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Abstract:
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In this talk, we consider the incompressible Navier-Stokes equations with free boundary, which has a finite-time splash singularity for a large class of specially prepared initial data. The approach is founded upon the Lagrangian description of the free-boundary problem, and a sequence of domains ?ε used as the initial data for the splash singularity are defined, wherein the boundary Γε of these domains is close to self-intersection between two approaching portions of Γε. We will present some preliminary lemmas which show that the constant appearing in elliptic estimates and the Sobolev embedding theorem is independent of ε and define the sequence of initial divergence-free velocity fields that are guaranteed to satisfy the required single compatibility condition, and whose norm is independent of ε.
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