On effects of stochastic regularization for the pressureless gas dynamics

TL;DR

This paper investigates how stochastic regularization influences pressureless gas dynamics by linking perturbed Burgers equations to viscous balance laws, revealing pressure effects during shock formation.

Contribution

It extends previous work by connecting stochastic perturbations of Burgers equations to viscous balance laws and pressure effects in gas dynamics during shock formation.

Findings

Viscous balance laws relate to stochastic Burgers equations.

Pressure effects emerge during shock formation.

Limit systems resemble pressure-influenced gas dynamics.

Abstract

We extend our result of [1] and show that one can associate with the stochastically perturbed non-viscid Burgers equation a system of viscous balance laws. The Cauchy data for the Burgers equation generates the data for this system. Till the moment of the shock formation in the solution to the Burgers equation the above system of viscous balance laws can be reduced to the pressureless gas dynamics system (in a limit as the parameters of perturbation tend to zero). If the solution to the Burgers equation contains shocks, the limit system is equivalent to the system with a specific pressure, in some sense analogous to the pressure of barotropic monoatomic gas.