The Existence And Uniqueness Of Global Weak Solutions And The Large Time Behavior For Two Kinds Of Incompressible Fluids Coupled With The Vlasov Equation

In this paper,we mainly discuss two kinds of fluid-particle coupled mod-els,focusing on the existence and uniqueness of global weak solutions and the large-time behavior of strong solutions.Firstly,the existence and uniqueness of global weak solutions for an incompressible non-Newtonian fluid coupled with the Vlasov equation are studied.Secondly,the large-time behavior of strong solutions to the coupled system for the incompressible Navier-Stokes equation and the Vlasov equation in a periodic domain of two dimensional spaces is es-tablished.The structure of the paper is as follows:In the first chapter,the background of fluid-particle coupled system and the research progress are introduced.In the second chapter,the basic symbols and important lemmas are given and some basic properties of stress tensor are introduced.In the third chapter,constructing the approximation system,combining the Schaulder fixed point theorem and the weak convergence method,we proved the existence and uniqueness of global weak solutions for an incompressible non-Newtonian fluid coupled with the Vlasov equation.In the fourth chapter,by constructing the Lyapunov function,it is proved that the energy of strong solutions to the coupled model for the incompressible Navier-Stokes equation and the Vlasov equation in a periodic domain of two dimensional spaces decays exponentially as time evolves.