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

Diego Alonso-Orán, Aythami Bethencourt de León, So Takao

Published 2018-08-23Version 1

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.