A representation of solutions to a scalar conservation law in several dimensions

TL;DR

This paper presents a stochastic perturbation approach to represent smooth solutions of multidimensional scalar conservation laws, aiding in understanding singularity formation and linking to a continuum system with a pressure term.

Contribution

It introduces a novel stochastic representation of solutions and an associated balance law system that captures the transition from smoothness to singularity.

Findings

Representation of solutions as small diffusion limits of stochastic processes

Insight into the formation of singularities in conservation laws

Connection to a continuum system with a pressure term after loss of smoothness

Abstract

We find a representation of smooth solutions to the Cauchy problem for a scalar multidimensional conservation law as small diffusion limit of a stochastic perturbation along characteristics. It helps, in particular, to study the process of singularities formation. Further, we introduce an associated system of balance laws that can be interpreted as describing the motion of a continuum with some specific pressure term. This term arises only after the instant when the solution to the initial Cauchy problem looses its smoothness. Before this instant the system coincides partly with the one known as pressure free gas dynamics.