The Stability Of Nonlinear Waves To The Compressible Non-isentropic Navier-Stokes-Maxwell Equations

This paper mainly considers the compressible nonisentropic Navier-Stokes-Maxwell equations,which are obtained from the Navier-Stokes equations coupling with the Maxwell equations through the Lorentz force.It is a basic model to describe the motion of conductive fluid(gas)under the interaction of electric field and magnetic field,and has a very wide range of physical and engineering background.The mathematical theory of Navier-Stokes-Maxwell equations is one of the research hotspots in the field of partial differential equations,and there are still many unsolved problems on this subject.Among them,there are few results about the large-time behavior of the solution toward some nonconstant states,especially the stability of nonlinear waves.This paper mainly studies the stability of related waves to the Cauchy problem and outflow problem of onedimensional compressible nonisentropic Navier-Stokes-Maxwell equations.Specifically,the main contents of this paper are as follows:·In Chapter Two,we study the global solution to the Cauchy problem for compressible nonisentropic Navier-Stokes-Maxwell equations can converge to the rarefaction wave,under the assumption of some smallness conditions and bounded dielectric coefficient.The dissipative structure of Navier-Stokes-Maxwell equations is more complex than that of compressible Navier-Stokes equations,because the effect of electromagnetic fields should also be considered on the basis of dealing with the hyperbolic-parabolic system of fluid part.We try to use the structure of the Maxwell equations and package extra terms together with the bad term to produce a compound time-space integrable good term ∫0t∫R(E+ub)2 dxdτ,so as to overcome the difficulty caused by the loss of the regularity for time-space integrability of electric field and magnetic field.·In Chapter Three,we study the large-time-asymptotic behavior of the global solution to the Cauchy problem for compressible nonisentropic Navier-Stokes-Maxwell equations toward the composite wave of viscous contact wave and two rarefaction waves.Based on a new observation of the specific structure of the Maxwell equation in the Lagrangian coordinates,we prove that for the Navier-Stokes-Maxwell equations this typical composite wave pattern is time-asymptotically stable under some smallness conditions on the initial perturbations and wave strength,and also under the assumption that the dielectric constant is bounded.The proof of the main result is based on the weighted energy inequality of electromagnetic fields and elementary L2 energy methods.This is the first result about the nonlinear stability of the combination of two different wave patterns for the compressible nonisentropic Navier-Stokes-Maxwell equations.·In Chapter Four,we investigate the large-time-asymptotic behavior of the global solution toward the combination of the boundary layer and 3-rarefaction wave to the outflow problem for the compressible nonisentropic Navier-Stokes-Maxwell equations in the quarter plane R+×[0,+∞).We make full use of the special structure of the Maxwell equation and the composite boundary condition of the electromagnetic fields to obtain some boundary good terms.It is proved that this typical composite wave pattern is time-asymptotically stable under some smallness conditions and the assumption that the dielectric coefficient is bounded.This result can be viewed as the first result about the nonlinear stability of the combination of two different wave patterns for the IBVP of nonisentropic Navier-Stokes-Maxwell equations.