Regularity for Fluid-Structure Interactions and Its Relation to Uniqueness

In the lecture I will survey recent progress on regularity estimates for solutions of Navier-Stokes equations interacting with an elastic shell. The shell is assumed to be perfectly elastic, which means that it is governed by a hyperbolic evolution. The deformation of the shell prescribes a part of the fluid domain, which makes the problem inherently non-linear. It will be shown how the ''parabolic effect'' of the fluid suffices to show results for weak solutions to the coupled fluid-structure interactions previously known for the Navier-Stokes equations in fixed domains. In particular I will discuss the validity of the so-called Ladyzhenskaya-Prodi-Serrin condition for regularity and weak-strong uniqueness for 3D Navier Stokes in the context of fluid-structure interactions. The lecture is based on works in collaboration with D. Breit, P. R. Mensah, B. Muha, M. Sroczinski and P. Su.

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