Vanishing Pressure And Magnetic Field Limit For Magnetogasdynamics System

Magnetogasdynamics system is a kind of important Euler equations for compressible fluid,which can be used to describe the motion of compressible fluid with transverse magnetic field.When the adiabatic exponent in the pressure is different,accordingly,polytropic gas Euler equations with magnetic field and Chaplygin gas Euler equations with magnetic field can be obtained.Based on the introduction of Riemann solutions of a class of transport equations and a class of Chaplygin gas equations including deltashock and vacuum.Firstly,the Riemann problem of these two types of Euler equations is solved by using the characteristic analysis method and the analysis method in phase plane.Secondly,the limit behavior of Riemann solutions is studied when the pressure and magnetic field vanish.For polytropic gas Euler equations with magnetic field,according to solving the Riemann problem of the system,the Riemann solutions containing five different structures are constructed.Furthermore,as the pressure and magnetic field vanish,we rigorously prove that the Riemann solution containing two shock waves tends to a delta-shock solution of the transport equations.The Riemann solution including two rarefaction waves tends to the a vacuum solution of the transport equations,and the nonvacuum intermediate state between the rarefaction waves tends to a vacuum state.Finally,we produce some numerical results to examine formation process of delta-shocks and vacuum states as pressure and magnetic field vanish.For Chaplygin gas Euler equations with magnetic field,the Riemann problem is firstly solved,and the Riemann solutions containing four different structures are constructed.Secondly,we show that when both the pressure and magnetic field disappear,the Riemann solution including two shock waves tends to a delta-shock solution of the transport equations,the Riemann solution including two rarefaction waves tends to the a vacuum solution of the transport equations.Then,we rigorously analyze that,when only the magnetic field vanishes,the Riemann solution including two shock waves approaches the a delta-shock solution of the Chaplygin gas equations,the Riemann solution containing two rarefaction waves tends to the solution containing two contact discontinuities of the Chaplygin gas equations.Finally,the numerical results obtained are consistent with the theoretical analysis.