# DE Seminar: Boris Muha

### University of Zagreb

Location

Mathematics/Psychology : 401

Date & Time

April 17, 2023, 11:00 am – 12:00 pm

Description

**Title**: Existence and regularity of weak solutions for a fluid interacting with a non-linear shell in three dimensions

**Speaker**: Boris Muha

Department of Mathematics, Faculty of Science University of Zagreb, Bijenicka 30, Zagreb,
Croatia borism@math.hr

**Abstract**: We study the unsteady incompressible Navier-Stokes equations in three dimensions interacting with a non-linear flexible shell of Koiter type. This leads to a coupled system of non-linear PDEs where the moving part of the boundary is an unknown of the problem. The known existence theory for weak solutions is extended to non-linear Koiter shell models. We introduce a-priori estimates that reveal higher regularity of the shell displacement beyond energy estimates. These are essential for non-linear Koiter shell models, since such shell models are non-convex (w.r.t. terms of highest order). The estimates are obtained by introducing new analytical tools that allow to exploit dissipative effects of the fluid for the (non-dissipative) solid. The regularity result depends on the geometric constitution alone and is independent of the approximation procedure; hence it holds for arbitrary weak solutions. The developed tools are further used to introduce a generalized Aubin-Lions type compactness result suitable for fluid-structure interactions. This is a joint work with S. Schwarzacher.

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