# Analysis Seminar

#### Existence, Uniqueness and Regularity of Solutions for the Thin-Film Equation

**Speaker:**
Mohamed Majdoub, FacultÃ© des sciences de Tunis

**Location:**
Warren Weaver Hall 1302

**Date:**
Thursday, December 18, 2014, 11 a.m.

**Synopsis:**

We show the uniqueness of strong solution for the thin-film equation: \(u_t + (u u_{xxx})_x =0, t>0\) with initial data \(u(0)=m\delta, m>0\) where \(\delta\) is the Dirac mass at the origin. We also investigate the boundary regularity of source-type self-similar solutions to the thin-film equation with gravity: \(h_t=-(h^nh_{xxx})_x+(h^{n+3})_{xx}, t>0, h(0)= m \delta\) where \(n \in (3/2,3)\). The existence of these solutions has been established by E. Beretta. It is also shown that the leading order expansion near the edge of the support coincides with that of a travelling-wave solution for the standard TFE: \(h_t=-(h^n h_{xxx})_x\). We sharpen this result by proving that the higher order corrections are analytic with respect to three variables: the first one is just the spacial variable, whereas the second and third (except for \(n = 2\)) are irrational powers of it. We also prove the uniqueness of solutions.

This is a joint work with Nader Masmoudi and Slim Tayachi.