Stability Of Rarefaction Waves Of One-dimensional Isentropic Compressible Navier-Stokes-Maxwell Equations

Navier-Stokes-Maxwell equations is mainly used to describe the motion of charged fluids in electromagnetic fields.The equations has a wide range of applications in the fields of aerodynamics,geophysics,astrophysics,etc..In this paper,we mainly study the nonlinear stability of the rarefaction waves for the Cauchy problem of one-dimensional isentropic compressible Navier-Stokes-Maxwell equations.It is expected that the large-time asymptotic behavior of the solution of the Navier-Stokes-Maxwell equations is con-sistent with the Navier-Stokes equations when b₊=b_-=0 and the dielectric constant is bounded,that is,the solution of Navier-Stokes-Maxwell equations(?,u,E,b)converges to the Riemannian solution of Euler equations(?^R,u^R,0,0).It can be proved that the so-lution of Cauchy problem of one-dimensional isentropic Navier-Stokes-Maxwell equations is asymptotically stable with respect to time under the initial perturbation of rarefaction waves is small.In the process of proof,since Maxwell equations has no dissipative structure for the magnetic field b,it is impossible to directly deal with the non-linear bad term caused by the coupling of the fluid and the electromagnetic field.We fully excavate equations of structure,found on the basis of the nonlinear item bad good item:?₀^t(||?+?b+(?)b||)²d? and get estimate about it to solve the problem.This article is divided into the following three chapters:In the first chapter,we introduce the problems to be studied and the related back-ground,summarize the main work of this paper and give the main theorem,and finally analyze the key points and difficulties encountered in the research process.In the second chapter,we explain the symbols,notations and some common inequal-ities used in this article,and then introduce the properties of rarefaction waves for the Euler equations and its smooth approximation wave,paving the way for the following proof.In the third chapter,under the condition of ?<(?)((?)depending only on,?,?_±,u_±),the problem is transformed into a Cauchy problem of perturbation equations,and then a priori hypothesis is proposed and proved by using the basic L²-energy method.Thus,the stability of rarefaction waves for the Cauchy problem of one-dimensional isentropic Navier-Stokes-Maxwell equations is obtained.