The Burgers' equation with stochastic transport: shock formation, local and global existence of smooth solutions

TL;DR

This paper investigates the stochastic Burgers' equation, analyzing shock formation, deriving a stochastic Rankine-Hugoniot condition, and establishing local and global existence of smooth solutions in viscous and inviscid cases.

Contribution

It provides new theoretical results on shock formation, solution existence, and uniqueness for the stochastic Burgers' equation, including a stochastic shock condition and blow-up criteria.

Findings

Shock formation in stochastic Burgers' equation is characterized.

Global existence of smooth solutions is proved in the viscous case.

A stochastic Rankine-Hugoniot condition for shocks is derived.

Abstract

In this work, we examine the solution properties of the Burgers' equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine-Hugoniot condition that the shocks satisfy. Next, we establish the local existence and uniqueness of smooth solutions in the inviscid case and construct a blow-up criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions.