The Low Mach Limit Of Solutions Of Non-Viscous Compressible Navier-Stokes Equations

Navier-Stokes equations are established on the motion state of fluid.They may be used to model the weather,ocean currents,water flow in a pipe and air flow around a wing.Under the usual conditions,the explicit expression of solutions of the Navier-Stokes equations can not be obtained,thus people usually study the asymptotic proper-ty of solutions to some given waves.The low Mach limit is one of the powerful methods.It describes the change law of the state of the fluid when the Mach number,that is,the speed of sound,goes to zero.Most of the previous studies are centered on the case with viscosity.In this paper,however,we study the non-viscous compressible Navier-Stokes equations in the case of one-dimension by using the method of energy estimate,and have proved that the solu-tion of the system converges to a nonlinear diffusion wave globally in time as the Mach number goes to zero.Furthermore,as the velocity of diffusion wave is proportional with the variation of temperature,which means the fluid would flow from low temperature to high,it is shown that the solution of the Navier-Stokes system also has the same phenomenon when Mach number is suitably small.Because the momentum equation is not parabolic in the non-viscous case,we can not use energy estimate directly,but also need the compensation matrix technique to complete the proof.