Existence of global strong solution for the compressible Navier-Stokes system and the Korteweg system in two-dimension

TL;DR

This paper proves the global existence of strong solutions for 2D compressible Navier-Stokes and Korteweg systems with large initial data, extending previous results by weakening viscosity conditions and employing advanced commutator estimates.

Contribution

It extends known results on global strong solutions by relaxing viscosity coefficient conditions and applying new analytical techniques for the 2D compressible fluid systems.

Findings

Global strong solutions exist for 2D compressible Navier-Stokes with β>2.

Existence of solutions for Korteweg system with degenerate viscosity and friction.

Improved conditions on viscosity coefficients compared to previous studies.

Abstract

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N=2. We address the question of the global existence of strong solutions with large initial data for compressible Navier-Stokes system and Korteweg system. In the first case we are interested by slightly extending a famous result due to V. A. Vaigant and A. V. Kazhikhov in \cite{VG} concerning the existence of global strong solution in dimension two for a suitable choice of viscosity coefficient ($\mu(\rho)=\mu>0$ and $\lambda(\rho)=\lambda\rho^{\beta}$ with $\beta>3$) in the torus. We are going to weaken the condition on $\beta$ by assuming only $\beta>2$ essentially by taking profit of commutator estimates introduced by Coifman et al in \cite{4M} and using a notion of \textit{effective velocity} as in \cite{VG}. In the second case we study the existence of global strong solution with large initial data in the sense of the scaling of the equations for Korteweg system with degenerate viscosity coefficient and with friction term.