Introduction \`a l'\'equation de Burgers stochastique et \`a la burgulence

TL;DR

This paper introduces the stochastic Burgers equation, proving fundamental results on solutions, their regularity, and turbulence behavior, serving as an accessible entry point to modern stochastic PDE methods.

Contribution

It provides an elementary introduction to stochastic Burgers equation theory, including existence, uniqueness, regularity, and turbulence analysis, with focus on the zero-viscosity limit.

Findings

Existence and uniqueness of solutions established

Analysis of solution regularity and properties

Description of turbulence phenomena in the zero-viscosity limit

Abstract

This paper is an introduction to the theory of 1d stochastic Burgers equation under periodic boundary conditions and with a stochastic force, sufficiently smooth in the space variable. We prove the classical results on the existence and uniqueness of solutions, study their regularity and discuss their properties when the time goes to infinity or the viscosity goes to zero. The latter limit describes the turbulence in the Burgers equation, named by U. Frish "the burgulence". Our paper may be used as an elementary introduction to the modern methods of stochastic PDE.