Uniqueness & Weak-BV Stability in the Large for Isothermal Gas Dynamics

TL;DR

This paper demonstrates the stability of large-BV entropic solutions to the 1D isothermal Euler system under significant perturbations, using a novel modified front tracking method with shock contraction theory.

Contribution

It introduces a new weight construction in the front tracking algorithm applicable to large-BV solutions, extending stability analysis.

Findings

Large-BV solutions are stable under large perturbations.

The modified front tracking algorithm effectively handles large total variation.

The method applies shock contraction theory to stability analysis.

Abstract

For the $1$-d isothermal Euler system, we consider the family of entropic BV solutions with possibly large, but finite, total variation. We show that these solutions are stable with respect to large perturbations in a class of weak solutions to the system which may not even be BV. The method is based on the construction of a modified front tracking algorithm, in which the theory of $a$-contraction with shifts for shocks is used as a building block. The main contribution is to construct the weight in the modified front tracking algorithm in a large-BV setting.