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Interface singularities for the Euler equations

Wednesday 23rd July 2014 - 14:00 to 14:45
INI Seminar Room 1
In fluid dynamics, a "splash" singularity occurs when a locally smooth interface self-intersects in finite-time. It is now well-known that solutions to the water waves equations (and a host of other one-phase fluid interface models) has a finite-time splash singularity. By means of elementary arguments, we prove that such a singularity cannot occur in finite-time for vortex sheet evolution (or two-fluid interfaces). This means that the evolving interface must lose regularity prior to self-intersection. We give a proof by contradiction: we assume that such a singularity does indeed occur in finite-time. Based on this assumption, we find precise blow-up rates for the components of the velocity gradient which, in turn, allows us to characterize the geometry of the evolving interface just prior to self-intersection. The constraints on the geometry then lead to an impossible outcome, giving the contradiction. This is joint work D. Coutand.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons