The Limit Analysis Of The Solutions Of Two Kinds Of Fluid Mechanical Equations

In this dissertation,two kinds of fluid mechanical equations,namely compressible Navier-Stokes-Poisson equations and compressible Euler-Korteweg equations,are taken into account.The compressible Navier-Stokes-Poisson equations describe the motion-s of charged particles(e.g.electrons)in the electric field generated by electrostatic force without magnetic effect.The compressible Euler-Korteweg equations represent the phenomenon of phase change in nature,and consider the capillary effect of the re-gion with large density change,especially the interface with liquid-steam phase change.This dissertation is concerned with the zero-electron-mass limit of the global solution for the initial-boundary value problem of the two-dimensional compressible Navier-Stokes-Poisson equations in the bounded domain,the zero-electron-mass limit of the local classic solution for the initial value problem of the multi-dimensional compressible Navier-Stokes Poisson equations,and the zero-Mach-number limit of the local classic solution for the initial value problem of the three-dimensional compressible Euler-Korteweg equations.In Chapter 1,the author introduces the relevant background,research status,re-search objectives,research ideas,and necessary preliminaries for compressible Navier-Stokes-Poisson equations and compressible Euler-Korteweg equations.In Chapter 2,the author studies the zero-electron-mass limit of the global solution for the initial-boundary value problem of the two-dimensional compressible Navier-Stokes-Poisson equations in the bounded domain.First,the local existence of the solution for the initial-boundary value problem of the two-dimensional compressible Navier-Stokes-Poisson equations in the bounded domain is obtained by using Schauder fixed point the-orem.Then,the estimates of the solution for initial-boundary value problem are proved by using energy estimations,and are uniform on the time and electron mass.Finally,the global existence of the solution can be obtained by using uniformly a prior estimates,the local existence theorem,and standard continuity method.At the same time,It can be justified by uniform estimates,and compactness method that the global solution for the initial-boundary problem of the compressible Navier-Stokes-Poisson equations converges to that for the initial-boundary problem of the incompressible Navier-Stokes equations as the electron mass tends to zero.In Chapter 3,the author investigates the zero-electron-mass limit of the local clas-sic solution for the initial problem of the multi-dimensional compressible Navier-Stokes-Poisson equations in torus.First,by the ratio of the electron/ion mass,after some variable displacements,the original equations are turned into the symmetry style.Second,the uniformly a prior estimates of the solution in local time(time interval is related to dimen-sionless parameters)are obtained by using energy estimations,Sobolev space embedding theorems and Moser-type inequalities.Then,the local-in-time existence which is uniform on electron mass is proved with the help of the results on hyperbolic-parabolic system studied by Kawashima.In addition,the uniform estimates of the solution about the time derivative are constructed,and it is justified by the Aubin-Lions compactness lemma that the local classic solution for the initial problem of the compressible Navier-Stokes-Poisson equations converges to that for the initial problem of the incompressible Navier-Stokes equations as the electron mass tends to zero.In Chapter 4,the author researches the zero-Mach-number limit of the local classic solution for the initial problem of the three-dimensional compressible Euler-Korteweg equations in whole space or torus.First,by the Mach number,after some variable displacements,the original equations are turned into the symmetry style.Second,based on the local existence theorem,the convergence-stability criterion of the local classic solution for the compressible Euler-Korteweg equations is constructed.What's more,the error estimate between the solution of the compressible Euler-Korteweg equations and the one of corresponding incompressible equations is proved by using the method of energy estimation,and then it is justified that the local classic solution for the initial problem of the compressible Euler-Korteweg equations converges to that for the initial problem of the incompressible Euler equations as the Mach number tends to zero.In Chapter 5,the author talks about the general initial data,and the related research prospect is put forward.