The thin film equation close to self-similarity

TL;DR

This paper analyzes the well-posedness and regularity of the multidimensional thin film equation near self-similar solutions, demonstrating smoothness and analyticity of solutions and free boundaries through a variable transformation.

Contribution

It introduces a transformation that converts the free boundary problem into a well-posed parabolic equation, establishing regularity and analyticity results.

Findings

Solutions are smooth and analytic in time and angular variables.

Level sets and free boundary are analytic.

The transformed equation is well-posed.

Abstract

In the present work, we study well-posedness and regularity of the multidimensional thin film equation with linear mobility in a neighborhood of the self-similar Smyth--Hill solutions. To be more specific, we perform a von Mises change of dependent and independent variables that transforms the thin film free boundary problem into a parabolic equation on the unit ball. We show that the transformed equation is well-posed and that solutions are smooth and even analytic in time and angular direction. The latter entails the analyticity of level sets of the original equation, and thus, in particular, of the free boundary.