On Existence and Blowup Criterion of Strong Solutions to Cauchy Problem of 2D Full Compressible Navier-Stokes System with Vacuum

TL;DR

This paper establishes local existence, uniqueness, and a blowup criterion for strong solutions to the 2D full compressible Navier-Stokes equations with vacuum, using weighted estimates and approximation schemes.

Contribution

It introduces new weighted a priori estimates and a blowup criterion independent of temperature for the 2D compressible Navier-Stokes system with vacuum.

Findings

Established local existence and uniqueness of strong solutions with vacuum.

Derived a blowup criterion based solely on divergence of velocity.

Obtained new weighted estimates for velocity and temperature.

Abstract

This paper investigates the Cauchy problem of two-dimensional full compressible Navier-Stokes system with density and temperature vanishing at infinity. For the strong solutions, some a priori weighted $L^2(R^2)$-norm of the gradient of velocity is obtained by the techniques of $A_p$ weights and cancellation of singularity. Based on this key weighted estimate and some basic weighted analysis of velocity and temperature, we establish the local existence and uniqueness of strong solutions with initial vacuum by means of a two-level approximation scheme. Meanwhile, for $q_1$ and $q_2$ as in (1.8), the $L^\infty_tL^{q_1}_x$-norm of velocity and the $L^2_tL^{q_2}_x$-norm of temperature are derived originally. Moreover, we obtain a blowup criterion only in terms of the temporal integral of the maximum norm of divergence of velocity, which is independent of the temperature.