We study inertial motions of the coupled system, (Formula presented.) , constituted by a rigid body containing a cavity entirely filled with a viscous liquid. We show that for arbitrary initial data having only finite kinetic energy, every corresponding weak solution (à la Leray–Hopf) converges, as time goes to infinity, to a uniform rotation, unless two central moments of inertia of (Formula presented.) coincide and are strictly greater than the third one. This corroborates a famous “conjecture” of N.Ye. Zhukovskii in several physically relevant cases. Moreover, we show that, in a known range of initial data, this rotation may only occur along the central axis of inertia of (Formula presented.) with the larger moment of inertia. We also provide necessary and sufficient conditions for the rigorous nonlinear stability of permanent rotations, which improve and/or generalize results previously given by other authors under different types of approximation. Finally, we present results obtained by a targeted numerical simulation that, on the one hand, complement the analytical findings, whereas, on the other hand, point out new features that the analysis is yet not able to catch, and, as such, lay the foundation for interesting and challenging future investigation

On bounded domains Ω ⊂ R 3 , we consider divergence-type operators −∇ • µ∇, including mixed homogeneous Dirichlet and Neumann boundary conditions on ∂Ω \ Γ and Γ ⊂ ∂Ω, respectively, and discontinuous coefficient functions µ. We develop a general geometric framework for Ω, Γ and µ in which it is possible to prove that −∇ • µ∇ + 1 provides an isomorphism from W 1,q Γ (Ω) to W −1,q Γ (Ω) for some q > 3. We indicate relevant examples from real-world applications.

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This article develops for the first time a rigorous analysis of Hibler’s model of sea ice dynamics. Identifying Hibler’s ice stress as a quasilinear second-order operator and regarding Hibler’s model as a quasilinear evolution equation, it is shown that a regularized version of Hibler’s coupled sea ice model, i.e., the model coupling velocity, thickness and compactness of sea ice, is locally strongly well-posed within the $$L_q$$ L q -setting and also globally strongly well-posed for initial data close to constant equilibria.

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