Oberseminar AG Analysis - Felix Brandt, TU Darmstadt

Analysis of Hibler's Sea Ice Model and Time Periodic Quasilinear Evolution Equations

The first part of the talk is dedicated to the rigorous analysis of several models in the context of Hibler’s viscous-plastic sea ice model. The latter model was introduced in 1979 and represents a large-scale dynamic-thermodynamic model. Even though there is a plethora of literature on
numerical analysis and modeling, a rigorous mathematical analysis for this model has only been developed quite recently.
First, we consider a fully parabolic variant of Hibler’s model and establish
the local strong well-posedness and global strong well-posedness close to
equilibria. Key steps here are the reformulation as a quasilinear abstract
Cauchy problem as well as the study of the so-called Hibler operator arising
from the internal ice stress.
Sea ice is subject to atmospheric wind and oceanic forces. We thus take into account a coupled atmosphere-sea ice-ocean model, where Hibler’s model is coupled to two viscous incompressible primitive equations via atmospheric drag force, shear stress and continuity of the ocean and ice velocity.
A decoupling argument involving a hydrostatic Dirichlet and Dirichlet-to-Neumann operator allows us to deduce the bounded H∞-calculus of the linearized operator matrix with non-diagonal domain. This paves the way for similar well-posedness results as for the fully parabolic Hibler model.
We also investigate a parabolic-hyperbolic variant of Hibler’s model. Employing Lagrangian coordinates, we are able to handle the hyperbolic effects in the balance laws and obtain maximal Lp-regularity of the linearization and then also local strong well-posedness of the parabolic-hyperbolic problems.
In the second part of the talk, we present approaches to time periodic quasilinear equations by the Arendt-Bu theorem on maximal periodic Lp-regularity and by a time periodic version of the classical Da Prato-Grisvard theorem. We also provide a framework tailored to bilinear problems. Finally,
a previously developed framework is used to obtain the existence of time periodic strong solutions to Hibler’s fully parabolic model for small forces.
The talk is based on joint work with Tim Binz, Karoline Disser, Robert
Haller and Matthias Hieber.