Interface singularities for the Euler equations
Update: 2014-07-25
Description
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.
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