Beale-Kato-Majda type condition for Burgers equation

TL;DR

This paper establishes conditions for global solutions to multidimensional Burgers equations on different domains, using novel probabilistic methods, and analyzes their long-term behavior.

Contribution

It introduces a Beale-Kato-Majda type criterion for global existence of solutions on  space, employing new probabilistic techniques.

Findings

Unique global solutions on the torus with large time estimates

Existence of solutions on  space under BKM-type condition

Novel probabilistic approach for Burgers equations

Abstract

We consider a multidimensional Burgers equation on the torus $\mathbb{T}^d$ and the whole space $\Rd$. We show that, in case of the torus, there exists a unique global solution in Lebesgue spaces. For a torus we also provide estimates on the large time behaviour of solutions. In the case of $\Rd$ we establish the existence of a unique global solution if a Beale-Kato-Majda type condition is satisfied. To prove these results we use the probabilistic arguments which seem to be new.