Hydrodynamic Limit For 1D Compressible Navier-Stokes-Vlasov Equations

In this paper,we consider the hydrodynamic limit of weak solutions to the compressible Navier-Stokes-Vlasov equations without the diffusive term?or the Brownian effect?in one dimension.This work extends in some sense the previ-ous work,[Mellet and Vasseur?2008?],which provided the hydrodynamic limit of weak solutions to the Vlasov-Fokker-Planck/compressible Navier-Stokes systems in three dimensions with₂³<<2?where the pressure??=A?.In this s-tudy,we improve the results by taking advantage of the one space dimension.By using the Sobolev embedding inequality₀¹???????????₀¹?????^?????,we obtain the hydrodynamic limit for 1compressible Navier-Stokes-Vlasov e-quations based on the relative entropy method and weak convergence argument.Furthermore,we obtain the strong convergence of corresponding physical quan-tities.In Chapter 1,we introduce some related backgrounds about the existence results of weak solutions and hydrodynamic limit of weak solutions to the fluid-particle model both at home and abroad;In Chapter 2,we combine the conservative theory with the relative entropy theory to derive the relative entropy inequality.Next,we use the basic energy estimate and related technique to get the relative entropy estimate.Finally,by passing to the limit as?0,we obtain the corresponding strong convergence and get the limit equation.In Chapter 3,we give some basic facts in the Appendix.