Well-posedness of sudden directional diffusion equations

TL;DR

This paper proves existence and analyzes properties of solutions to a class of one-dimensional parabolic equations with monotone flux functions, focusing on regularity and geometric features like facets.

Contribution

It establishes existence results for solutions with general initial data and explores their regularity and geometric properties, including the formation of facets.

Findings

Solutions exist for a broad class of initial data.

Solutions exhibit maximal regularity under certain conditions.

Flat regions (facets) naturally form in solutions.

Abstract

Our goal is to establish existence with suitable initial data of solutions to general parabolic equation in one dimension, $u_t = L(u_x)_x$, where $L$ is merely a monotone function. We also expose the basic properties of solutions, concentrating on maximal possible regularity. Analysis of solutions with convex initial data explains why we may call them {\it almost classical}. Some qualitative aspects of solutions, like facets -- flat regions of solutions, are studied too.