Local-in-time well-posedness for the regular solution to the 2D full compressible Navier-Stokes equations with degenerate viscosities and heat conductivity
Yue Cao, Xun Jiang
TL;DR
The paper proves local-in-time well-posedness for the two-dimensional full compressible Navier–Stokes equations with degenerate viscosities and heat conductivity that depend on temperature, in the presence of far-field vacuum. By reformulating the system in density-entropy variables via $\phi=\frac{A\gamma}{\gamma-1}\rho^{\gamma-1}$ and $l=e^{S/c_v}$ and introducing singular-weighted estimates (notably using $h^{-1}\psi$), the authors overcome 2D embedding challenges and control nonlinear, degenerate terms. They construct regular solutions through an ε-approximation (non-vacuum) and then pass to the vacuum limit, obtaining a unique strong solution with precise regularity and weighted bounds on a short time interval. This work extends the local theory for degenerate, temperature-dependent viscous flows from 3D to 2D and provides a framework for handling vacuum and entropy dynamics in singular-weighted energy spaces. The results have significance for mathematical fluid dynamics models with physically motivated degeneracies near vacuum boundaries and non-isentropic effects.
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 $θ$ in the form of $θ^ν$ with $ν>0$. 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 $ρ$) estimates of the solution, we establish the local-in-time well-posedness of the regular solution with far-field vacuum in terms of $ρ$, the velocity $u$ and the entropy $S$.