# Global well-posedness of non-heat conductive compressible Navier-Stokes equations in 1D

## TL;DR

This paper proves the global existence and uniqueness of solutions for the 1D compressible Navier-Stokes equations without heat conduction, allowing initial vacuum and nonnegative density, thus extending previous heat-conductive results.

## Contribution

It establishes the global well-posedness for the 1D full compressible Navier-Stokes equations with zero heat conductivity, generalizing prior results to include initial vacuum and non-heat conductive cases.

## Key findings

- Global well-posedness for any $H^1$ initial data.
- Initial vacuum is permitted, density need not be uniformly positive.
- Extension of Kazhikhov--Shelukhin's result to non-heat conductive case.

## Abstract

In this paper, the initial-boundary value problem of the 1D full compressible Navier-Stokes equations with positive constant viscosity but with zero heat conductivity is considered. Global well-posedness is established for any $H^1$ initial data. The initial density is required to be nonnegative, which is not necessary to be uniformly away from vacuum. This not only generalizes the well-known result of Kazhikhov--Shelukhin (Kazhikhov, A.~V.; Shelukhin, V.~V.: \emph{Unique global solution with respect to time of initial boundary value problems for one-dimensional equations of a viscous gas}, J.\,Appl.\,Math.\,Mech., \bf41 \rm(1977), 273--282.) from the heat conductive case to the non-heat conductive case, and the initial vacuum is allowed.