Existence and Blowup Behavior of Global Strong Solutions to the Two-Dimensional Baratropic Compressible Navier-Stokes System with Vacuum and Large Initial Data

TL;DR

This paper proves the global existence and analyzes the blowup behavior of strong solutions to the 2D compressible Navier-Stokes equations with vacuum and large initial data, under specific viscosity conditions.

Contribution

It extends previous results by allowing a broader range of the bulk viscosity coefficient, specifically for b2 > 4/3, and establishes uniform density bounds and long-term solution behavior.

Findings

Global strong and weak solutions exist for b2 > 4/3.

Uniform upper bounds for the density are obtained.

Long-time behavior of solutions is characterized.

Abstract

For periodic initial data with initial density allowed to vanish, we establish the global existence of strong and weak solutions for the two-dimensional compressible Navier-Stokes equations with no restrictions on the size of initial data provided the bulk viscosity coefficient is $\lambda = \rho^{\beta}$ with $\beta>4/3$. These results generalize and improve the previous ones due to Vaigant-Kazhikhov([Sib. Math. J. (1995), 36(6), 1283-1316]) which requires $\beta>3$. Moreover, both the uniform upper bound of the density and the large-time behavior of the strong and weak solutions are also obtained.