Global weak solution to the chemotaxis-fluid system
Received:May 05, 2018  Revised:September 11, 2018
Key Word: Chemotaxis-fluid system   Logistic source   Global solution.
Fund ProjectL:National Natural Science Foundation of China(11701399)and NSF of Sichuan Science and Technology Department(2015JY0125),
 Author Name Affiliation E-mail Mei Liu Sichuan normal university 1256996637@qq.com Mengling Yu Sichuan normal university 1938070290@qq.com Hong Luo Sichuan normal university lhscnu@163.com
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We investigate the existence of the global weak solution to the coupled Chemotaxis-fluid system \begin{eqnarray*} \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&gt;0, \c_{t}+u\cdot\nabla c=\triangle c+n-c,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in \Omega,t&gt;0, \u_{t}+\nabla P=\triangle u+n\nabla \phi+g(x,t),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in \Omega,t&gt;0, \\nabla\cdot u=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in \Omega,t&gt;0, \end{array} \right. \end{eqnarray*} in a bounded smooth domain $\Omega\subset \mathds{R}^{2}$. Here, $r\geq 0$ and $\mu&gt;0$ 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.