On Almost Entire Solutions Of The Burgers Equation

TL;DR

This paper proves that solutions to the Burgers equation that are smooth except on a set of potential singularities with zero Hausdorff 1-measure, constrained to a union of graphs, must be constant.

Contribution

It establishes that almost entire solutions with singularities on a specific set are necessarily trivial, extending understanding of solution regularity for the Burgers equation.

Findings

Solutions with singularities on sets of zero Hausdorff 1-measure are constant.

Singularities confined to countable unions of graphs imply trivial solutions.

Abstract

Solutions that satisfy classically the Burgers equation except, perhaps, on a closed set S of the plane of potential singularities whose Hausdorff 1-measure is zero, $H^1(S) = 0$, are necessarily identically constant. We show this under the additional hypothesis that $S$ is a subset of a countable union of ordered graphs.