Vanishing viscosity limit to the planar rarefaction wave with vacuum for 3-D full compressible Navier-Stokes equations with temperature-dependent transport coefficients

TL;DR

This paper constructs global solutions to 3-D compressible Navier-Stokes equations with temperature-dependent coefficients that converge to planar rarefaction waves with vacuum as viscosity vanishes, including periodic perturbations and degeneracies.

Contribution

It introduces a novel approach to handle temperature-dependent viscosity and heat conduction with vacuum degeneracies in 3-D Navier-Stokes equations, establishing convergence and decay rates.

Findings

Solutions converge to rarefaction waves as viscosity vanishes.

Established convergence rates concerning viscosity and heat conductivity.

Proved exponential decay for non-zero modes of solutions.

Abstract

In this paper, we construct a family of global-in-time solutions of the 3-D full compressible Navier-Stokes (N-S) equations with temperature-dependent transport coefficients (including viscosity and heat-conductivity), and show that at arbitrary times {and arbitrary strength} this family of solutions converges to planar rarefaction waves connected to the vacuum as the viscosity vanishes in the sense of $L^\infty(\R^3)$. We consider the Cauchy problem in $\R^3$ with perturbations of the infinite global norm, particularly, periodic perturbations. To deal with the infinite oscillation, we construct a suitable ansatz carrying this periodic oscillation such that the difference between the solution and the ansatz belongs to some Sobolev space and thus the energy method is feasible. The novelty of this paper is that the viscosity and heat-conductivity are temperature-dependent and degeneracies caused by vacuum. Thus the a priori assumptions and two Gagliardo-Nirenberg type inequalities are essentially used. Next, more careful energy estimates are carried out in this paper, by studying the zero and non-zero modes of the solutions, we obtain not only the convergence rate concerning the viscosity and heat conductivity coefficients but also the exponential time decay rate for the non-zero mode.