniqueness of solutions for a complex 2D stochastic chemotaxis and fluid dynamics system, advancing mathematical understanding of such coupled stochastic PDEs.

Contribution

It introduces a novel approach to establish both mild and weak solutions for the 2D stochastic chemotaxis-Navier-Stokes system, including fixed point and martingale methods.

Findings

Existence of mild/variational solutions via fixed point theorem.

Existence of martingale weak solutions.

Pathwise uniqueness of solutions.

Abstract

In this paper, we establish the existence and uniqueness of both mild(/variational) solutions and weak (in the sense of PDE) solutions of coupled system of 2D stochastic Chemotaxis-Navier-Stokes equations. The mild/variational solution is obtained through a fixed point argument in a purposely constructed Banach space. To get the weak solution we first prove the existence of a martingale weak solution and then we show that the pathwise uniqueness holds for the martingale solution.

Equations

$$dn+u\cdot \nabla ndt=\delta \Delta ndt-\nabla \cdot (\chi (c)n\nabla c)dt,$$

$$dc+u\cdot \nabla cdt=\mu \Delta cdt-k(c)ndt,$$

$$du+(u\cdot \nabla )udt+\nabla Pdt=\nu \Delta udt-n\nabla \varphi dt+\sigma (u)d{W}_t,$$

$$\nabla \cdot u=0,t>0,x\in \mathcal{O}.$$

$$\frac{\partial n}{\partial v}=\frac{\partial c}{\partial v}=0\text{ and }u=0\text{ for }x\in \partial \mathcal{O}\text{ and }t>0,$$

$$n(0,x)=n_0(x),c(0,x)=c_0(x),u(0,x)=u_0(x),x\in \mathcal{O}.$$

$$\Vert \cdot {\Vert }_{\infty }:=\Vert \cdot {\Vert }_{L^{\infty }(\mathcal{O})},\Vert \cdot {\Vert }_{L^q}:=\Vert \cdot {\Vert }_{L^q(\mathcal{O})},\Vert \cdot {\Vert }_{\alpha }:=\Vert \cdot {\Vert }_{D(A^{\alpha })},\Vert \cdot {\Vert }_{k,q}:=\Vert \cdot {\Vert }_{W^{k,q}(\mathcal{O})},\Vert \cdot {\Vert }_{H^k}:=\Vert \cdot {\Vert }_{W^{k,2}(\mathcal{O})}.$$

$$\Vert \sigma (u_1)-\sigma (u_2){\Vert }_{{\mathcal{L}}_0^2}^2\le K\Vert u_1-u_2{\Vert }_H^2,\text{ and }\Vert \sigma (u){\Vert }_{{\mathcal{L}}_0^2}^2\le K(1+\Vert u{\Vert }_H^2),$$

$$\Vert \sigma (u_1)-\sigma (u_2){\Vert }_{{\mathcal{L}}_{\alpha }^2}^2\le K\Vert u_1-u_2{\Vert }_{\alpha }^2,\text{ and }\Vert \sigma (u){\Vert }_{{\mathcal{L}}_{\alpha }^2}^2\le K(1+\Vert u{\Vert }_{\alpha }^2),$$

$$\Vert \sigma (u_1)-\sigma (u_2){\Vert }_{{\mathcal{L}}_{\frac{1}{2}}^2}^2\le K\Vert u_1-u_2{\Vert }_{\frac{1}{2}}^2,\text{ and }\Vert \sigma (u){\Vert }_{{\mathcal{L}}_{\frac{1}{2}}^2}^2\le K(1+\Vert u{\Vert }_{\frac{1}{2}}^2).$$

$$\displaystyle n(t)=e^{t\delta\Delta}n_{0}-\int_{0}^{t}e^{(t-s)\delta\Delta}\Big{\{}u(s)\cdot\nabla n(s)\Big{\}}ds-\int_{0}^{t}e^{(t-s)\delta\Delta}\Big{\{}\nabla\cdot\Big{(}\chi(c(s))n(s)\nabla c(s)\Big{)}\Big{\}}ds,$$

$$\displaystyle c(t)=e^{t\mu\Delta}c_{0}-\int_{0}^{t}e^{(t-s)\mu\Delta}\Big{\{}u(s)\cdot\nabla c(s)\Big{\}}ds-\int_{0}^{t}e^{(t-s)\mu\Delta}\Big{\{}k(c(s))n(s)\Big{\}}ds,$$

$$\displaystyle u(t)=e^{-t\nu A}u_{0}-\int_{0}^{t}e^{-(t-s)\nu A}\mathcal{P}\Big{\{}(u(s)\cdot\nabla)u(s)\Big{\}}ds-\int_{0}^{t}e^{-(t-s)\nu A}\mathcal{P}\Big{\{}n(s)\nabla\phi\Big{\}}ds$$

$$+{\int }_0^t{e}^{-(t-s)\nu A}\sigma (u(s))d{W}_s,$$

$$\displaystyle n(t)+\int_{0}^{t}u(s)\cdot\nabla n(s)ds=n_{0}+\delta\int_{0}^{t}\Delta n(s)ds-\int_{0}^{t}\nabla\cdot\Big{(}\chi(c(s))n(s)\nabla c(s)\Big{)}ds,$$

$$c(t)+{\int }_0^{t}u(s)\cdot \nabla c(s)ds=c_0+\mu {\int }_0^t\Delta c(s)ds-{\int }_0^{t}k(c(s))n(s)ds,$$

$$\displaystyle u(t)+\int_{0}^{t}\mathcal{P}\Big{\{}(u(s)\cdot\nabla)u(s)\Big{\}}ds=u_{0}-\nu\int_{0}^{t}Au(s)ds-\int_{0}^{t}\mathcal{P}\Big{\{}(n(s)\nabla\phi)\Big{\}}ds$$

$$+{\int }_0^t\sigma (u(s))d{W}_s,$$

$$n_0\in C^0(\bar{\mathcal{O}}),n_0>0\text{in}\bar{\mathcal{O}},$$

$$c_0\in W^{1,q}(\mathcal{O}),\text{for some }q>2,c_0>0\text{in}\bar{\mathcal{O}},$$

$$u_0\in D(A^{\alpha }),\text{for some}\alpha \in (1/2,1),$$

$$\Vert u{\Vert }_V:=\Vert A^{1/2}u{\Vert }_H=\Vert \nabla u{\Vert }_{L^2}.$$

$$\Psi (c)={\int }_0^c\sqrt{\frac{\chi (s)}{k(s)}}ds.$$

$$\Vert \sigma (u){\Vert }_{{\mathcal{L}}_0^2}^2\le C(1+\Vert u{\Vert }_H^2)\text{ and }\Vert \sigma (u_1)-\sigma (u_2){\Vert }_{{\mathcal{L}}_0^2}^2\le C\Vert u_1-u_2{\Vert }_H^2.$$

$$n(1+|x|+|\ln n|)\in L^{\infty }([0,T],L^1(\mathcal{O})),\nabla \sqrt{n}\in L^2([0,T],L^2(\mathcal{O})),$$

$$c\in L^{\infty }([0,T],L^{\infty }(\mathcal{O})\cap H^1(\mathcal{O}))\cap L^2([0,T],H^2(\mathcal{O})),$$

$$u\in C([0,T],H)\cap L^2([0,T],V);$$

$${\int }_{\mathcal{O}}{\psi }_1(0,x)n_{0}dx={\int }_0^T{\int }_{\mathcal{O}}n[{\partial }_t{\psi }_1+\nabla {\psi }_1\cdot u+\delta \Delta {\psi }_1+\nabla {\psi }_1\cdot (\chi (c)\nabla c)]dxdt,$$

$${\int }_{\mathcal{O}}{\psi }_2(0,x)c_{0}dx={\int }_0^T{\int }_{\mathcal{O}}c[{\partial }_t{\psi }_2+\nabla {\psi }_2\cdot u+\mu \Delta {\psi }_2]-nk(c){\psi }_{2}dxdt,$$

$$⟨u(t),e{⟩}_{H,H}$$

$$P-a.s.$$

$${\int }_{\mathcal{O}}n(t,x)dx={\int }_{\mathcal{O}}{n}_0(x)dx,$$

$$\Vert c(t,\cdot ){\Vert }_{\infty }\le \Vert c_0{\Vert }_{\infty },n(t,x)\ge 0,c(t,x)\ge 0.$$

$$\Vert n(t){\Vert }_{L^2}$$

$${{\rm Y}}_t^n:=L^{\infty }([0,t],C^0(\bar{\mathcal{O}})),{{\rm Y}}_t^c:=L^{\infty }([0,t],W^{1,q}(\mathcal{O})),{{\rm Y}}_t^u:=L^{\infty }([0,t],D(A^{\alpha }))$$

$$\Vert n{\Vert }_{{{\rm Y}}_t^n}=\underset{s\in [0,t]}{\sup }\Vert n(s){\Vert }_{\infty },\Vert c{\Vert }_{{{\rm Y}}_t^c}=\underset{s\in [0,t]}{\sup }\Vert c(s){\Vert }_{1,q},\Vert u{\Vert }_{{{\rm Y}}_t^u}=\underset{s\in [0,t]}{\sup }\Vert u(s){\Vert }_{\alpha }.$$

$$dn+{\theta }_m(\Vert u{\Vert }_{{{\rm Y}}_t^u}){\theta }_m(\Vert n{\Vert }_{{{\rm Y}}_t^n})u\cdot \nabla ndt=\delta \Delta ndt-{\theta }_m(\Vert n{\Vert }_{{{\rm Y}}_t^n}){\theta }_m(\Vert c{\Vert }_{{{\rm Y}}_t^c})\nabla \cdot (\chi (c)n\nabla c)dt,$$

Full text

1. 1 Introduction

2. 2 Framework and Statement of the Main Results

3. 3 Existence and Uniqueness of Mild/Variational Solutions

4. 3.1 Existence of Local Solutions

5. 3.2 Uniqueness of Local Solutions

6. 3.3 Global Existence

7. 4 Existence and Uniqueness of Weak Solutions

8. 4.1 Entropy Function

9. 4.2 Energy Estimates for Approximating Solutions

10. 4.3 Existence of Martingale Weak Solutions

11. 4.4 Pathwise Weak Solution

1 Introduction

The purpose of this paper is to establish the existence and uniqueness of the solution of the coupled 2D stochastic Chemotaxis-Navier-Stokes system:

[TABLE]

The system arises in the modeling of bacterial suspensions in fluid drops and describes the spontaneous emergence of patterns in populations of oxygen-driven swimming bacteria. Here, $\mathcal{O}\subset\mathbb{R}^{2}$ is a bounded convex domain with smooth boundary $\partial\mathcal{O}$, which will be the spatial domain where the moving cells and the fluid interact. The unknowns are $n=n(t,x):\mathbb{R}^{+}\times\mathcal{O}\rightarrow\mathbb{R}^{+}$, $c(t,x):\mathbb{R}^{+}\times\mathcal{O}\rightarrow\mathbb{R}^{+}$, $u(t,x):\mathbb{R}^{+}\times\mathcal{O}\rightarrow\mathbb{R}^{2}$ and $P=P(t,x):\mathbb{R}^{+}\times\mathcal{O}\rightarrow\mathbb{R}$, which represent respectively the cell density, chemical concentration, velocity field and pressure of the fluid. Positive constants $\delta,\mu,\nu$ are the corresponding diffusion coefficients for the cells, chemical and fluid. The gravitational potential $\phi=\phi(x)$, the chemotactic sensitivity $\chi(c)$ and the per-capita oxygen consumption rate $k(c)$ are supposed to be given sufficiently smooth functions. $\{W_{t},\ t\geq 0\}$ is a cylindrical Wiener process representing the external random driving force.

System (1) is considered with the boundary conditions

[TABLE]

and the initial conditions

[TABLE]

The deterministic models of system (1) (i.e. $\sigma=0$) was proposed by Tuval et al. in [17]. In [25], the authors suggest a wider variants to describe more complicated interaction neighborhood environment around cells. The well-posedness of the deterministic models of system (1) (and its variants) is a highly non-trivial problem. In the past several years, the main focus of the existing literature is on the solvability of the system, see [2, 3, 5, 7, 9, 10, 11, 14, 20, 21, 22, 23, 27] and reference therein. We like to mention a few of them which are relevant to our work. In [12], local (in time) weak solutions (in the sense of PDE) were constructed in a bounded domain in $\mathbb{R}^{d}$, $d=2,3$ with no-flux boundary condition and in $\mathbb{R}^{2}$ for a special case. Based on some nice energy estimates, if the convective term $(u\cdot\nabla)u$ is neglected, global weak solutions were obtained in [4] provided the initial data or $\nabla\phi$ is small. Our work is motivated and influenced by the recent papers [11] and [20]. In [11], for the models in $\mathbb{R}^{2}$, Liu and Lorz developed some nice entropy estimates to prove the global existence of weak solutions to the deterministic models of system (1) for large initial data. In [20], when $\mathcal{O}\subset\mathbb{R}^{2}$ is a bounded convex domain with smooth boundary $\partial\mathcal{O}$, the author managed to establish the existence and uniqueness of global strong (in the sense of PDE) solution of system (1) without the restriction of the smallness of either the initial data or the coefficients. There are many other interesting results on this topic, we refer to the references mentioned above. Finally, we refer the reader to [19, 24] for the stabilization and convergence rate of solutions of the deterministic models of system (1) and its variants.

Taking into account the random environment the bacteria are in and the effect of random external forces, it is natural to consider the coupled 2D stochastic Chemotaxis-Navier-Stokes system (1). Adding the singular random noise to the system changes the mathematical analysis significantly. In this paper we seek for probabilistically the so called pathwise/strong solutions. While in sense of PDE, we consider both the mild/variational solutions and the weak solutions under two different sets of conditions. From now on, the term of weak solutions are reserved for the weak solutions in the sense of PDE. The paper is divided into two parts. In the first part, we establish the existence and uniqueness of mild/variational solutions to system (1). To this end, we first appropriately cut off the coefficients of the system and construct a local (in time) mild/variational solution using fixed point arguments in a certain Banach space and we then show that the mild/variational solution is global by providing some energy estimates. In the second part, we obtain the existence and uniqueness of pathwise weak solution of the system (1). For this purpose, we first establish the existence of a martingale weak solution. In order to do so, we define a sequence of approximating systems and prove that a subsequence of the approximate solutions converges in law to a martingale weak solution of system (1). Then we prove that the pathwise uniqueness of weak solutions holds. As an application of Watanable and Yamada Theorem we obtain both the pathwise existence and uniqueness of the weak solution. Because the proofs of the main results are involved, we will state the main results in next section and leave the details of the arguments in the rest of the paper.

The paper is organized as follows. In Section 2, we spread out the precise assumptions and the framework. We also state the main results. Section 3 consists of several subsections. It is devoted to establishing the existence and uniqueness of mild/variational solution. The entire Section 4 is to prove the existence and uniqueness of the pathwise weak solution.

2 Framework and Statement of the Main Results

Let $L^{q}(\mathcal{O})$ denote the $L^{q}$ space with respect to the Lebesgue measure. $W^{k,q}(\mathcal{O})$ denotes the Sobolev space of functions whose distributional derivatives of order up to $k$ belong to $L^{q}$. Let $A$ be the realization of the Stokes operator $-\mathcal{P}\Delta$, where $\mathcal{P}$ denotes the Helmholtz projection from $L^{2}(\mathcal{O})$ into the space $H=\{\varphi\in L^{2}(\mathcal{O})|\nabla\cdot\varphi=0\}$. In the sequel, \Big{(}e^{t\Delta}\Big{)}_{t\geq 0}, \Big{(}e^{-tA}\Big{)}_{t\geq 0} will denote respectively the Neumann heat semigroup and the Stokes semigroup with Dirichlet boundary condition.

For simplicity, we set $H^{k}(\mathcal{O}):=W^{k,2}(\mathcal{O})$,

[TABLE]

We introduce the following conditions on the parameters and functions involved in the system (1):

• (H.1)

• (a)

$\chi\in C^{2}([0,\infty)),\ \chi>0$ in $[0,\infty)$,

• (b)

$k\in C^{2}([0,\infty)),\ k(0)=0,\ k>0$ in $(0,\infty)$,

• (c)

$\phi\in C^{2}(\bar{\mathcal{O}})$,

• (H.2)

$(\frac{k(c)}{\chi(c)})^{\prime}>0,\ (\frac{k(c)}{\chi(c)})^{\prime\prime}\leq 0,\ (\chi(c)\cdot k(c))^{\prime}\geq 0$ on $[0,\infty)$.

Let $U$ be a real Hilbert space and $\{W_{t},\ t\geq 0\}$ a $U$-cylindrical Wiener process on a given complete, filtered probability space $(\Omega,\mathcal{F},\mathcal{F}_{t},\ t\geq 0,\mathbb{P})$, representing the driving external random force. Let ${\mathcal{L}^{2}}(U,D(A^{\beta}))$ denote the space of Hilbert-Schmidt operators $g$ from $U$ into $D(A^{\beta})$ and its norm is denoted by $\|g\|_{{\mathcal{L}^{2}_{\beta}}}$. For a mapping $\sigma:D(A^{\beta})\rightarrow\mathcal{L}^{2}(U,D(A^{\beta}))$, we introduce the following hypothesis:

• (H.3)

there exists a positive constant $K$ such that for all $u_{1},u_{2},u\in H$,

[TABLE]

where $\mathcal{L}^{2}_{0}={\mathcal{L}^{2}}(U,H)$,

• (H.4)

there exists a positive constant $K$ such that for all $u_{1},u_{2},u\in D(A^{\alpha})$,

[TABLE]

• (H.5)

there exists a positive constant $K$ such that for all $u_{1},u_{2},u\in D(A^{\frac{1}{2}})$,

[TABLE]

Set: $u(t)=u(t,\cdot)$, $n(t)=n(t,\cdot)$ and $c(t)=c(t,\cdot)$. Let $q>2$.

Definition 2.1

We say that $(n,c,u)$ is a mild solution of system (1) if $(n,c,u)$ is a progressively measurable stochastic processes with values in $C^{0}(\bar{\mathcal{O}})\times W^{1,q}({\mathcal{O}})\times D(A^{\alpha})$, which satisfies

[TABLE]

$P$-a.s.

Remark 2.1

Note that $u(s)\cdot\nabla n(s)=\nabla\cdot(u(s)n(s))$ because $\nabla\cdot u(s)=0$. Under the setting in the above definition, actually $n(\cdot)\in L^{2}_{loc}([0,\infty),W^{1,2}({\mathcal{O}}))$ $P$-a.s., and $(n,c,u)$ is equivalent to a variational solution of the system in the Gelfand triple $W^{1,2}({\mathcal{O}})\subset L^{2}({\mathcal{O}})\subset W^{1,2}({\mathcal{O}})^{*}$, that is, $(n,c,u)$ satisfies

[TABLE]

$P$-a.s..

Here is our first main result.

Theorem 2.1

Assume

[TABLE]

and the assumptions (H.1)-(H.5) hold. Then there exists a unique mild/variational solution to the system (1).

Define $V:=D(A^{1/2})$ and its norm

[TABLE]

Its dual space will be denoted by $V^{*}$.

Introduce the following conditions:

• (A)

• (a)

$\chi(\cdot)$ and $k(\cdot)$ are smooth with $k(0)=0$, $k(c)>0$ in $(0,\infty)$ and $k^{\prime}(c)\geq 0$, $\chi(c)>0$ for every $c\in\mathbb{R}$,

• (b)

$\chi^{\prime}(c)\geq 0,\ (\frac{k(c)}{\chi(c)})^{\prime}>0,\ (\frac{k(c)}{\chi(c)})^{\prime\prime}<0$, $(\chi(c)\cdot k(c))^{\prime}>0$ on $[0,\infty)$,

• (c)

$\phi\in C^{2}(\bar{\mathcal{O}})$,

• (B)

$(n_{0},c_{0},u_{0})$ satisfies

• (B1)

$n_{0}(x)\geq 0$, $0\leq c_{0}(x)\leq C_{M}<\infty$, $\nabla\cdot u_{0}(x)=0$ on $x\in\mathcal{O}$,

• (B2)

$u_{0}\in H$,

• (B3)

$n_{0}(1+|x|+|\ln n_{0}|)\in L^{1}(\mathcal{O})$,

• (B4)

$\nabla c_{0}\in L^{2}(\mathcal{O})$, $\nabla\Psi(c_{0})\in L^{2}(\mathcal{O})$ where

[TABLE]

• (C)

for any $u,u_{1},u_{2}\in H$,

[TABLE]

Definition 2.2

We say that $({n},{c},{u})$ is a weak solution to the system (1) if $({n},{c},{u})$ is a progressively measurable process that satisfies, for any $T>0$,

• (1)

$P$-a.s.

[TABLE]

• (2)

For all $\psi_{1}$, $\psi_{2}\in C^{\infty}([0,T]\times\mathcal{O})$ with compact supports in the space variable, and $\psi_{1}(T,\cdot)=\psi_{2}(T,\cdot)=0$, $P$-a.s.

[TABLE]

[TABLE]

• (3)

For all $e\in V$, $0\leq t\leq T$,

[TABLE]

The following is our second main result.

Theorem 2.2

Assume the assumptions (A)-(C) hold, and the function $\chi(\cdot)$ is a positive constant. Then there exists a unique weak solution to the system (1).

We end this section by recalling the following two properties of the solution (see Lemma 2.2 in [20]). The first property follows by integrating the first equation in the system (1). The second one is a consequence of the comparison theorem/maximum principle.

Lemma 2.1

The solution of (1) satisfies, for all $t\geq 0$,

[TABLE]

and

[TABLE]

Using (2.5) and the Gagliardo-Nirenberg-Sobolev inequality, we also have

[TABLE]

3 Existence and Uniqueness of Mild/Variational Solutions

In this section, we assume that conditions (H.1)-(H.5) hold. Our aim is to prove Theorem 2.1.

3.1 Existence of Local Solutions

Introduce the following spaces

[TABLE]

with the corresponding norms given by

[TABLE]

Definition 3.1

We say that $(n,c,u,\tau)$ is a local mild/variational solution of system (1) if

(1) $\tau$ is a stopping time and $(n,c,u)$ is a progressively measurable stochastic processes with values in $C^{0}(\bar{\mathcal{O}})\times W^{1,q}(\mathcal{O})\times D(A^{\alpha})$,

(2) there exists a nondecreasing sequence of stopping times $\{\tau_{l},l\geq 1\}$ with $\tau_{l}\uparrow\tau$ a.s. as $l\uparrow\infty$, such that $\{(n(t\wedge\tau_{l}),c(t\wedge\tau_{l}),u(t\wedge\tau_{l})),t\geq 0\}$ is a mild/variational solution to system (1).

Theorem 3.1

There exists a local mild/variational solution to the system (1).

Proof. To use a cut off argument, we will modify the coefficients in system (1). Fix a function $\theta\in C^{2}([0,\infty),[0,1])$ such that

• (1)

$\theta(r)=1$, $r\in[0,1]$,

• (2)

$\theta(r)=0$, $r>2$,

• (3)

$\sup_{r\in[0,\infty)}|\theta^{\prime}(r)|\leq C<\infty$.

Set $\theta_{m}(\cdot)=\theta(\frac{\cdot}{m})$. For every $m\geq 1$, consider the following system of SPDEs

[TABLE]

To simplify the exposition, we assume $\delta=\mu=\nu=1$, $\chi(c)=1$, and $k(c)=c$. The general case is entirely similar.

Let $S_{T}$ be the space of all $\{\mathcal{F}_{t}\}_{t\in[0,T]}$-adapted, $C^{0}(\bar{\mathcal{O}})\times W^{1,q}(\mathcal{O})\times D(A^{\alpha})$-valued stochastic processes $(n(t),c(t),u(t)),t\geq 0$ such that

[TABLE]

Then $S_{T}$ equipped with the norm $\|\cdot\|_{S_{T}}$ is a Banach space.

We introduce a mapping $\Phi=(\Phi_{1},\Phi_{2},\Phi_{3})$ on $S_{T}$ by defining

[TABLE]

[TABLE]

and

[TABLE]

Let $B$ denote the operator $-\Delta+1$ in $L^{q}(\mathcal{O})$ ($q>2$) equipped with Neumann boundary condition. Then, for $\beta\in(\frac{1}{q},\frac{1}{2})$, we have the continuous imbedding $D(B^{\beta})\hookrightarrow C^{0}(\bar{\mathcal{O}})$. Using a similar argument as that in [20] ( page 325), we have

[TABLE]

here we have used the continuous imbedding $D(A^{\alpha})\hookrightarrow C^{0}(\bar{\mathcal{O}})$.

Fix any $\gamma\in(\frac{1}{2},1)$,

[TABLE]

For $\Phi_{3}$, we have

[TABLE]

Noticing

[TABLE]

we have

[TABLE]

For $I_{2}$, we have

[TABLE]

To estimate $I_{3}$, let $Z(t):=\int_{0}^{t}e^{-(t-s)A}A^{\alpha}\sigma(u(s))dW_{s}$. Then $Z$ is the solution of the evolution equation

[TABLE]

Applying $\rm It\hat{o}^{\prime}s$ Formula, and then the BDG inequality, we have

[TABLE]

here we have used Assumption (H.4).

Hence

[TABLE]

Combining (3.7)–(3.10), we get

[TABLE]

(3.5), (3.6) and (3.11) together show that $\Phi$ maps $S_{T}$ into itself.

Next we will prove that if $T>0$ is small enough then $\Phi$ can be made a contraction on $S_{T}$.

Let $(n_{1},c_{1},u_{1}),(n_{2},c_{2},u_{2})\in S_{T}$. We have

[TABLE]

where $\varrho$ is some generic constant. Similar to the proof of (3.5), we have

[TABLE]

Set

[TABLE]

We will distinguish six cases to bound $J$. By the property of $\theta$ and the Minkowski inequality, we have the following estimates.

(J1) Suppose $\|n_{1}\|_{\Upsilon^{n}_{s}}\vee\|c_{1}\|_{\Upsilon^{c}_{s}}\vee\|n_{2}\|_{\Upsilon^{n}_{s}}\vee\|c_{2}\|_{\Upsilon^{c}_{s}}\leq 2m$. We have

[TABLE]

(J2) Suppose $\|n_{1}\|_{\Upsilon^{n}_{s}}\vee\|c_{1}\|_{\Upsilon^{c}_{s}}>2m$ and $\|n_{2}\|_{\Upsilon^{n}_{s}}\vee\|c_{2}\|_{\Upsilon^{c}_{s}}>2m$. We have

[TABLE]

(J3) Suppose $\|n_{1}\|_{\Upsilon^{n}_{s}}>2m$ and $\|n_{2}\|_{\Upsilon^{n}_{s}}\vee\|c_{2}\|_{\Upsilon^{c}_{s}}\leq 2m$.

[TABLE]

(J4) Suppose $\|c_{1}\|_{\Upsilon^{n}_{s}}>2m$ and $\|n_{2}\|_{\Upsilon^{n}_{s}}\vee\|c_{2}\|_{\Upsilon^{c}_{s}}\leq 2m$.

[TABLE]

The proofs of the following two cases are similar as (J3) and (J4).

(J5) If $\|n_{2}\|_{\Upsilon^{n}_{s}}>2m$ and $\|n_{1}\|_{\Upsilon^{n}_{s}}\vee\|c_{1}\|_{\Upsilon^{c}_{s}}\leq 2m$, then

[TABLE]

(J6) Suppose $\|c_{2}\|_{\Upsilon^{n}_{s}}>2m$ and $\|n_{1}\|_{\Upsilon^{n}_{s}}\vee\|c_{1}\|_{\Upsilon^{c}_{s}}\leq 2m$. Then

[TABLE]

Putting (J1)–(J6) together, we get

[TABLE]

Substituting (3.14) into (3.1), we get

[TABLE]

Using the similar arguments as in the proof of (3.14), we can show

[TABLE]

Thus, similar to (3.15), we have

[TABLE]

Substitute (3.15) and (3.16) into (3.12) to get

[TABLE]

for $t\leq T$.

By a similar reasoning, we can show that

[TABLE]

for $t\leq T$, here $\gamma$ is a number in $(\frac{1}{2},1)$.

Now we estimate $\|\Phi_{3}(n_{1},c_{1},u_{1})-\Phi_{3}(n_{2},c_{2},u_{2})\|_{\Upsilon^{u}_{T}}$. We have

[TABLE]

Using the similar arguments as in the proof of (3.15), it can be shown that

[TABLE]

for $t\leq T$. Note that $\Gamma_{3}(t)=\|Z(t)\|_{L^{2}}$, where $Z$ satisfies the following SPDE

[TABLE]

Using $\rm It\hat{o}$’s Formula and the BDG inequality, we have

[TABLE]

here we have used Assumption (H.4). Hence,

[TABLE]

To estimate $\Gamma_{1}$, set

[TABLE]

We will bound $J_{1}$ in four different cases. Set $B(u)=(u\cdot\nabla)u$.

• (1)

Suppose $\|u_{1}\|_{\Upsilon^{u}_{s}}\vee\|u_{2}\|_{\Upsilon^{u}_{s}}\leq 2m$. From the definition of $\theta_{m}$, we get

[TABLE]

• (2)

Suppose $\|u_{1}\|_{\Upsilon^{u}_{s}}\leq 2m$ and $\|u_{2}\|_{\Upsilon^{u}_{s}}>2m$. We have

[TABLE]

• (3)

Suppose $\|u_{1}\|_{\Upsilon^{u}_{s}}>2m$ and $\|u_{2}\|_{\Upsilon^{u}_{s}}\leq 2m$. Similar to case (2), we have

[TABLE]

• (4)

Suppose $\|u_{1}\|_{\Upsilon^{u}_{s}}\wedge\|u_{2}\|_{\Upsilon^{u}_{s}}>2m$. Then

[TABLE]

Hence, it follows that for all the cases,

[TABLE]

Combining (3.19) (3.20) (3.21) and (3.22) together we arrive at

[TABLE]

By virtue of (3.17) (3.1) and (3.23), one can find constants $\rho,\ C_{m}>0$ such that

[TABLE]

Choose $T=T_{m}$ such that $C_{m}T^{\rho}_{m}=\frac{1}{2}$. Then $\Phi$ is a contraction on the space $S_{T_{m}}$. Applying the Banach fixed point theorem, we conclude that there exists a unique element $(n_{m},c_{m},u_{m})\in S_{T_{m}}$ such that $(n_{m},c_{m},u_{m})$ is a solution of (3.1) for $t\in[0,T_{m}]$.

Let $S^{1}_{T}$ be the space of all $\{\mathcal{F}_{t}\}_{t\in[0,T]}$-adapted, $C^{0}(\bar{\mathcal{O}})\times W^{1,q}(\mathcal{O})\times D(A^{\alpha})$-valued stochastic processes $(n(t),c(t),u(t)),t\geq 0$ such that

[TABLE]

and

[TABLE]

Then $(S^{1}_{T},\|\cdot\|_{S^{1}_{T}})$ is a Banach space.

We introduce a mapping $\Phi^{1}=(\Phi^{1}_{1},\Phi^{1}_{2},\Phi^{1}_{3})$ on $S^{1}_{T}$ by defining

[TABLE]

[TABLE]

and

[TABLE]

Observe that the constant $T_{m}$ does not depend on the initial datum. Repeating the above arguments, we can solve (3.1) for $t\in[T_{m},2T_{m}]$, $[2T_{m},3T_{m}]$,… and we finally obtain a unique solution $(n_{m},c_{m},u_{m})\in S_{T}$ of (3.1) for any $T>0$.

Define

[TABLE]

The $\tau_{m}$ is a stopping time. When $m>>\|n_{0}\|_{\infty}\vee\|c_{0}\|_{1,q}\vee\|u_{0}\|_{\alpha}$, we have

[TABLE]

By the definition of $\theta_{m}$, it is seen that $(n_{m}(t),c_{m}(t),\ u_{m}(t))_{t\in[0,\tau_{m})}$ is a local variational solution to the system (1). $\Box$

3.2 Uniqueness of Local Solutions

Theorem 3.2

Suppose that $(n_{1},c_{1},u_{1},\tau^{1})$ and $(n_{2},c_{2},u_{2},\tau^{2})$ are two local mild/variational solutions of system (1.1). Set $\tau=\tau^{1}\wedge\tau^{2}$. Then we have

[TABLE]

Proof. Define

[TABLE]

and set $\tau_{R}=\tau_{R}^{1}\wedge\tau_{R}^{2}$.

Notice that $\nabla\cdot(u_{i}(t)n_{i}(t))=u_{i}(t)\cdot\nabla n_{i}(t)$ because $\nabla\cdot u_{i}=0$, $i=1,2$. For all $t\in[0,T\wedge\tau_{R})$, we have

[TABLE]

Here for the last inequality, we have used Ehrling’s lemma and the compact embedding $W^{1,2}(\mathcal{O})\hookrightarrow L^{\frac{2q}{q-2}}(\mathcal{O})$.

Recall that $W^{1,q}(\mathcal{O})$ is continuously embedded into $C^{0}(\bar{\mathcal{O}})$. For all $t\in[0,T\wedge\tau_{R})$, we have

[TABLE]

By $\rm It\hat{o}$’s formula, for all $t\in[0,T\wedge\tau_{R})$,

[TABLE]

For the first inequality of (3.29), we have used

[TABLE]

Combining (3.27) (3.28) and (3.29), we get

[TABLE]

Let

[TABLE]

Apply BDG inequality, Assumption (H.3) and Gronwall’s lemma to conclude from (3.30) that

[TABLE]

We obtain the uniqueness by noting $\tau_{R}\uparrow\tau$ as $R\uparrow\infty$. $\Box$

3.3 Global Existence

Definition 3.2

Let $(n,c,u,\tau)$ be a local mild/variational solution of system (1). If $\limsup_{t\nearrow\tau}(\|n\|_{\Upsilon^{n}_{t}}+\|c\|_{\Upsilon^{c}_{t}}+\|u\|_{\Upsilon^{u}_{t}})=\infty$ on $\{\omega,\,\tau<\infty\}$ a.s., then the local solution $(n,c,u,\tau)$ is called a maximal local solution.

Recall the stopping times $\{\tau_{m},\,m\in\mathbb{N}\}$ defined in (3.25). By the uniqueness of local solution we proved in Section 3.2, we infer that $\tau_{m}\leq\tau_{m+1}$ a.s. and

[TABLE]

Introduce a stopping time:

[TABLE]

and define a stochastic process $(n,c,u)$ on $[0,\tau)$ by

[TABLE]

Since $\|n_{m}\|_{\Upsilon^{n}_{\tau_{m}}}\vee\|c_{m}\|_{\Upsilon^{c}_{\tau_{m}}}\vee\|u_{m}\|_{\Upsilon^{u}_{\tau_{m}}}\geq m$ on $\{\omega,\,\tau<\infty\}$, we have

[TABLE]

Therefore $(n,c,u,\tau)$ is a maximal local solution of system (1).

To obtain the global existence of the solution of the system (1), we will establish some a priori estimates for $(n,c,u)$ in the space $L^{\infty}([0,T\wedge\tau);C^{0}(\bar{\mathcal{O}})\times W^{1,q}(\mathcal{O})\times D(A^{\alpha}))$ for any $T>0$. We first recall the following results from Corollary 4.2 and Lemma 4.5 in [20].

Lemma 3.1

Let $p>1$ and $r\in[1,\frac{p}{p-1}]$. Then there exists a constant $C_{T}$ and $C_{p}$ such that

[TABLE]

and

[TABLE]

We start with an estimate of the $L^{2}$ norm of $u$ and $\nabla u$.

Lemma 3.2

Let $\theta\in(0,1)$. Then there exists a constant $C_{T,\|u_{0}\|_{L^{2}}}$ such that

[TABLE]

Moreover,

[TABLE]

Proof. By Ito’s formula,

[TABLE]

Let $p:=\frac{4}{4-\theta}$. As in the proof of Lemma 4.3 in [20], writing $p^{\prime}:=\frac{p}{p-1}$, by the Sobolev imbedding $W^{1,2}(\mathcal{O})\hookrightarrow L^{p^{\prime}}(\mathcal{O})$, Hölder’s inequality, we have, for $t\in[0,{T\wedge\tau})$,

[TABLE]

where Lemma 3.1 was used. Substitute (3.36) into (3.35) to obtain

[TABLE]

Squaring the above inequality and taking expectation, by the BDG inequality and Assumption (H.3), we get

[TABLE]

To complete the proof (3.33), we apply the Gronwall’s inequality. By the Gagliardo-Nirenberg inequality we have

[TABLE]

The assertion (3.34) follows from (3.33).$\Box$

Corollary 3.1

Let $\theta\in(0,1)$ . The following statements hold:

[TABLE]

[TABLE]

*Proof * From the proof of Corollary 4.4 in [20], we know that

[TABLE]

and

[TABLE]

where $g(c)=\frac{k(c)}{\chi(c)}$, $\rho(c)=\int_{0}^{c}\frac{d\sigma}{g(\sigma)}$.

Both (3.40) and (3.41) now follows from Lemma 3.2. $\Box$

To proceed, we recall the following inequality obtained in [20]. For any $p>1$,

[TABLE]

For $R>0$, define the stopping time $T_{R}$ by

[TABLE]

Note that $T_{R}\rightarrow\infty$ a.s. as $R\rightarrow\infty$. Set $u^{R}(t,x):=u(t\wedge T_{R},x)$, $c^{R}(t,x):=c(t\wedge T_{R},x)$ and $n^{R}(t,x):=n(t\wedge T_{R},x)$. The following result is crucial for establishing the global existence.

Proposition 3.1

For $R>0$ and $T>0$, there exists some constant $C_{R,T}>0$ such that

[TABLE]

Proof. We will prove the proposition along the same lines as in the proof of Theorem 1.1 in [20]. In view of (3.43), for any $p>1$ we have

[TABLE]

Applying Ito’s formula and following the similar arguments as in the proofs of (4.16) and (4.17) in [20], we can show that

[TABLE]

By Gronwall’s inequality, it follows that

[TABLE]

From the definition of $T_{R}$,

[TABLE]

Hence, it follows from the Burkholder inequality and (3.48) that for any $p>1$,

[TABLE]

where we have used the fact that $\|A^{\frac{1}{2}}u\|_{L^{2}}$ is equivalent to $\|\nabla u^{R}(s)\|_{L^{2}}$. By Gronwall’s inequality, we get from (3.49) that

[TABLE]

Next we show that

[TABLE]

By the variation of constants formula we have

[TABLE]

Clearly,

[TABLE]

By virtue of (3.46), we have

[TABLE]

By Hölder’s inequality and Gagliardo-Nirenberg inequality, it holds that

[TABLE]

This along with (3.50) yields

[TABLE]

where $C_{T,R}$ is some constant.

To estimate $u_{3}$ in (3.51), we notice that $u_{3}$ satisfies the SPDE:

[TABLE]

Applying the Ito formula, we get

[TABLE]

By Burkholder inequality we get from (3.56) that

[TABLE]

which leads to

[TABLE]

Combining (3.51), (3.53),(3.55) and (3.58) together we deduce that

[TABLE]

An application of Gronwall’s inequality yields

[TABLE]

To bound $\|\nabla c^{R}(t)\|_{L^{q}}$, we use the variation of constants formula

[TABLE]

to obtain

[TABLE]

We note that $\|n^{R}(s)\|_{L^{2}}\leq C_{R}$ according to (3.46). On the other hand, by Lemma 3.4 in [20] and Gagliardo-Nirenberg inequality, we have

[TABLE]

where the last inequality follows from the definition of $T_{R}$. (3.62) and (3.63) together yields

[TABLE]

To estimate $\|n^{R}(t)\|_{L^{\infty}}$, we fix $\beta\in(\frac{1}{q},\frac{1}{2})$ and then $r\in(\frac{1}{\beta},q)$. As the proof of (4.26) in [20], we have

[TABLE]

where (3.46),(3.64) and the imbedding $D(A^{\alpha})\hookrightarrow L^{\infty}$ have been used. Now, we can conclude from (3.60) that

[TABLE]

for some constant $C_{R,T}$. The proof is complete.$\Box$

Theorem 3.3

Suppose the conditions in Theorem 2.1 are met. Then, the system (1.1) admits a unique global mild/variational solution.

Proof. Let $(n,c,u,\tau)$ be the maximal local solution of system (1.1) obtained in Section 3.3. From Proposition 3.1 we see that for any $T>0$, $R>0$,

[TABLE]

Send $R$, $T$ go to infinity to get the global existence. Uniqueness was proved in Section 3.2 $\Box$

Remark 3.1

We notice that the unique global mild/variational solution $(n,c,u)$ obtained in Theorem 3.3 is a weak solution in the sense of Definition 2.2. We only need to verify the statement (1) in Definition 2.2.

In the proof of Theorem 3.3(see (3.45)), we have, for any $T>0$,

[TABLE]

Theorem 3.3, Lemma 3.2 and (3.40) imply that

[TABLE]

[TABLE]

and

[TABLE]

Combining (3.67)–(3.70) with Lemma 2.1 and the fact that $\mathcal{O}$ is bounded, it is not difficult to deduce that $P$-a.s.

[TABLE]

For example, to prove ${c}\in L^{\infty}([0,T],L^{\infty}(\mathcal{O})\cap H^{1}(\mathcal{O}))\cap L^{2}([0,T],H^{2}(\mathcal{O}))$, one can follow the proof of Lemma 4.2 below.

We now estimate $\int_{\mathcal{O}}n|\ln n|dx$. Since

[TABLE]

and, in view of (2.5),

[TABLE]

it follows that

[TABLE]

Recall $\Psi(c)=\int_{0}^{c}\sqrt{\frac{\chi(s)}{k(s)}}ds$ in Assumption (B). Lemma 3.4 in [20] implies that

[TABLE]

The assumption on $n_{0},c_{0}$ (see (2.1)) implies that

[TABLE]

Putting (3.69) and (3.72)–(3.74) together,

[TABLE]

(3.1) and (3.75) show that the statement (1) in Definition 2.2 holds.

4 Existence and Uniqueness of Weak Solutions

In this part, we assume that conditions (A)-(C) introduced in Section 2 are in place. Our aim is to prove the existence and uniqueness of a weak solution to the system (1.1). Because the operator $A$ is positive self-adjoint with compact resolvent, there is a complete orthonormal basis $\{e_{i},\ i\in\mathbb{N}\}$ in $H$ made of eigenvectors of $A$, with corresponding eigenvalues $0<\beta_{i}\uparrow\infty$, that is

[TABLE]

4.1 Entropy Function

Let $y\in L^{2}(\Omega;\ L^{\infty}([0,T],H))$ be an adapted process. Let $(n,c,u)$ be a solution to the following system

[TABLE]

Recall $\Psi(c)=\int_{0}^{c}\sqrt{\frac{\chi(s)}{k(s)}}ds$ and $C_{M}$ is the constant appeared in Condition (B). Set

[TABLE]

here

[TABLE]

are positive by Condition (A).

We have the following result.

Proposition 4.1

It holds that

[TABLE]

Proof. Lemma 2.1 and Condition (B) imply that $c$ and $n$ preserve the nonnegativity of the initial data, moreover, $\|c(t,\cdot)\|_{\infty}\leq C_{M}$.

Keeping in mind the boundary condition (1.2), as (3.5) in [11], we can show that

[TABLE]

Now applying $\rm It\hat{o}^{\prime}s$ formula to $\|u\|^{2}_{L^{2}}$ and using (2.7), it follows that

[TABLE]

Adding the above inequality to (4.3), we obtain

[TABLE]

$\Box$

4.2 Energy Estimates for Approximating Solutions

In this section, we consider a sequence of approximating solutions and establish some necessary energy estimates for the proof of the tightness.

Let $H_{m}=\text{span}\{e_{1},\cdots,e_{m}\}$ and define $P_{m}:H\rightarrow H_{m}$ as

[TABLE]

Set $\sigma_{m}=P_{m}\sigma$. Then

[TABLE]

Similarly, we can prove that

[TABLE]

This shows that $\sigma_{m}$ satisfies Conditions (H.3) (H.4) and (H.5).

For any $(n_{0},c_{0},u_{0})$ satisfying Condition $(B)$ in Section 2, it is easy to see that one can find $(n^{m}_{0},c^{m}_{0},u^{m}_{0})$ such that

(1) $(n^{m}_{0},c^{m}_{0},u^{m}_{0})$ satisfies (2.1),

(2)

[TABLE]

By Theorem 3.3 and Remark 3.1, we know that there exists an adapted $C^{0}(\bar{\mathcal{O}})\times W^{1,q}(\mathcal{O})\times D(A^{\alpha})$-valued stochastic process $(n^{m},c^{m},u^{m})$ satisfying the following SPDE:

[TABLE]

with initial value $(n^{m}_{0},c^{m}_{0},u^{m}_{0})$.

In the rest of the this section, we will provide a number of estimates for $(n^{m},c^{m},u^{m})$.

Lemma 4.1

There exists a constant $C_{T}$ independent of $m$ such that

[TABLE]

Proof. By (4.4), we have the following estimates

[TABLE]

In particular, together with (3.72), we have

[TABLE]

By the BDG inequality and the growth condition (C) on $\sigma$,

[TABLE]

and

[TABLE]

Substituting (4.10) and (4.11) into (4.9), and applying the Gronwall’s Lemma, we obtain

[TABLE]

(4.7) and (4.12) together imply that

[TABLE]

$\Box$

Corollary 4.1

There exists a constant $C$ independent of $m$ such that

(a) $0\leq n^{m}(t,x)$ and $c^{m}(t,x)\in[0,C_{M}]$, for all $t\geq 0$, $x\in\mathcal{O}$,

(b)

[TABLE]

and

[TABLE]

(c)

[TABLE]

(d)

[TABLE]

(e)

[TABLE]

Proof. (a) follows from the comparison theorem, see Lemma 2.1. (b) is a consequence of (4.13). (3.72) and (4.14) imply (c). By (2.7) and (4.15), we have

[TABLE]

(e) is the statement of (4.12). $\Box$

For $N\geq 1$, put

[TABLE]

By Corollary 4.1 and the Chebyshev’s inequality, we find that

[TABLE]

Lemma 4.2

We have

[TABLE]

where the constant $C_{\mu,T,N,C_{M},\|c(0)\|_{H^{1}}}$ is independent of $m$.

Proof.

By the chain rule, we have

[TABLE]

where we have used (a) of Corollary 4.1 and $u^{m}(s)\cdot\nabla c^{m}(s)=\nabla\cdot(u^{m}(s)c^{m}(s))$ (due to $\nabla\cdot u^{m}(s)=0$). Hence

[TABLE]

By the Gronwall’s lemma,

[TABLE]

Hence we have for $\omega\in\Omega_{N}^{m}$,

[TABLE]

Next we estimate $\|\nabla c^{m}(t)\|^{2}_{L^{2}}$. Again by the chain rule,

[TABLE]

Noticing

[TABLE]

it follows that

[TABLE]

By the Gagliardo-Nirenberg inequality, we have

[TABLE]

Recall also the Gagliardo-Nirenberg-Sobolev inequality:

[TABLE]

Hence we can find a constant $C>0$ such that

[TABLE]

here

[TABLE]

According to Proposition 7.2 in [15](Page 404), for any $f\in H^{2}$ satisfying the Neumann boundary condition one has

[TABLE]

By (4.24) (4.26) and (4.27), for any $\omega\in\Omega_{N}^{m}$, we have

[TABLE]

For $\omega\in\Omega_{N}^{m}$, substituting (4.28) into (4.2) we obtain

[TABLE]

Hence by Gronwall’s lemma and (4.22), it follows that

[TABLE]

Thus, in view of (4.27) (4.22) and (4.29), we can conclude that

[TABLE]

The constant $C_{\mu,T,N,C_{M},\|c(0)\|_{H^{1}}}$ is independent of $m$.$\Box$

Corollary 4.2

There exists a constant $C_{\mu,T,N,C_{M},\|c(0)\|_{H^{1}}}$ such that for all $\omega\in\Omega_{N}^{m}$,

[TABLE]

Proof. Combining (4.28) and (4.30), for $\omega\in\Omega_{N}^{m}$, we have

[TABLE]

Hence, for $\omega\in\Omega_{N}^{m}$, by (4.30), we have

[TABLE]

$\Box$

Lemma 4.3

There exists a constant $C(T,N)$ such that for all $\omega\in\Omega_{N}^{m}$ and $m\geq 1$,

[TABLE]

and

[TABLE]

Proof.

We first prove (4.33). According to the Sobolev inequalities, we have

[TABLE]

Hence, for $\varphi\in H^{3}(\mathcal{O})$,

[TABLE]

Therefore, for $\omega\in\Omega_{N}^{m}$,

[TABLE]

(4.30) has been used in the last inequality. This proves (4.33).

By the chain rule,

[TABLE]

By (4.24) and (4.25),

[TABLE]

In view of (4.25) and (4.26), we have

[TABLE]

Combining (4.36)–(4.2), and applying the Gronwall’s lemma, we obtain

[TABLE]

where

[TABLE]

From the definition of $\Omega_{N}^{m}$ and (4.30), we deduce that

[TABLE]

$\Box$

Remark 4.1

(4.2) and (4.39) imply that

[TABLE]

4.3 Existence of Martingale Weak Solutions

Definition 4.1

We say that there exists a martingale weak solution to the system (1) if there exists a stochastic basis $({\Omega},{\mathcal{F}},\{{\mathcal{F}}_{t}\}_{t\in[0,T]},{P})$ and, on this basis, a $U$- cylindrical Wiener process ${W}$, a progressively measurable process $({n},{c},{u})$ satisfying

• (1)

$P$-a.s.

[TABLE]

• (2)

For all $\psi_{1}$, $\psi_{2}\in C^{\infty}([0,T]\times\mathcal{O})$ with compact supports with respect to the space variable, and $\psi_{1}(T,\cdot)=\psi_{2}(T,\cdot)=0$, $P$-a.s.

[TABLE]

[TABLE]

• (3)

For $e\in V$, $0\leq t\leq T$,

[TABLE]

holds $P$-a.s..

Theorem 4.1

Suppose the assumptions (A)-(C) in Section 2 hold. Then, there exists a martingale weak solution to the stochastic Chemotaxis-Navier-Stokes system (1.1).

Proof. Let $(n^{m},c^{m},u^{m})$ be the solution constructed in Section 4.2. We will prove that the family $\{(n^{m},c^{m},u^{m});m\geq 1\}$ is tight in the space $L^{2}([0,T],L^{2}(\mathcal{O}))\times L^{2}([0,T],H^{1}(\mathcal{O}))\times L^{2}([0,T],H)$. To this end, it suffices to show that the families $\{n^{m};m\geq 1\}$, $\{c^{m};m\geq 1\}$, $\{u^{m};m\geq 1\}$ are respectively tight in the spaces $L^{2}([0,T],L^{2}(\mathcal{O}))$, $L^{2}([0,T],H^{1}(\mathcal{O}))$ and $L^{2}([0,T],H)$.

Define

[TABLE]

and the norm

[TABLE]

By Chapter III Theorem 2.1 in [16](Page 271) and the Kondrachov embedding theorem, it is known that the embedding of $\mathcal{Y}$ into $L^{2}([0,T],H^{1}(\mathcal{O}))$ is compact. By (4.30) and (4.31), we get that

[TABLE]

Since we can choose the integer $N$ as large as we wish, we conclude that the family $\{c^{m}\}$ is tight in $L^{2}([0,T],H^{1}(\mathcal{O}))$. Similarly, (4.35) and (4.39) imply that $\{n^{m}\}$ is tight in $L^{2}([0,T],L^{2}(\mathcal{O}))$.

Given $\kappa\in(0,1)$, let $W^{\kappa,2}([0,T],V^{*})$ be the Sobolev space of all $g\in L^{2}([0,T],V^{*})$ such that

[TABLE]

endowed with the norm

[TABLE]

Set

[TABLE]

Recall

[TABLE]

It is known (see e.g. [18]) that $B$ can be extended to a continuous operator

[TABLE]

for some $\varrho>1$.

Using the equation satisfied by $u^{m}$, applying the similar arguments as in the proof of Theorem 3.1 in [6], we can show that

[TABLE]

Recall that the embedding of $L^{2}([0,T],V)\cap W^{\kappa,2}([0,T],V^{*})$ into $L^{2}([0,T],H)$ is compact (see e.g. Theorem 2.1 in [6]). (4.41) and (4.12) imply that $\{u^{m}\}$ is tight in $L^{2}([0,T],H)$.

On the other hand, by (d) and (e) in Corollary 4.1, applying Theorem 1 in [1] and Corollary 5.2 in [8], as the proof of Lemma 4.5 in [26], we can prove that $\{u^{m}\}$ is tight in $D([0,T],D(A^{-\varrho}))$, here $\varrho$ is the constant appeared in (4.40) and $D([0,T],D(A^{-\varrho}))$ denotes the space of right continuous functions with left limits from $[0,T]$ into $D(A^{-\varrho})$ equipped with the Skorokhod topology. Moreover, since $u^{m}$ takes values in $C([0,T],H)$, Proposition 1.6 in [8] implies that $\{u^{m}\}$ is also tight in $C([0,T],D(A^{-\varrho}))$ equipped with the usual uniform topology.

Now we have proved that $\{(n^{m},c^{m},u^{m})\}$ is tight in the space:

[TABLE]

By the Skorohod embedding theorem, there exist a stochastic basis $(\widetilde{\Omega},\widetilde{\mathcal{F}},\{\widetilde{\mathcal{F}}_{t}\}_{t\in[0,T]},\widetilde{\mathbb{P}})$ and, on this basis, $\Pi$-valued random variables $(\tilde{n}^{m},\tilde{c}^{m},\tilde{u}^{m})$, $(\tilde{n},\tilde{c},\tilde{u})$ such that

• (I)

$(\tilde{n}^{m},\tilde{c}^{m},\tilde{u}^{m})$ has the same law as $(n^{m},c^{m},u^{m})$,

• (II)

$(\tilde{n}^{m},\tilde{c}^{m},\tilde{u}^{m})\rightarrow(\tilde{n},\tilde{c},\tilde{u})$ in $\Pi$, $\widetilde{\mathbb{P}}$-a.s.

By a similar argument as in the proof of Theorem 3.1 in [6], we can show that

[TABLE]

and there exists a $U$-cylindrical Wiener process $\widetilde{W}$ on the stochastic basis $(\widetilde{\Omega},\widetilde{\mathcal{F}},\{\widetilde{\mathcal{F}}_{t}\}_{t\in[0,T]},\widetilde{\mathbb{P}})$ such that $\widetilde{\mathbb{P}}$-a.s., the identity

[TABLE]

holds for all $t\in[0,T]$ and all $e\in D(A^{\varrho})$. Furthermore, it follows from (4.42) that $\widetilde{u}(\cdot)\in C([0,T],H)$ $\widetilde{\mathbb{P}}$-a.s.. This can be seen as follows. Let $\widetilde{L}(t)$ be the solution of the stochastic evolution equation:

[TABLE]

Then it is well known (see e.g. [13]) that $\widetilde{L}(\cdot,\tilde{\omega})\in C([0,T],H)\cap L^{2}([0,T],V)$ $\widetilde{\mathbb{P}}$-a.s.. On the other hand, using classical PDE arguments, Theorem 3.1 and Theorem 3.2 in [16], we can show that there exists a unique process $\widetilde{Z}\in C([0,T],H)\cap L^{2}([0,T],V)$ satisfying the random PDE:

[TABLE]

From the equation (4.42), it is easy to see that $\widetilde{u}=\widetilde{L}+\widetilde{Z}$. Hence

[TABLE]

By a density argument, it is easy to see that the identity (4.42) holds for all $e\in V$.

Using the equations satisfied by $(n^{m},c^{m},u^{m})$, we see that, for all $\psi_{1}$, $\psi_{2}\in C^{\infty}([0,T]\times\mathcal{O})$ with compact supports with respect to the space variable, and $\psi_{1}(T,\cdot)=\psi_{2}(T,\cdot)=0$,

[TABLE]

[TABLE]

Taking $m$ into $\infty$ in the above two equations, we see that $(\tilde{n},\tilde{c},\tilde{u})$ satisfies (2) in Definition 4.1.

Finally, by (a) (b) (c) (d) in Corollary 4.1, (I) (II), (4.30) and (4.44), we see that $(\tilde{n},\tilde{c},\tilde{u})$ satisfies (1) in Definition 4.1. Hence, $(\tilde{n},\tilde{c},\tilde{u})$ is a martingale weak solution. $\Box$

4.4 Pathwise Weak Solution

Theorem 4.2

Assume, in addition, that the function $\chi(\cdot)$ is a positive constant. Then there exists a unique pathwise weak solution to the stochastic Chemotaxis-Navier-Stokes system (1.1).

Proof. From Theorem 4.1, we already know that there exists a martingale weak solution to system (1.1). By the Watanable and Yamada Theorem, we will complete the proof of the theorem if we can show the pathwise uniqueness of the solutions. That is what we will do in the remaining part of the proof. Without loss of generality, we assume $\chi(\cdot)\equiv 1$.

Assume that $(n_{1},c_{1},u_{1})$ and $(n_{2},c_{2},u_{2})$ are two solutions of the system (1.1) on the same probability basis $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\in[0,T]},\mathbb{P})$, with a same $U$-valued cylindrical Wiener process $W$. We will prove that

[TABLE]

For simplicity, set

[TABLE]

By chain rule,

[TABLE]

By (4.24) and (4.25), for $I^{n}_{1}(t)$, we have for $\varepsilon>0$,

[TABLE]

By (4.25) and (4.26), for $\varepsilon>0$ we have

[TABLE]

Combining (4.45)–(4.4), we have

[TABLE]

Now we estimate $\|c^{\Delta}\|^{2}_{H^{1}}$. By the chain rule, we have

[TABLE]

and

[TABLE]

By the similar arguments as in the proof of (4.4), we have

[TABLE]

Furthermore, for $\varepsilon>0$,

[TABLE]

and

[TABLE]

By (4.27), we have

[TABLE]

Combining (4.49)–(4.54), we arrive at

[TABLE]

Since

[TABLE]

by $\rm It\hat{o}$’s formula, we have

[TABLE]

Set

[TABLE]

Choosing $\epsilon$ sufficiently small, by (4.4)(4.54) (4.55) and (4.56), we get that

[TABLE]

here

[TABLE]

By the Gronwall’ lemma, we arrive at

[TABLE]

Define

[TABLE]

Put $\tau_{N}=\tau^{N}_{1}\bigwedge\tau^{N}_{2}$. Because $(n_{i},c_{i},u_{i}),\ i=1,2$ satisfy (1) in Definition 4.1, we see that

[TABLE]

Repeating the arguments in Subsection 4.2, we can get the following result (see Remark 4.1):

[TABLE]

(4.59) and (4.60) imply that

[TABLE]

Replace $t$ by $t\wedge\tau_{N}$ in (4.58) to get

[TABLE]

By BDG inequality,

[TABLE]

By the Gronwall’s lemma, we obtain

[TABLE]

Let $N\rightarrow\infty$ to obtain

[TABLE]

which implies the uniqueness. $\Box$

Acknowledgements   This work is partly supported by National Natural Science Foundation of China (No.11671372, No.11431014, No.11401557), the Fundamental Research Funds for the Central Universities (No. WK 3470000008), and Key Research Program of Frontier Sciences CAS(No. QYZDB-SSW-SYS009).