$L^2$ Stability of Simple Shocks for Spatially Heterogeneous Conservation Laws

In this paper, we consider scalar conservation laws with smoothly varying spatially heterogeneous flux that is convex in the conserved variable.

We identify structural assumptions under which a single shock wave connecting two constant states emerges in finite time for all $L^{\infty}$ initial data satisfying the same far-field conditions.

Under a further condition on the mixed partial derivative of the flux, we establish the $L^2$-stability of these simple shock profiles: perturbations of the Riemann initial data yield solutions whose $L^2$ distance from the corresponding simple shock wave is non-increasing in time, up to a time-dependent spatial shift.

We further show that these conditions are sharp: we construct explicit counterexamples demonstrating that, in particular, the existence of a contractive shift function holds only under our assumptions and fails otherwise.

The main tools we use are Dafermos' generalised characteristics for the evolution analysis and the relative entropy method for stability.