Analytical and Computational Challenges of Incompressible Flows at High Reynolds Number

CSIC Building (#406), Seminar Room 4122.
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Existence and uniqueness for the fluid-structure interaction problem

Dr. Steve Shkoller

Department of Mathematics at University of CA, Davis

Abstract:   The motion of an elastic solid inside of an incompressible viscous fluid is ubiquitous in nature. Mathematically, such motion is described by a PDE system that couples the parabolic (fluid) and hyperbolic (solid) phases, the latter inducing a loss of regularity. In this talk, I will sketch the proof of existence and uniqueness of such motions (locally in time), when the elastic solid is the linear Kirchhoff elastic material. The solution is found using a topological fixed-point theorem that requires the analysis of a linear problem consisting of the coupling between the time-dependent Navier-Stokes equations set in Lagrangian variables and the linear equations of elastodynamics, for which the existence of a unique weak solution is proven. Then regularity of the weak solution is obtained in function spaces that scale in a hyperbolic fashion in both the fluid and solid phases. The functional framework appears to be optimal, and provides the a priori estimates necessary to employ the fixed-point procedure. This is joint work with Daniel Coutand at UC Davis.