2025/02/19 by Shyam Sundar Ghoshal, Parasuram Venkatesh, Ghoshal, Shyam Sundar +1 · 1 citation
Mathematics · Engineering · #Navier-Stokes equation solutions #Computational Fluid Dynamics and Aerodynamics #Gas Dynamics and Kinetic Theory
paper · pdf · doi:10.48550/arxiv.2502.13687
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∞ initial data satisfying the same far-field conditions. Under a further condition on the mixed partial derivative of the flux, we establish the L2-stability of these simple shock profiles: perturbations of the Riemann initial data yield solutions whose L2 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.