Global regularity and sharp decay rates to the 1D hypo-viscous compressible Navier-Stokes equations

TL;DR

This paper establishes global regularity and sharp decay rates for the 1D hypo-viscous compressible Navier-Stokes equations, revealing new stability results and decay phenomena under minimal initial data assumptions.

Contribution

It proves global stability for small initial data near equilibrium, establishes critical regularity in Sobolev spaces, and derives optimal decay rates, including decay without $L^2$ smallness.

Findings

Global stability for small initial data near equilibrium

Optimal decay rates under low-frequency smallness assumptions

Decay of $L^2$ norm even without initial $L^2$ smallness

Abstract

In this paper, we study the global regularity and sharp decay rates for the isentropic hypo-viscous compressible Navier-Stokes equations in 1D. Firstly, we prove the global stability for the small initial data near a stable equilibrium. Especially, we establish the global critical regularity in the Sobolev space $H^{\beta}$ with $\frac{1}{2}<\beta<1$. Furthermore, by bootstrap argument, Fourier splitting method and energy method, we then establish the optimal time decay rates under the extra low-frequency smallness assumption. We find the $L^2$ energy is self-closed, which motivates us to obtain the existence of global large solutions for initial data with high regularity. By a pure energy method, we also derive the optimal time decay rates when $\frac{1}{2}\le\beta<\frac{3}{4}$. We find a phenomenon that $\|(a,u)\|_{L^2}$ still decays even if the initial data does not possess $L^2$ smallness. Notably, the low-frequency smallness assumption is removed in the case with $\frac{1}{2}\le\beta<\frac{3}{4}$.