# Analysis Seminar

#### Well-Posedness and Regularity for a Class of Thin-Film Free Boundary Problems

**Speaker:**
Manuel Victor Gnann, University of Michigan

**Location:**
Warren Weaver Hall 1302

**Date:**
Tuesday, January 20, 2015, 11 a.m.

**Synopsis:**

We investigate a free boundary problem for a thin-film equation with quadratic mobility and a zero contact angle condition at the triple point where air, liquid, and solid meet. This problem can be derived by a lubrication approximation from the Navier-Stokes system with a Navier-slip condition at the substrate. By treating the model problem of source-type solutions, we motivate why general solutions to this problem are generically singular. The method for proving well-posedness therefore requires to subtract the leading-order singular expansion at the free boundary in the maximal regularity estimates for the linearized evolution. We also discuss the regularizing effect of the degenerate-parabolic operator to arbitrary orders of the singular expansion. Many of the presented results are joint with Lorenzo Giacomelli, Hans Knüpfer, and Felix Otto.