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A Maximal Regularity Approach to the Study of Motion of a Rigid Body with a Fluid-Filled Cavity

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Abstract

We consider the inertial motion of a rigid body with an interior cavity that is completely filled with a viscous incompressible fluid. The equilibria of the system are characterized and their stability properties are analyzed. It is shown that equilibria associated with the largest moment of inertia are normally stable, while all other equilibria are normally hyperbolic. We show that every Leray–Hopf weak solution converges to an equilibrium at an exponential rate. In addition, we determine the critical spaces for the governing evolution equation, and we demonstrate how parabolic regularization in time-weighted spaces affords great flexibility in establishing regularity of solutions and their convergence to equilibria.

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Notes

  1. In this case the inertia tensor of \({\mathcal {S}}\), calculated with respect to the center of mass G, is a multiple of the identity tensor.

  2. These are rigid body motions around the principal axes of inertia, with constant angular velocity. For this motion, the fluid is at relative rest with respect to the solid.

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Acknowledgements

We would like to thank Martin and Burga Simonett for their hospitality while visiting Lohn, where part of this manuscript was written.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada

    Giusy Mazzone

  2. Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, Theodor-Lieser-Strasse 5, 06120, Halle, Germany

    Jan Prüss

  3. Department of Mathematics, Vanderbilt University, Nashville, TN, USA

    Gieri Simonett

Authors
  1. Giusy Mazzone
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  2. Jan Prüss
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  3. Gieri Simonett
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Correspondence to Giusy Mazzone.

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This work was supported by a grant from the Simons Foundation (#426729, Gieri Simonett).

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Mazzone, G., Prüss, J. & Simonett, G. A Maximal Regularity Approach to the Study of Motion of a Rigid Body with a Fluid-Filled Cavity. J. Math. Fluid Mech. 21, 44 (2019). https://doi.org/10.1007/s00021-019-0449-y

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