Small dispersion approximation of shock wave dynamics

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.