Entropy-bounded solutions to the compressible Navier-Stokes equations: with far field vacuum

TL;DR

This paper proves that ideal gases can maintain bounded entropy near vacuum regions in the compressible Navier-Stokes equations, given certain decay conditions at the far field, and establishes local and global existence of such solutions.

Contribution

It introduces new mathematical results demonstrating entropy boundedness and solution existence for the compressible Navier-Stokes equations with far field vacuum and slowly decaying initial density.

Findings

Entropy remains bounded near vacuum at the far field.

Existence and uniqueness of entropy-bounded solutions are established.

Regularities propagate in inhomogeneous Sobolev spaces with slowly decaying initial density.

Abstract

The entropy is one of the fundamental states of a fluid and, in the viscous case, the equation that it satisfies is highly singular in the region close to the vacuum. In spite of its importance in the gas dynamics, the mathematical analyses on the behavior of the entropy near the vacuum region, were rarely carried out; in particular, in the presence of vacuum, either at the far field or at some isolated interior points, it was unknown if the entropy remains its boundedness. The results obtained in this paper indicate that the ideal gases retain their uniform boundedness of the entropy, locally or globally in time, if the vacuum occurs at the far field only and the density decays slowly enough at the far field. Precisely, we consider the Cauchy problem to the one-dimensional full compressible Navier-Stokes equations without heat conduction, and establish the local and global existence and uniqueness of entropy-bounded solutions, in the presence of vacuum at the far field only. It is also shown that, different from the case that with compactly supported initial density, the compressible Navier-Stokes equations, with slowly decaying initial density, can propagate the regularities in the inhomogeneous Sobolev spaces.