Classical solvability of the multidimensional free boundary problem for the thin film equation in the case of partial wetting

TL;DR

This paper establishes local in time existence and uniqueness of smooth solutions with smooth interfaces for a multidimensional free boundary problem related to the thin film equation under partial wetting conditions, including Schauder estimates.

Contribution

It provides the first proof of classical solvability for this complex free boundary problem in multiple dimensions, along with Schauder estimates for related linear degenerate parabolic equations.

Findings

Existence and uniqueness of smooth solutions with smooth interfaces.

Schauder estimates for linear degenerate parabolic equations.

Solvability results for Dirichlet and Neumann problems.

Abstract

We prove locally in time the existence of the unique smooth solution (including smooth interface) to the multidimensional free boundary problem for the thin film equation in the case of partial wetting. We also obtain the Schauder estimates and solvability for the Dirichlet and the Neumann problem for a linear degenerate parabolic equation of fourth order. The final expanded version of this paper is available at AIMS Journals, Discrete and Continuous Dynamical Systems - A at http://aimsciences.org/article/doi/10.3934/dcds.2017156