University of Toronto
                  PDE/Applied Math/Analysis Seminar
                    Friday October 20, 12:10-1:00pm
                          2195 Bahen Center      



SPEAKER:  Chongchun Zeng (Georiga Tech)

TITLE:    Hydrodynamical instabilities and energy estimates for 
          the free boundary problems of the Euler equation

ABSTRACT: We consider the evolution of free surfaces of incompressible 
          and inviscid fluids. Neglecting the gravity, we are interested 
          in the cases of 1.) the motion of a droplet in the vacuum with 
          or without surface tension and 2.) the motion of the interface 
          between two fluids with surface tension. The evolution of these 
          fluid boundaries and the velocity fields is determined by the 
          free boundary problem of the Euler's equation.

          Each of these problems can be considered in a Lagrangian 
          formulation on an infinite dimensional Riemannian manifold of 
          volume preserving diffeomorphisms. On the one hand, a scale of 
          functionals as "energies" are defined and they bound high Sobolev 
          norms of the velocity field as well as the mean curvature of the 
          fluid boundary. Thus we establish the regularity of the solutions 
          for a short time depending on the initial data. On the
          other hand, in the absence of surface tension, the well-known
          Rayleigh-Taylor and Kelvin-Helmholtz instablities appear naturally 
          related to the signs of the curvatures of those infinite 
          dimensional manifolds. The surface tension produces differential 
          operators stronger than the instabilities and thus regularizes 
          the surface evolution.


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University of Toronto PDE/Applied Math/Analysis Seminar 
http://www.math.toronto.edu/appmath/

2006-2007 organizers:
          Pieter Blue           pblue@ math.toronto.edu
          Almut Burchard        almut@ math.toronto.edu
          Robert McCann		mccann@ math.toronto.edu
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