## Seminars

# Research Unit FOR 5528 Seminars 2024

Speaker/Titel | Date | Location |

tba |
| Room 401 (Schlossgartenstraße 7 (S2|15), 64289 Darmstadt) and online (Zoom) |

### Abstract:

tba

Speaker/Titel | Date | Location |

The shallow water equations with contact angle |
| Room 401 (Schlossgartenstraße 7 (S2|15), 64289 Darmstadt) and online (Zoom) |

### "The shallow water equations with contact angle"

The shallow water equations are approximations of the water wave equations when the depth of the flow is small. The shallow water equations are one of the fundamental models for large-scale geophysics flow. In this work, we use the shallow water equations to model small scale water wave equations, where the surface tension becomes significant. Moreover, on the contact point of the flow with the bottom surface, the surface tension induces a contact angle between the upper surface and the bottom wall surface. At the level of shallow water equations, this resembles the free boundary problem of compressible flow with physical vacuum. In this talk, we formally derive the shallow water equations with contact angle, and investigate the asymptotic stability of the stationary equilibrium. This is joint work with J. Li and D. Peschka.

Speaker/Titel | Date | Location |

Singularity formation of the inviscid primitive equations | Feb 6, 2024 16h00 | Room 315 (Schlossgartenstraße 7 (S2|15), 64289 Darmstadt) and online (Zoom) |

### "Singularity formation of the inviscid primitive equations"

The inviscid primitive equations, also known as the hydrostatic Euler equations, describe the motion of inviscid fluid flow in a thin domain, such as the ocean and atmosphere on a planetary scale. Unlike the viscous primitive equations whose solutions exist for all time, certain solutions to the inviscid model blow up in finite time. In this talk, I will first revisit some earlier results on the singularity formation, and then discuss the effect of rotation on these blowup. Finally, I will talk about some recent progress toward the stability of the singularity formation. These results are joint works with Charles Collot, Slim Ibrahim, and Edriss Titi.

Speaker/Titel | Date | Location |

On the inviscid and non-diffusive primitive equations with Gent and McWilliams parametrization |
| Room 315 (Schlossgartenstraße 7 (S2|15), 64289 Darmstadt) and online (Zoom) |

### "On the inviscid and non-diffusive primitive equations with Gent and McWilliams parametrization"

In this talk I will present a work (arXiv:2206.01058) in collaboration with Roberta Bianchini (CNR, Roma), which proposes a naive approach to the inviscid and non-diffusive primitive equations.

The key observations that I would like to discuss are that

- isopycnal coordinates provide interesting insights on the structure of the primitive equations in idealized frameworks (stable stratification, flat bottom...)

- the Gent and McWilliams parametrization alone provides sufficient regularization of the equations so as to offer robust stability estimates for sufficiently regular data.As a result, our work provides a (local-in-time) well-posedness theory of the inviscid and non-diffusive primitive equations with Gent and McWilliams parametrization, and the strong convergence of solutions to the corresponding non-hydrostatic system as the shallow water aspect ratio vanishes.

# Research Unit FOR 5528 Seminars 2023

Speaker/Titel | Date | Location |

On energy conservation for inviscid hydrodynamic equations: analogues of Onsager's conjecture | Dec 12, 2023 16h00 | Room 315 (Schlossgartenstraße 7 (S2|15), 64289 Darmstadt) and online (Zoom) |

### "On energy conservation for inviscid hydrodynamic equations: analogues of Onsager's conjecture"

Onsager's conjecture states that 1/3 is the critical spatial (Hölder) regularity threshold for energy conservation by weak solutions of the incompressible Euler equations. We consider an analogue of Onsager's conjecture for the inviscid primitive equations of oceanic and atmospheric dynamics. The anisotropic nature of these equations allows us to introduce new types of weak solutions and prove a range of independent sufficient criteria for energy conservation. Therefore there probably is a 'family' of Onsager conjectures for these equations.

Furthermore, we employ the method of convex integration to show the nonuniqueness of weak solutions to the inviscid and viscous primitive equations (and also the Prandtl equations), and to construct examples of solutions that do not conserve energy in the inviscid case. Finally, we present a regularity result for the pressure in the Euler equations, which is of relevance to the Onsager conjecture in the presence of physical boundaries. As an essential part of the proof, we introduce a new weaker notion of pressure boundary condition which we show to be necessary by means of an explicit example. These results are joint works with Claude Bardos, Simon Markfelder and Edriss S. Titi.

Speaker/Titel | Date | Location |

Non-uniqueness and Inadmissibility of the Vanishing Viscosity Limit of the Passive Scalar Transport Equation | Nov 28, 2023 16h00 | Room 315 (Schlossgartenstraße 7 (S2|15), 64289 Darmstadt) and online (Zoom) |

A multi-cloud and multi-mode model for convectively driven coastal flows | Nov 28, 2023 17h00 | Room 315 (Schlossgartenstraße 7 (S2|15), 64289 Darmstadt) and online (Zoom) |

### "Non-uniqueness and Inadmissibility of the Vanishing Viscosity Limit of the Passive Scalar Transport Equation"

We consider the linear transport of a passive scalar along a given bounded background incompressible flow. In this setting the viscous problem (the advection-diffusion equation) is globally well-posed, while the non-viscous problem is ill-posed unless the background velocity field is assumed to be more regular. We study the limit as viscosity goes to zero for the advection diffusion equation and construct two background velocity fields for which the vanishing viscosity/diffusion limit behaves unphysically. For the first of these, any initial data admits two different vanishing viscosity subsequences which converge to two different (renormalised) weak solutions to the transport equation. The second construction has for any initial data a unique vanishing viscosity limit, but this limit is perfectly mixed to its average, and after a short delay subsequently unmixes to its initial state, disobeying entropy admissibility for every initial data.

### "A multi-cloud and multi-mode model for convectively driven coastal flows"

Coastal areas have the peculiarity of being affected by the land-sea contrast that drives sea and land breezes oscillating back a forth between the day and night, respectively. In return, these flows characterize the weather and local climates of coastal areas especially through the associated overturning circulation that regulates local convection and precipitation both over land and over the ocean. This sometimes leads to multiple scale precipitation systems that propagate in either direction across the coastline (i.e., landward and oceanward) that are believe to interact with weather systems on synoptic and intraseanal systems that are important globally for both weather predictions on a fews weeks to a few months as well as climate variability. Despite numerous theoretical and observational efforts to understand coastal convection, global climate models still fail to represent it adequately, mainly because of limitations in spatial resolution and shortcomings in the underlying cumulus parameterization schemes. Here we use a simplified model of intermediate complexity to simulate coastal convection under the influence of the diurnal cycle of solar heating.

A key mathematical aspect of the models is that it systematically couples the dynamics in the mixed planetary boundary layer and the barotropic and first two baroclinic modes of vertical structure. Convection is parameterized via a stochastic multicloud model (SMCM), which mimics the subgrid dynamics of organized convection. In particular, the boundary layer dynamics are represented by a bulk model coupling the changing surface conditions in both space and time and free tropospheric flow through the exchange of heat and momentum fluxes at respectively the bottom and top of the boundary layer.

Numerical results demonstrate that the model is able to capture some key modes of coastal convection variability, such as the diurnal cycle of convection and the accompanying sea and land breeze reversals, the slowly propagating mesoscale convective systems that move from land to ocean and vice-versa, and numerous moisture-coupled gravity wave modes. The physical features of the simulated modes, such as their propagation speeds, the timing of rainfall peaks, the penetration of the sea and land breezes, and how they are affected by the latitudinal variation of the Coriolis force, are generally consistent with existing theoretical and observational studies.

(Joint work with Abigail Dah and Courtney Schumacher).

Speaker/ Titel | Date | Location |

Well-posedness and long-time behaviour of the Stokes-transport equation | Oct 31, 2023 16h00 | Room 315 (Schlossgartenstraße 7 (S2|15), 64289 Darmstadt) and online (Zoom) |

Rigorous analysis of Hibler's sea ice model - recent results and related problems | Oct 31, 2023 17h00 | Room 315 (Schlossgartenstraße 7 (S2|15), 64289 Darmstadt) and online (Zoom) |

### "Well-posedness and long-time behaviour of the Stokes-transport equation"

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"

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.