Solutions to conservation laws are H\"older-stable in $L^2$ in the weak-BV setting

TL;DR

This paper establishes a novel H"older-stability estimate in $L^2$ for solutions to hyperbolic conservation laws, including those with large BV norms, extending stability and uniqueness results to physically relevant systems like isentropic Euler.

Contribution

It proves the first $L^2$ H"older stability estimate for weak solutions with large BV data, independent of the solution's BV norm, and demonstrates uniqueness for solutions with large initial data.

Findings

H"older stability estimate in $L^2$ for weak solutions with large BV data

Applicability to physical systems like isentropic Euler

First stability result that ensures uniqueness with large initial data

Abstract

We consider hyperbolic systems of conservation laws in one spatial dimension. For any limit of front tracking solutions $v$, and for a general weak solution $u\in L^\infty$ with no BV assumption, we prove the following H\"older-type stability estimate in $L^2$: $$||u(\cdot,\tau)-v(\cdot,\tau)||_{L^2} \leq K \sqrt{||u( \cdot,0)-v( \cdot,0)||_{L^2}}$$ for all $\tau$ without smallness and for a universal constant $K$. Our result holds for all limits of front tracking solutions $v$ with BV bound, either for general systems with small-BV data, or for special systems (isothermal Euler, Temple-class systems) with large-BV data. Our results apply to physical systems such as isentropic Euler. The stability estimate is completely independent of the BV norm of the potentially very wild solution $u$. We use the $L^2$ theory of shock stability modulo an artificial shift of position (Vasseur [Handbook of Differential Equations: Evolutionary Equations, 4:323 -- 376, 2008]) but our stability results do not depend on an artificial shift. Moreover, we give the first result within this framework which can show uniqueness of some solutions with large $L^\infty$ and infinite BV initial data. We apply these techniques to isothermal Euler.