Global strong solutions of 3D Compressible Navier-Stokes equations with short pulse type initial data

TL;DR

This paper proves the global existence and behavior of solutions to 3D compressible Navier-Stokes equations with short pulse initial data, showing the initial density bump dissipates quickly and solutions remain bounded.

Contribution

It introduces a novel analysis for large amplitude initial density in compressible Navier-Stokes equations with short pulse data, establishing global well-posedness and decay estimates.

Findings

Large initial density amplitude dissipates rapidly

Solutions are globally well-posed and bounded

Gradient of velocity remains integrable over time

Abstract

Short pulse initial datum is referred to the one supported in the ball of radius $\delta$ and with amplitude $\delta^{\frac12}$ which looks like a pulse. It was first introduced by Christodoulou to prove the formation of black holes for Einstein equations and also to catch the shock formation for compressible Euler equations. The aim of this article is to consider the same type initial data, which allow the density of the fluid to have large amplitude $\delta^{-\frac{\alpha}{\gamma}}$ with $\delta\in(0,1],$ for the compressible Navier-Stokes equations. We prove the global well-posedness and show that the initial bump region of the density with large amplitude will disappear within a very short time. As a consequence, we obtain the global dynamic behavior of the solutions and the boundedness of $\|\nabla u\|_{L^1([0,\infty);L^\infty)}$. The key ingredients of the proof lie in the new observations for the effective viscous flux and new decay estimates for the density via the Lagrangian coordinate.