In this CRC colloquium we're happy to welcome:
Edriss Titi (location: WIAS) - Please note the additional date 7.7.2023, 10:00 am, via Zoom-Meeting:
Title: The Inviscid Primitive Equations and the Effect of Fast Rotation
Large scale dynamics of the oceans and the atmosphere are governed by the primitive equation (PE). It is well-known that the three-dimensional viscous PE is globally well-posed in Sobolev spaces. In this talk, I will discuss the ill-posedness in Sobolev spaces, the local well-posedness in the space of analytic functions, and the finite-time blowup of solutions to the three-dimensional inviscid PE (also known as the hydrostatic Euler equations) with rotation (Coriolis force). Eventually, I will also show, in the case of “well-prepared” analytic initial data, the regularizing effect of the Coriolis force by providing a lower bound for the life-span of the solutions which grows toward infinity with the rotation rate. The latter is achieved by a delicate analysis of a simple limit resonant system whose solution approximates the corresponding solution of the 3D inviscid PE with the same initial data. In addition, and if time allows I will discuss Onsager’s conjecture in the context of the inviscid primitive equations.
Title: Transfer operators on graphs: Spectral clustering and beyond
While spectral clustering algorithms for undirected graphs are well established and have been successfully applied to unsupervised machine learning problems ranging from image segmentation and genome sequencing to signal processing and social network analysis, clustering directed graphs remains notoriously difficult. We will first exploit relationships between the graph Laplacian and transfer operators and in particular between clusters in undirected graphs and metastable sets in stochastic dynamical systems and then use a generalization of the notion of metastability to derive clustering algorithms for directed and time-evolving graphs. The resulting clusters can be interpreted as coherent sets, which play an important role in the analysis of transport and mixing processes in fluid flows. We will illustrate the results with the aid of guiding examples and simple benchmark problems.
Title: A revisit to the rigorous justification of the quasi-geostrophic approximation
The quasi-geostrophic approximation is used to model large-scale atmospheric/oceanic flows close to the geostrophic balance, i.e., the Coriolis force, the pressure, and the gravity are in balance. Such an approximation for inviscid flows has been investigated in the case without boundary or without oscillating fast waves. In this talk, I will (1) review the classical mathematical results of the QG approximation, (2) point out the possible boundary layer when fast rotation is not present, and (3) show that with fast rotation, there is no boundary layer. In particular, we rigorously justify the QG approximation with both boundary and oscillatory fast waves. This is done by introducing a new generalized potential vorticity, obtaining uniform estimates, and passing the weak limit. Our result demonstrates the stabilizing effect of rotation by suppressing the boundary layer. This is joint work with C. Bardos and E. Titi.
Title: Semi-explicit discretization of elliptic-parabolic equations
Within this talk, we discuss the time discretization of coupled elliptic-parabolic systems, which include the equations of poroelasticity. Fully implicit methods exhibit inefficiency due to the high dimensionality of the coupled problem. As such, we consider a semi-explicit approach, meaning that the mechanics and flow equations are solved sequentially. In contrast to classical iterative methods such as the fixed-stress scheme, no relaxation parameter or inner iteration is needed. On the other hand, the semi-explicit approach restricts the class of possible applications due to stability issues. To overcome this problem, we present a novel time integration scheme which combines the iterative idea with the semi-explicit Euler approach. For this, we are able to prove first-order convergence for an a priori specified number of inner iteration steps, only depending on the coupling strength.