Small dispersion approximation of shock wave dynamics

TL;DR

This paper presents a dispersion approximation method for multidimensional scalar conservation laws, demonstrating how small and large scale discontinuities propagate and converge to classical solutions as the scale parameter approaches zero.

Contribution

It introduces a novel variational kinetic representation for weak entropy solutions, linking small scale discontinuity propagation with classical shock wave speeds.

Findings

Small scale discontinuities propagate with characteristic velocities.

Large scale shocks propagate near classical shock speeds.

Approximate solutions converge to unique entropy solutions as scale tends to zero.

Abstract

We introduce a dispersion approximation of weak, entropy solutions of multidimensional scalar conservation laws using variational kinetic representation, where equilibrium densities satisfy the Gibb's entropy minimization principle for a piecewise linear, convex entropy. For such solutions, we show that small scale discontinuities, measured by the entropy increments, propagate with characteristic velocities, while the large scale, shock-type discontinuities propagate with speeds close to the speeds of classical shock waves. In the zero-limit of the scale parameter, approximate solutions converge to a unique, entropy solution of a scalar conservation law.