Global well-posedness of the 1d compressible Navier-Stokes system with rough data

TL;DR

This paper proves the global well-posedness of the 1D compressible Navier-Stokes system with rough initial data, extending previous results to broader function spaces using endpoint smoothing estimates.

Contribution

It improves existing global well-posedness results for 1D compressible Navier-Stokes equations by considering initial data in more general and rougher function spaces.

Findings

Established global well-posedness for initial velocity in $W^{2 heta,1}$ space.

Extended results to initial velocity in $L^2 \cap W^{2 heta,1}$ and temperature data in specific Besov spaces.

Developed endpoint smoothing estimates for 1D parabolic equations.

Abstract

In this paper, we study the global well-posedness problem for the 1d compressible Navier-Stokers system (cNSE) in gas dynamics with rough initial data. Frist, Liu- Yu (2022) established the global well-posedness theory for the 1d isentropic cNSE with initial velocity data in BV space. Then, it was extended to the 1d cNSE for the polytropic ideal gas with initial velocity and temperature data in BV space by Wang-Yu-Zhang (2022). We improve the global well-posedness result of Liu-Yu with initial velocity data in $W^{2\gamma,1}$ space; and of Wang-Yu-Zhang with initial velocity data in $ L^2\cap W^{2\gamma,1}$ space and initial data of temperature in $\dot W^{-\frac{2}{3},\frac{6}{5}}\cap \dot W^{2\gamma-1,1}$ for any $\gamma>0$ \textit{arbitrary small}. Our essential ideas are based on establishing various "end-point" smoothing estimates for the 1d parabolic equation.