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

TL;DR

This paper introduces a novel method to establish local well-posedness for the one-dimensional compressible Navier-Stokes equations with rough initial data, utilizing smoothing and Lipschitz estimates for parabolic equations with discontinuous coefficients.

Contribution

It develops a new analytical approach based on smoothing and Lipschitz estimates to handle rough initial data in 1D compressible Navier-Stokes systems.

Findings

Proves local well-posedness for rough initial data

Establishes smoothing estimates for parabolic equations with discontinuous coefficients

Provides a framework for analyzing rough data in compressible fluid models

Abstract

This paper presents a new approach to the local well-posedness of the $1d$ compressible Navier-Stokes systems with rough initial data. Our approach is based on establishing some smoothing and Lipschitz-type estimates for the $1d$ parabolic equation with piecewise continuous coefficients.