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SUMMARY:Sharp Hadamard well-posedness for the incompressible free boundary
  Euler equations - Mitchell A. Taylor (ETH Zürich)
DTSTART:20240226T140000Z
DTEND:20240226T150000Z
UID:TALK212584@talks.cam.ac.uk
CONTACT:Zexing Li
DESCRIPTION:We will discuss a recent preprint in which we establish an opt
 imal local well-posedness theory in $H^^^s$ based Sobolev spaces for the f
 ree boundary incompressible Euler equations on a connected fluid domain. S
 ome components of this result include: (i) Local well-posedness in the Had
 amard sense\, i.e.\, local existence\, uniqueness\, and the first proof of
  continuous dependence on the data\, all in low regularity Sobolev spaces\
 ; (ii) Enhanced uniqueness: A uniqueness result which holds at the level o
 f the Lipschitz norm of the velocity and the $C^^^{1\,\\frac{1}{2}}$ \n re
 gularity of the free surface\; (iii) Stability bounds: We construct a nonl
 inear functional which measures\, in a suitable sense\, the distance betwe
 en two solutions (even when defined on different domains) and we show that
  this distance is propagated by the flow\; (iv) Energy estimates: We prove
  essentially scale invariant energy estimates for solutions\, relying on a
  newly constructed family of refined elliptic estimates\; (v) Continuation
  criterion: We give the first proof of a continuation criterion at the sam
 e scale as the classical Beale-Kato-Majda criterion for the incompressible
  Euler equations on fixed domains. Roughly speaking\, we show that solutio
 ns can be continued as long as the velocity is in $L^^^1_T W^^^{1\,\\infty
 }$\n and the free surface is in $L^^^1_T C^^^{1\,\\frac{1}{2}}$\; (vi) A n
 ovel proof of the construction of regular solutions.\n\nOur entire approac
 h is in the Eulerian framework and can be adapted to work in relatively ge
 neral fluid domains. This is based on joint work with Mihaela Ifrim\, Ben 
 Pineau and Daniel Tataru.
LOCATION:MR13
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