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SUMMARY:On the fully nonlinear 2D Peskin problem - Robert Strain (Universi
 ty of Pennsylvania)
DTSTART:20220113T160000Z
DTEND:20220113T170000Z
UID:TALK166702@talks.cam.ac.uk
DESCRIPTION:The Peskin problem models the dynamics of a closed elastic str
 ing immersed in an incompressible 2D stokes fluid. This set of equations w
 as proposed as a simplified model to study blood flow through heart valves
 . The immersed boundary formulation of this problem has proven very useful
  in particular giving rise to the widely used immersed boundary method in 
 numerical analysis. Proving the existence and uniqueness of smooth solutio
 ns is vitally useful for this system in particular to guarantee that numer
 ical methods based upon different formulations of the problem all converge
  to the same solution. In this project ``Critical local well-posedness for
  the fully nonlinear Peskin problem''\, which is a joint work with Stephen
  Cameron\, we consider the case with equal viscosities but with a fully no
 n-linear tension law. This situation has been called the fully nonlinear P
 eskin problem. In this case we prove local wellposedness for arbitrary ini
 tial data in the scaling critical Besov space $\\dot{B}^{3/2}_{2\,1}(\\mat
 hbb{T}\; \\mathbb{R}^2)$. We additionally prove the high order smoothing e
 ffects for the solution. To prove this result we derive a new formulation 
 of the equation that describes the parametrization of the string\, and we 
 crucially utilize a new cancellation structure.
LOCATION:Seminar Room 1\, Newton Institute
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