## Global weak solution for kinetic models of active swimming and passive suspensions

**报告****题目**：Global weak solution for kinetic models of active swimming and passive suspensions

**报告人**：陈秀卿 副教授 （北京邮电大学）

**时间**：2012年9月13日(星期四)16：00-17：00

**地点**：理科楼数学系A304

**摘要**：Collaborator: Jian-Guo Liu (Department of Physics and Department of Mathematics, Duke University) Bacterium swimming, acting as a force dipole, in fluid is modeled by a coupled Fokker-Planck equation and incompressible (Navier-)Stokes equation. According to the mechanism for swimming, a bacterium can be classified into pusher and puller. The local flow generated by the pusher outward force dipole increases the local straining flow, and hence reduces the effective viscosity and enhance flow-mixture.Therefore, a kind of instability appears for pusher swimming, which has been extensively studied numerically in physics literature. This instability can be explained by the fact that there is no entropy-dissipation relation for the pusher suspensions of coupled Fokker-Planck-(Navier-)Stokes system. Nonetheless, with some careful estimates, we are able to control the entropy and obtain the existence of global weak entropy solution for both pusher and puller systems. Polymer suspensions is commonly modeled by suspension of extensible rods. There exists a spring force resisting to the rod extension. It is well known for this system that there is a relative entropy-dissipation relation with maxwellian weight. The main difficult of establishing global weak solution is the weak compactness of stress tensor exerted by the rod particles on the fluid. We will present a compactness embedding theorem. This compactness embedding theorem enables us to establish the existence of global weak solution to the coupled Fokker-Planck-(Navier-)Stokes equations for polymers. We will also prove that this compactness embedding theorem does not hold for linear spring potential, which indicating that the super-linear condition is sharp.

**联系人**：陈丽