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The Interaction between Quasilinear Elastodynamics and the Navier-Stokes Equations

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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. Unlike our approach in [5] 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 strong norms of our sequence of regularized problems.

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Authors and Affiliations

  1. Department of Mathematics, University of California at Davis, Davis, CA, 95616, U.S.A.

    Daniel Coutand & Steve Shkoller

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  1. Daniel Coutand
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  2. Steve Shkoller
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Correspondence to Steve Shkoller.

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Communicated by V.Šverák

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Coutand, D., Shkoller, S. The Interaction between Quasilinear Elastodynamics and the Navier-Stokes Equations. Arch. Rational Mech. Anal. 179, 303–352 (2006). https://doi.org/10.1007/s00205-005-0385-2

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  • DOI: https://doi.org/10.1007/s00205-005-0385-2

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