On Almost Entire Solutions Of The Burgers Equation
  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. 
Nicholas Alikakos and Dimitrios Gazoulis
math.AP