Generic regularity in time for solutions of the Stefan problem in 4+1 dimensions

Abstract

We show that the free boundary of a solution of the Stefan problem in $\mathbb R^{4+1}$ is a $3$-dimensional manifold of class $C^\infty$ in $\mathbb R^4$ for almost every time. This is achieved by showing that for all dimensions $n$ the singular set $\Sigma\subset \mathbb R^{n+1}$ can be decomposed in two parts $\Sigma=\Sigma^\infty\cup \Sigma^*$, where $\Sigma^\infty$ is covered by one $(n-1)$-dimensional manifold of class $C^\infty$ in $\mathbb R^{n+1}$ and its projection onto the time axis has Hausdorff dimension 0, while $\Sigma^*$ is parabolically countably $(n-2)$-rectifiable.