On the interaction between quasilinear elastodynamics and the Navier-Stokes equations
Daniel Coutand, Steve Shkoller
TL;DR
This work analyzes the interaction between a viscous incompressible fluid and a large-deformation solid, coupling $Navier\text{-}Stokes$ in the fluid with quasilinear elastodynamics governed by the $St.\ Venant\text{-}Kirchhoff$ model for the solid along a moving interface where velocity and traction are continuous. Because a fixed-point argument on the nonlinear system is not applicable, the authors regularize the solid with a parabolic artificial viscosity term and derive estimates that are independent of the regularization parameter. They formulate the problem in a Lagrangian frame, solve a $\kappa$-regularized time-dependent linear fixed-point problem, and establish $\kappa$-independent energy bounds; by passing to the limit $\kappa\to 0$, they obtain a solution of the original system and characterize the interface motion as the weak limit of the regularized solutions, with uniqueness shown via a Gronwall-type energy estimate. The main contributions include local-in-time well-posedness in Sobolev spaces for the moving-interface fluid–structure problem, κ-independent a priori bounds, and a rigorous limit procedure that yields a global weak solution in both phases along with the interface evolution, with extensions to incompressible elasticity.
Abstract
The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations along a moving interface. Unlike our approach for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial viscosity term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in srong norms of our sequence of regularized problems.