We prove that the 3-D free-surface Euler equations with regular initial
geometries and velocity fields have solutions which can form a finite-time
"splash" (or "splat") singularity, wherein the evolving 2-D
hypersurface, the moving boundary of the fluid domain, self-intersects at a
point (or on a surface). Such singularities can occur when the crest of a
breaking wave falls unto its trough, or in the study of drop impact upon
liquid surfaces. Our approach is founded upon the Lagrangian description of
the free-boundary problem, combined with a novel approximation scheme of a
finite collection of local coordinate charts; as such we are able to
analyze a rather general set of geometries for the evolving 2-D
free-surface of the fluid. We do not assume the fluid is irrotational, and
as such, our method can be used for a number of other fluid interface
problems, including compressible flows, plasmas, as well as the inclusion
of surface tension effects. This is joint work with D. Coutand.
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