MATHEMATICS COLLOQUIUM

SPEAKER: Prof. Almut Burchard, University of Toronto

TITLE: "Convergence to equilibrium for a thin-film equation on a horizontal cylinder"

ABSTRACT: In this talk, I will discuss recent work with Marina Chugunova and Ben Stephens on the long-term evolution of a thin liquid film on a horizontal cylinder, modeled by the degenerate parabolic equation

u_t + [u^3(u_xxx + u_x - sin x)]_x=0 .

For each given mass, we find that the unique steady state is a droplet that hangs from the bottom of the cylinder and meets the dry part of surface at zero contact angle. The steady state attracts all strong solutions, but the distance decays no faster than a power law.

One difficulty of thin-film equations is the lack of a uniqueness theorem for a good class of non-negative solutions. The key to our results is an energy (made up from the surface tension and gravitational potential energy) that decreases along solutions, and an entropy that increases at most linearly with time. Time permitting, I will discuss how the evolution can be viewed as the

gradient flow of the energy, by realizing it on a space of measures with a distance function related to mass transportation. This promises to open new approaches to the long-standing uniqueness problem.

Refreshments will be available at 3:15 p.m.