Motion of an Elastic Solid inside an Incompressible Viscous Fluid

By Daniel Coutand, Steve Shkoller. 2004

230 citations recorded by the source corpus.

Abstract.The motion of an elastic solid inside an incompressible viscous fluid is ubiquitous in nature. Mathematically, such motion is described by a PDE system that couples the parabolic and hyperbolic phases, the latter inducing a loss of regularity which has left the basic question of existence open until now.In this paper, we prove the 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 we prove the existence of a unique weak solution. We then establish the regularity of the weak solution; this regularity is obtained in function spaces that scale in a hyperbolic fashion in both the fluid and solid phases. Our functional framework is optimal, and provides the a priori estimates necessary for us to employ our fixed-point procedure.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/03bc8ee1cd9c006da465e0d8725b7855a5ffb1df

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In Applied Mathematics

In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

The Interaction between Quasilinear Elastodynamics and the Navier-Stokes Equations (2006, methodology): “(19c) By proceding as in [5], we can establish the existence of a fixed-point for system (14).”