From Navier-Stokes to BV solutions of the barotropic Euler equations

TL;DR

This paper proves that small BV solutions to the one-dimensional barotropic Euler equations can be obtained as inviscid limits of the compressible Navier-Stokes equations, advancing understanding of fluid dynamics limits.

Contribution

It establishes the inviscid limit for small BV solutions of the 1D barotropic Euler equations from Navier-Stokes solutions and proves well-posedness in a broader class.

Findings

BV solutions are inviscid limits of Navier-Stokes solutions.

Well-posedness of BV solutions in the inviscid limit context.

Extension to solutions with locally bounded energy initial data.

Abstract

In the realm of mathematical fluid dynamics, a formidable challenge lies in establishing inviscid limits from the Navier-Stokes equations to the Euler equations, wherein physically admissible solutions can be discerned. The pursuit of solving this intricate problem, particularly concerning singular solutions, persists in both compressible and incompressible scenarios. This article focuses on small $BV$ solutions to the barotropic Euler equation in one spatial dimension. Our investigation demonstrates that these solutions are inviscid limits for solutions to the associated compressible Navier-Stokes equation. Moreover, we extend our findings by establishing the well-posedness of such solutions within the broader class of inviscid limits of Navier-Stokes equations with locally bounded energy initial values.