PDE & GEOMETRIC ANALYSIS SEMINAR
The seminar will be held weekly on Monday afternoon at 3:30pm in
room B115 (Van Vleck Hall.)
Scheduled speakers for Spring 2010
April 12:
Einstein metrics and exotic smooth structures on 4-manifolds
The existence or non-existence of Einstein metrics on a
topological 4-manifold is strongly related to the differential
structure considered. We analyze this dependency for manifolds with
small topology. We also consider 4-manifolds with the canonical smooth
structure, and for a large class of manifolds we prove non-existence
theorems of group invariant metrics. The main techniques used come
from Seiberg-Witten theory, the geometry of complex surfaces and
symplectic topology.
February 1:
Well-posedness and Lubrication Approximation for Spreading Droplets
Consider a 2-d droplet spreading on a plate. Our main interest is
the moving triple point where air, liquid and solid meet. A simple
model for the evolution of the liquid is given by the Darcy flow. In
the regime of lubrication approximation, i.e. when the liquid film
is very thin, the evolution can be described by a class of reduced
1--d models, the so called thin--film equations. We present
well-posedness results for both the Darcy Flow and for certain
thin-film equations. One difficulty is that the regularity at the
moving triple point depends strongly on the imposed slip condition
at the liquid-solid interface. In the regime of thin films, we show
that solutions of the Darcy flow converge to solutions of the
corresponding thin-film equation.
January 25:
Vacuum in Gas and Fluid dynamics
The study of vacuum is important in understanding the
motion of a gaseous star or shallow water. The mathematical
difficulty is the degeneracy caused by vacuum. In this talk,
I will review some interesting problems of vacuum states
arising in gas and fluid dynamics, and present the current
status in the understanding of compressible Euler flows
near vacuum. My contribution to the subject is joint with
Nader Masmoudi.