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

• Published in: Journal of mathematical fluid mechanics Vol. 21; no. 3; pp. 1 - 20
• Main Authors: Mazzone, Giusy, Prüss, Jan, Simonett, Gieri
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.09.2019; Springer Nature B.V
• Subjects: Classical and Continuum Physics; Convergence; Economic models; Equilibrium; Fluid dynamics; Fluid flow; Fluid mechanics; Fluid- and Aerodynamics; Incompressible flow; Incompressible fluids; Mathematical Methods in Physics; Moments of inertia; Physics; Physics and Astronomy; Regularity; Regularization; Rigid structures; Stability analysis; Theoretical mathematics; Normally stable; Critical spaces; Normally hyperbolic; Rigid body motion; Global existence; Primary 35Q35; 76D05; Fluid–solid interactions; 35Q30; 35B40; 35K58
• ISSN: 1422-6928; 1422-6952
• DOI: 10.1007/s00021-019-0449-y

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.