Research Unit Seminar Oct 31, 2023

  • Speaker:  Antoine Leblond (Max-Planck-Institut für Meteorologie) 16h00, Felix Brandt (Technische Universität Darmstadt) 17h00
  • Date: Oct 31, 2023 16h00 - 18h00
  • Location: Room 315 (Schlossgartenstraße 7 (S2|15), 64289 Darmstadt) and online (Zoom)
  • Abstracts:

"Well-posedness and long-time behaviour of the Stokes-transport equation" (Antoine Leblond)

The Stokes-transport equation models an incompressible, viscous and inhomogeneous fluid, subject to gravity. It is a reduced model for oceanography and sedimentation. The density is transported by the velocity field, satisfying at any time the momentum balance between viscosity, pressure and gravity effects, namely the Stokes equation. In the first part of this presentation, we establish the global well-posedness of this system in bounded domains and in the infinite channel, in the weak sense and for Lebesgue initial data. The unbounded channel case is solved in uniformly local Sobolev spaces, with solutions having infinite energy. These results are compared with previous works in the whole space and in the plane. In the second part, we focus on the long-time evolution of the solutions of the Stokes-transport equation in a periodic channel. We show that a class of monotonous stratified density profiles is stable for small and regular enough perturbations. We consider no-slip boundary conditions on the velocity field, which raises mathematical difficulties due to the presence of boundary effects. We obtain explicit algebraic convergence rates and show that the density rearranges vertically and monotonously, in line with the common intuition of sedimentation. We also give a refined description of the density profile, involving a boundary layer expansion in the vicinity of the boundaries. Besides, we extend a previous result obtained for a related problem, proving that any stationary profile is unstable in low regularity topologies. We also highlight properties compatible with the conjecture that the density always stratifies. In the last part, we undertake a numerical study of the evolution of graph density interfaces governed by the Stokes-transport equation. Several behaviours are observed, from the convergence toward the flat rest interface to the graph break. We compare our observations with existing theoretical results.

"Rigorous analysis of Hibler's sea ice model - recent results and related problems" (Felix Brandt)

Recently, Hibler's viscous-plastic sea ice model has been investigated from a rigorous mathematical point of view. In this talk, we give an overview of these results and some related problems. Under some regularization, viewing Hibler's model as a system of quasilinear evolution equations, we first discuss the local strong well-posedness as well as the global strong well-posedness for initial data close to constant equilibria. Furthermore, we elaborate on the sea ice equations in the time periodic setting and the interaction problem of sea ice with a rigid body. Another focal point is the study of a coupled atmosphere-sea ice-ocean model, where primitive equations are used to model the atmosphere and the ocean, and the ocean force on the sea ice is assumed to be proportional to the shear rate. We then explain the strategy to obtain the local strong well-posedness and the global strong well-posedness close to constant equilibria of this model. Finally, we consider some related problems and remaining questions. This talk is based on joint work with T. Binz, K. Disser, R. Haller, M. Hieber, A. Roy and T. Zöchling.

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