Well-posedness of entropy-bounded solutions of the compressible Navier-Stokes equations with vacuum

Abstract: The entropy is one of the fundamental physical states of a fluid. For the ideal gases, the entropy can be expressed as some linear combination of the logarithms of the density and temperature in the non-vacuum region, and, in the viscous case, the equation that it satisfies is highly singular in the region close to the vacuum. Due to the singularity of the logarithmic function at zero, which may lead to the singularity of the entropy, and the singularity of the entropy equation near the vacuum region, 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, it was unknown if the entropy remains its boundedness. We will show in this talk 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 full compressible Navier-Stokes equations, with or without heat conductivity, and establish the local and global existence and uniqueness of entropy-bounded solutions in the presence of vacuum at the far field only. These are joint works with Prof. Zhouping Xin.

Mathematics

Audience: researchers in the topic

ICCM 2020