Global regularity and sharp decay to the 2D Hypo-Viscous compressible Navier-Stokes equations

TL;DR

This paper proves global regularity and optimal decay rates for solutions to the 2D hypo-viscous compressible Navier-Stokes equations with small initial data, using advanced Fourier and Littlewood-Paley techniques.

Contribution

It establishes the existence of global strong solutions near equilibrium and derives the optimal decay rates for low regularity initial data.

Findings

Existence of global strong solutions for small initial data.

Optimal decay rates for solutions with low regularity.

Application of Fourier splitting and Littlewood-Paley methods.

Abstract

In this paper, we consider the global regularity and the optimal time decay rate for the 2D isentropic hypo-viscous compressible Navier-Stokes equations. Firstly, we prove that there exists a global strong solution with the small initial data are close to the constant equilibrium state in $H^s$ framework with $s>1$. Furthermore, by virtue of improved Fourier splitting method and the Littlewood-Paley decomposition theory, we then establish the optimal time decay rate for low regularity data.