Viscous shock solutions to the stochastic Burgers equation

TL;DR

This paper introduces a new concept of viscous shock solutions for the stochastic Burgers equation, demonstrating their properties, invariant measures, and convergence from initial conditions, advancing understanding of stochastic shock dynamics.

Contribution

It defines viscous shock solutions connecting stationary states, characterizes their invariant measures, and proves convergence from initial constant solutions.

Findings

Viscous shocks admit time-invariant measures in their reference frames.

Shock solutions are deterministic functions of boundary states and shock location.

Solutions converge to stationary shock solutions over time.

Abstract

We define a notion of a viscous shock solution of the stochastic Burgers equation that connects "top" and "bottom" spatially stationary solutions of the same equation. Such shocks generally travel in space, but we show that they admit time-invariant measures when viewed in their own reference frames. Under such a measure, the viscous shock is a deterministic function of the bottom and top solutions and the shock location. However, the measure of the bottom and top solutions must be tilted to account for the change of reference frame. We also show a convergence result to these stationary shock solutions from solutions initially connecting two constants, as time goes to infinity.