Global Weak Solution to the Chemotaxis-Fluid System
Received:May 05, 2018  Revised:August 01, 2018
Key Word: Chemotaxis-fluid system   logistic source   global solution
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11701399).
 Author Name Affiliation Mei LIU College of Mathematics and Software Science, Sichuan Normal University, Sichuan 610066, P. R. China Mengling YU College of Mathematics and Software Science, Sichuan Normal University, Sichuan 610066, P. R. China Hong LUO College of Mathematics and Software Science, Sichuan Normal University, Sichuan 610066, P. R. China
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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.