Interface Singularities for the Euler Equations
Interface Singularities for the Euler Equations
The fluid interface听 ``splash'' singularity was introduced by Castro, C\'{o}rdoba,听 Fefferman, Gancedo, \& G\'{o}mez-Serrano.听 A splash singularity occurs when a fluid interface remains locally smooth but self-intersects in finite time.听听 In this talk, I will very briefly discuss how we construct splash singularities for the one-phase 3-D Euler and Navier-Stokes equations.听 I will then discuss the problem of two-phase Euler flow.听听 Recently, Fefferman, Ionescu, and Lie have shown that a locally smooth vortex sheet cannot self-intersect in finite time.听听 I will explain our proof of this result, which is based on elementary arguments and some precise blow-up rates for the gradient of the velocity of the fluid through which the interfaces tries to self-intersect.听听 This is joint work with D. Coutand.听