Global Weak Solution to the Chemotaxis-Fluid System

Abstract:

      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.