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

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.