## Special Colloquium

### Friday, February 8th, 2019

**Time:** 2:30 p.m. **Place:** Jeffery Hall 234

**Speaker:** Giusy Mazzone (Vanderbilt)

**Title:** On the Stability and Long-time Behavior of Fluid-Solid Systems.

**Abstract: ** Consider the inertial motion of a coupled system constituted by a rigid body with a cavity completely filled by a viscous incompressible fluid. In 1885, Zhukovskii conjectured that "the motions of the coupled system about its center of mass will eventually be rigid motions and, precisely, permanent rotations, no matter the size and the shape of the cavity, the viscosity of the liquid and the initial movement of the system'". I will present a proof of Zhukovskii's conjecture for a very broad class of motions. We will see how the fluid stabilizing effect suggested by Zhukovskii occurs when Navier-type (no- and partial-slip) conditions are imposed on the fluid-solid interface. A nonlinear stability analysis shows that equilibria (permanent rotations with the fluid at a relative rest with respect to the solid) associated with the largest moment of inertia are asymptotically, exponentially stable. All other equilibria are normally hyperbolic and unstable in an appropriate topology. Moreover, every Leray-Hopf solution to the time-dependent problem converges to an equilibrium at an exponential rate in the $L_q$-topology, $ q\in (1,6) $, for every fluid-solid configuration.

**Giusy Mazzone** is an Assistant Professor (NTT) of Mathematics at Vanderbilt University. She has received a PhD in Mathematics at Universita del Salento, Lecce, Italy, in 2012 and a second PhD in Mechanical Engineering and Material Sciences from the University of Pittsburgh, Pennsylvania, in 2016. Her research interests include mathematical analysis of fluid dynamics, applications of partial differential equations in fluid mechanics, and the study of stability and asymptotic behavior of fluid-solid systems.