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Well posedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid

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Abstract

In this paper, we consider the interaction between a rigid body and an incompressible, homogeneous, viscous fluid. This fluid-solid system is assumed to fill the whole space ℝd, d = 2 or 3. The equations for the fluid are the classical Navier-Stokes equations whereas the motion of the rigid body is governed by the standard conservation laws of linear and angular momentum. The time variation of the fluid domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary value problem.

We improve the known results by proving a complete wellposedness result: our main result yields a local in time existence and uniqueness of strong solutions for d = 2 or 3. Moreover, we prove that the solution is global in time for d = 2 and also for d = 3 if the data are small enough.

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Authors and Affiliations

  1. Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Avda. Andrés Bello s/n, Casilla 447, Chillán, Chile

    Patricio Cumsille

  2. Institut Elie Cartan UMR 7502, INRIA, Nancy-Université, CNRS, Project-Team CORIDA, BP239, 54506, Vandœuvre-lès-Nancy, Cedex, France

    Takéo Takahashi

Authors
  1. Patricio Cumsille
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  2. Takéo Takahashi
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Correspondence to Takéo Takahashi.

Additional information

Patricio Cumsille’s research was partially supported by CONICYT-FONDECYT grant (No. 3070040) and Takéo Takahashi’s research was partially supported by Grant (JCJC06 137283) of the Agence Nationale de la Recherche.

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Cumsille, P., Takahashi, T. Well posedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid. Czech Math J 58, 961–992 (2008). https://doi.org/10.1007/s10587-008-0063-2

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