Local-in-time well-posedness for the regular solution to the 2D full compressible Navier-Stokes equations with degenerate viscosities and heat conductivity

Abstract

This paper considers the two-dimensional Cauchy problem of the full compressible Navier-Stokes equations with far-field vacuum in $\mathbb{R}^2$, where the viscosity and heat-conductivity coefficients depend on the absolute temperature $\theta$ in the form of $\theta^\nu$ with $\nu>0$. Due to the appearance of the vacuum, the momentum equation are both degenerate in the time evolution and spatial dissipation, which makes the study on the well-posedness challenged. By establishing some new singular-weighted (negative powers of the density $\rho$) estimates of the solution, we establish the local-in-time well-posedness of the regular solution with far-field vacuum in terms of $\rho$, the velocity $u$ and the entropy $S$.