Diffusion approximation for self-similarity of stochastic advection in Burgers' equation

TL;DR

This paper investigates the self-similar behavior of stochastic Burgers' equation, constructing stationary solutions and proving long-term convergence to these solutions using diffusion approximation techniques.

Contribution

It introduces a method to establish stochastically self-similar solutions for stochastic Burgers' equation and demonstrates their stability over time.

Findings

Existence of a stochastically self-similar solution.

Construction of a stationary solution in self-similar variables.

Proof of long-time convergence to the self-similar solution.

Abstract

Self-similarity of Burgers' equation with some stochastic advection is studied. In self-similar variables a stationary solution is constructed which establishes the existence of a stochastically self-similar solution for the stochastic Burgers' equation. The analysis assumes that the stochastic coefficient of advection is transformed to a white noise in the self-similar variables. Furthermore, by a diffusion approximation, the long time convergence to the self-similar solution is proved in the sense of distribution.