Diffusive Wave in the Low Mach Limit for Compressible Navier-Stokes Equations

TL;DR

This paper rigorously justifies the low Mach limit for 1D compressible Navier-Stokes equations with different asymptotic states, showing convergence to a nonlinear diffusion wave and revealing new properties of the solutions as Mach number approaches zero.

Contribution

It establishes the convergence of solutions to a nonlinear diffusion wave in the low Mach limit for both well-prepared and ill-prepared data, including new observations about velocity driven by temperature variation.

Findings

Solutions converge to a nonlinear diffusion wave as Mach number approaches zero.

Velocity of diffusion wave is driven solely by temperature variation.

Convergence rates are obtained for well-prepared data.

Abstract

The low Mach limit for 1D non-isentropic compressible Navier-Stokes flow, whose density and temperature have different asymptotic states at infinity, is rigorously justified. The problems are considered on both well-prepared and ill-prepared data. For the well-prepared data, the solutions of compressible Navier-Stokes equations are shown to converge to a nonlinear diffusion wave solution globally in time as Mach number goes to zero when the difference between the states at $\pm\infty$ is suitably small. In particular, the velocity of diffusion wave is only driven by the variation of temperature. It is further shown that the solution of compressible Navier-Stokes system also has the same property when Mach number is small, which has never been observed before. The convergence rates on both Mach number and time are also obtained for the well-prepared data. For the ill-prepared data, the limit relies on the uniform estimates including weighted time derivatives and an extended convergence lemma. And the difference between the states at $\pm\infty$ can be arbitrary large in this case.