Chemotaxis-fluild 系统的全局弱解

Global Weak Solution to the Chemotaxis-Fluid System

中文关键词:  Chemotaxis-fluid系统  逻辑增长  全局解

英文关键词:Chemotaxis-fluid system  logistic source  global solution

中文摘要:

      我们研究Chemotaxis-fluid 系统$$\left\{ \begin{array}{ll}n_{t}+u\cdot\nabla n=\triangle n-\nabla\cdot(n\nabla c)+rn-\mu n^{2}, &x\in \Omega,t>0, \\ c_{t}+u\cdot\nabla c=\triangle c+n-c, &x\in \Omega,t>0,\\ u_{t}+\nabla P=\triangle u+n\nabla \phi+g(x,t), &x\in \Omega,t>0,\\ \nabla\cdot u=0, &x\in \Omega,t>0,\end{array}\right.$$ 在光滑有界区域$\Omega\subset \mathds{R}^{2}$中全局弱解的存在性. 其中,$r\geq 0$和$\mu>0$是已知常数,$\nabla\phi\in L^{\infty}(\Omega)$和$g\in L^{2}((0,T);L^{2}_{\sigma}(\Omega))$是已知函数. 首先通过Schauder不动点定理我们得到该系统弱解的局部存在性. 进一步研究该系统的正则性估计. 再结合正则性估计, 我们得到耦合的Chemotaxis-fluid系统的初边值问题存在一个全局弱解.

英文摘要:

      We investigate the existence of the global weak solution to the coupled Chemotaxis-fluid system $$\left\{ \begin{array}{ll}n_{t}+u\cdot\nabla n=\triangle n-\nabla\cdot(n\nabla c)+rn-\mu n^{2}, &x\in \Omega,t>0, \\ c_{t}+u\cdot\nabla c=\triangle c+n-c, &x\in \Omega,t>0,\\ u_{t}+\nabla P=\triangle u+n\nabla \phi+g(x,t), &x\in \Omega,t>0,\\ \nabla\cdot u=0, &x\in \Omega,t>0,\end{array}\right.$$ in a bounded smooth domain $\Omega\subset \mathds{R}^{2}$. Here, $r\geq 0$ and $\mu>0$ are given constants, $\nabla\phi\in L^{\infty}(\Omega)$ and $g\in L^{2}((0,T);L^{2}_{\sigma}(\Omega))$ are prescribed functions. We obtain the local existence of the weak solution of the system by using the Schauder fixed point theorem. Furthermore, we study the regularity estimate of this system. Utilizing the regularity estimates, we obtain that the coupled Chemotaxis-fluid system with the initial-boundary value problem possesses a global weak solution.