Asymptotic behaviour of a rigid body with a cavity filled by a viscous liquid

We consider the system of equations modeling the free motion of a rigid body with a cavity filled by a viscous (Navier-Stokes) liquid. We give a rigorous proof of Zhukovskiy's Theorem, which states that in the limit of time going to infinity, the relative fluid velocity tends to zero and the rigid velocity of the full structure tends to a steady rotation around one of the principle axes of inertia. The existence of global weak solutions for this system was established previously. In particular, we prove that every weak solution of this type is subject to Zhukovskiy's Theorem. Independently of the geometry and of parameters, this shows that the presence of fluid prevents precession of the body in the limit. In general, we cannot predict which axis will be attained, but we show stability of the largest axis and provide criteria on the initial data which are decisive in special cases.Comment: 18 pages, 0 figure

Global existence, uniqueness and stability for nonlinear dissipative bulk-interface interaction systems

We show global well-posedness and exponential stability of equilibria for a general class of nonlinear dissipative bulk-interface systems. They correspond to thermodynamically consistent gradient structure models of bulk-interface interaction. The setting includes nonlinear slow and fast diffusion in the bulk and nonlinear coupled diffusion on the interface. Additional driving mechanisms can be included and non-smooth geometries and coefficients are admissible, to some extent. An important application are volume-surface reaction-diffusion systems with nonlinear coupled diffusion.Comment: 21 page

Global existence, uniqueness and stability for nonlinear dissipative systems of bulk-interface interaction

We consider a general class of nonlinear parabolic systems corresponding to thermodynamically consistent gradient structure models of bulk-interface interaction. The setting includes non-smooth geometries and e.g. slow, fast and "entropic'' diffusion processes under mass conservation. The main results are global well-posedness and exponential stability of equilibria. As a part of the proof, we show bulk-interface maximum principles and a bulk-interface Poincaré inequality. The method of proof for global existence is a simple but very versatile combination of maximal parabolic regularity of the linearization, a priori L∞-bounds and a Schaefer's fixed point argument. This allows us to extend the setting e.g. to Allen-Cahn dissipative dynamics and to include large classes of inhomogeneous boundary conditions and external forces

Well-posedness for coupled bulk-interface diffusion with mixed boundary conditions

In this paper, we consider a quasilinear parabolic system of equations describing coupled bulk and interface diffusion, including mixed boundary conditions. The setting naturally includes non-smooth domains. We show local well-posedness using maximal Ls-regularity in dual Sobolev spaces of type W 1,q (Omega) for the associated abstract Cauchy problem

A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces

In this paper we consider scalar parabolic equations in a general non-smooth setting with emphasis on mixed interface and boundary conditions. In particular, we allow for dynamics and diffusion on a Lipschitz interface and on the boundary, where diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we additionally take into account diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal regularity for the corresponding abstract Cauchy problem.Comment: 27 pages, 4 figure

The 3D transient semiconductor equations with gradient-dependent and interfacial recombination

We establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on chargecarrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergence-form operators

On gradient structures for Markov chains and the passage to Wasserstein gradient flows

We study the approximation of Wasserstein gradient structures by their finite-dimensional analog. We show that simple finite-volume discretizations of the linear Fokker-Planck equation exhibit the recently established entropic gradient-flow structure for reversible Markov chains. Then we reprove the convergence of the discrete scheme in the limit of vanishing mesh size using only the involved gradient-flow structures. In particular, we make no use of the linearity of the equations nor of the fact that the Fokker-Planck equation is of second order

The 3D transient semiconductor equations with gradient-dependent and interfacial recombination

We establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on charge-carrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergence-form operators

Long-time behaviour of solutions to a nonlinear system of fluid-structure interaction

We consider a non-linear system modelling the dynamics of a linearly elastic body immersed in an incompressible viscous fluid. We prove local existence of strong solutions and global existence and uniqueness for small data. The main result is the characterization of long-time behaviour of the elastic displacement. We show convergence either to a rest state or rigid motion, or to a time-periodic pressure wave that may occur only in specific geometric settings.Comment: 39 pages, 1 figur

A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces

In this paper we consider scalar parabolic equations in a general non-smooth setting with emphasis on mixed interface and boundary conditions. In particular, we allow for dynamics and diffusion on a Lipschitz interface and on the boundary, where diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we additionally take into account diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal regularity for the corresponding abstract Cauchy problem