On regular solutions for three-dimensional full compressible Navier-Stokes equations with degenerate viscosities and far field vacuum

TL;DR

This paper establishes the local existence of regular solutions to the 3D full compressible Navier-Stokes equations with temperature-dependent viscosities and vacuum at infinity, overcoming challenges posed by degeneracies and singularities.

Contribution

It introduces a reformulated approach and weighted energy estimates to prove local well-posedness for a class of initial data with vacuum in a complex degenerate setting.

Findings

Existence of local-in-time regular solutions with vacuum

Velocity remains in H^3 Sobolev space during the solution lifespan

Conservation laws of mass, momentum, and energy are satisfied

Abstract

In this paper, the Cauchy problem for the three-dimensional (3-D) full compressible Navier-Stokes equations (CNS) with zero thermal conductivity is considered. First, when shear and bulk viscosity coefficients both depend on the absolute temperature $\theta$ in a power law ($\theta^\nu$ with $\nu>0$) of Chapman-Enskog, based on some elaborate analysis of this system's intrinsic singular structures, we identify one class of initial data admitting a local-in-time regular solution with far field vacuum in terms of the mass density $\rho$, velocity $u$ and entropy $S$. Furthermore, it is shown that within its life span of such a regular solution, the velocity stays in an inhomogeneous Sobolev space, i.e., $u\in H^3(\mathbb{R}^3)$, $S$ has uniformly finite lower and upper bounds in the whole space, and the laws of conservation of total mass, momentum and total energy are all satisfied. Note that due to the appearance of the vacuum, the momentum equations are degenerate both in the time evolution and viscous stress tensor, and the physical entropy for polytropic gases behaves singularly, which make the study on corresponding well-posedness challenging. For proving the existence, we first introduce an enlarged reformulated structure by considering some new variables, which can transfer the degeneracies of the full CNS to the possible singularities of some special source terms related with $S$, and then carry out some singularly weighted energy estimates carefully designed for this reformulated system.