Shock formation, local and global existence of smooth solutions
In this work, we examine the solution properties of the Burgers’ equation with stochastic transport. First, we prove results on the forma- tion 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.