Zero dissipation limit of full compressible Navier-Stokes equations with Riemann initial data

TL;DR

This paper proves that solutions to the full compressible Navier-Stokes equations with Riemann initial data converge to the corresponding Euler solutions, composed of rarefaction waves and contact discontinuity, as viscosity and heat conductivity vanish.

Contribution

It establishes the zero dissipation limit for the full compressible Navier-Stokes equations with complex wave interactions and proves convergence to the Euler Riemann solution.

Findings

Existence of a unique global smooth solution for small viscosity and heat conductivity.

Uniform convergence of Navier-Stokes solutions to Euler Riemann solutions as viscosity tends to zero.

Validation of the zero dissipation limit in the presence of multiple wave interactions.

Abstract

We consider the zero dissipation limit of the full compressible Navier-Stokes equations with Riemann initial data in the case of superposition of two rarefaction waves and a contact discontinuity. It is proved that for any suitably small viscosity $\varepsilon$ and heat conductivity $\kappa$ satisfying the relation \eqref{viscosity}, there exists a unique global piecewise smooth solution to the compressible Navier-Stokes equations. Moreover, as the viscosity $\varepsilon$ tends to zero, the Navier-Stokes solution converges uniformly to the Riemann solution of superposition of two rarefaction waves and a contact discontinuity to the corresponding Euler equations with the same Riemann initial data away from the initial line $t=0$ and the contact discontinuity located at $x=0$.