### Research Seminar (Continuing Class)

Talks in winter 2023/24 |
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The talks are taking place in seminar room 2.066.

23.04.2024, 14:00 Uhr | Mathias Wilke (Halle) |

*On the evolution of fluid flows*

We consider systems of partial differential equations, modeling the dynamics of fluids on Riemannian manifolds. In particular, we will demonstrate, how the theory of evolution equations contributes to the analysis of those problems.

30.04.2024, 14:00 Uhr | David Ploß (Karlsruhe) |

*Fourth-order elliptic operators with Wentzell boundary conditions*

This talk investigates elliptic operators of order four with Wentzell (or dynamic) boundary conditions, which desribe exchange of free energy between interior and boundary of a domain. In the simplest case the system is given by

We show that the operator governing this equation on a suitable product space generates a strongly continuous, real, and analytic semigroup . In order to achieve this we follow two different approaches:

In the case of a Hilbert space we use two nested quadratic forms and generalized Neumann traces. This method works even for Lipschitz domains and we obtain Hölder-regular solutions, exponential stablity and eventual positivity.

In the -setting we use parabolic theory, instead. However, we do not employ the classical trace spaces but the base space which leads to a theory of boundary value problems with very rough data; also the boundary regularity is finite. Key ideas, here, are spaces with different regularities in tangential and normal direction.

07.05.2024, 14:00 Uhr | Hedong Hou (Paris) |

*Well-posedness and maximal regularity for parabolic Cauchy problems on weighted tent spaces*

Used by the work of Koch-Tataru on Navier-Stokes equations, the theory of tent spaces turns out to be useful to deal with evolution equations with very rough initial data. In this talk, we shall discuss the recent progress on studying linear parabolic equations with time-independent, uniformly elliptic, bounded measurable complex coefficients via tent spaces. The talk is based on a joint work with Pascal Auscher.

28.05.2024, 14:00 Uhr | Sebastian Bechtel (Delft) |

*Hardy spaces on open sets*

The classical Hardy spaces can be considered as spaces adapted to the negative Laplacian on the Euclidean space. Taking this as the point of departure, we motivate and introduce Hardy spaces that are adapted to some elliptic operator on a domain, subject to suitable boundary conditions. We are interested in deriving an atomic description for such spaces. We will review constructions in the case of pure Dirichlet or Neumann boundary conditions and show how the definition of an atom in these two cases is related to the underlying boundary conditions of the operator. We will then work towards generalisations to so-called mixed boundary conditions and other relaxations of geometric requirements. This is based on joint work with T. Böhnlein.

04.06.2024, 14:00 Uhr | Felix Brandt (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 -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 -regularity of the linearization and then also local strong well-posedness of the parabolic-hyperbolic problem.

In the second part of the talk, we present approaches to time periodic quasilinear equations by the Arendt-Bu theorem on maximal periodic -regularity and by a time periodic version of the classical Da Prato-Grisvard theorem. 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.

You can find previous talks in the archive of the research seminar.