Well-posedness for the diffusive 3D Burgers equations with initial data in $H^{1/2}$

TL;DR

This paper proves the existence and uniqueness of global solutions for the 3D diffusive Burgers equations with initial data in the fractional Sobolev space $H^{1/2}$, highlighting differences from Navier-Stokes analysis.

Contribution

It establishes well-posedness results for the 3D Burgers equations with low regularity initial data, adapting techniques from Navier-Stokes theory and addressing unique challenges.

Findings

Global classical solutions exist for initial data in $H^{1/2}$.

Standard $L^2$ estimates are insufficient due to lack of incompressibility.

Maximum principle ensures global regularity.

Abstract

In this note we discuss the diffusive, vector-valued Burgers equations in a three-dimensional domain with periodic boundary conditions. We prove that given initial data in $H^{1/2}$ these equations admit a unique global solution that becomes classical immediately after the initial time. To prove local existence, we follow as closely as possible an argument giving local existence for the Navier--Stokes equations. The existence of global classical solutions is then a consequence of the maximum principle for the Burgers equations due to Kiselev and Ladyzhenskaya (1957). In several places we encounter difficulties that are not present in the corresponding analysis of the Navier--Stokes equations. These are essentially due to the absence of any of the cancellations afforded by incompressibility, and the lack of conservation of mass. Indeed, standard means of obtaining estimates in $L^2$ fail and we are forced to start with more regular data. Furthermore, we must control the total momentum and carefully check how it impacts on various standard estimates.