8 - Well-posedness for the diffusive 3D Burgers equations with initial data in H½

Summary

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 & 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 L² 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.

Introduction

We consider the three-dimensional, vector-valued diffusive Burgers equations. The equations, for a fixed viscosity v > 0 and initial data u₀, are

Working on the torus T³ = R³/2πZ³, we will investigate the existence and uniqueness of solutions u to (8.1). Using the rescaling ũ(x,t) :=vu(x,vt), it suffices to prove well-posedness in the case v = 1.

This system is well known and is often considered to be ‘well understood’. However we have not found a self-contained account of its well–posedness in the literature, although for very regular data (with two Hölder continuous derivatives, for example) existence and uniqueness can be deduced from standard results about quasi–linear systems. We are particularly interested in an analysis parallel to the familiar treatment of the Navier–Stokes equations, which motivates our choice of function spaces here.

It is interesting to note that we find some essential difficulties in treating this system which do not occur when incompressibility is enforced, i.e. for the Navier–Stokes equations. These prevent us from making the usual estimates that would give existence of (L²-valued) weak solutions. We also find that taking initial data with zero average is not sufficient to ensure that the solution has zero average for positive times. This necessitates estimating the momentum and checking carefully that the methods applicable to the Navier–Stokes equations have a suitable analogue.