Diffusive wave in the singular limit for the relaxed compressible Navier-Stokes equations with Maxwell's law

In this paper, we investigate the combined low Mach number and relaxation time limits for one-dimensional full compressible Navier-Stokes equations with Maxwell's law, where the density and temperature have different asymptotic states at far fields. The problems are considered for both well-prepared and ill-prepared initial data. It is proved that, for the well-prepared initial data, as Mach number and relaxation time tend to zero simultaneously and the difference between the states at infinity is suitably small, the solution to the relaxed compressible system converges globally in time to a nonlinear diffusion wave of which the velocity is proportional with the variation of temperature. Furthermore, it is shown that the solution to the relaxed compressible system also has the same phenomenon when Mach number and relaxation time are suitably small. For the ill-prepared initial data, the difference between the states at infinity can be arbitrary large.