The hydrostatic primitive equations (HPEs) constitute a solid backbone for atmosphere-ocean flow modelling. Today, there is a substantial body of results on their well-posedness as well as on the local/global in time existence/uniqueness of weak/strong solutions. Moreover, they are the basic flow model for the highly successful global weather forecasts of the European Centre for Medium-Range Weather Forecasts (ECMWF), so that their validity is also well-established from the practitioners' perspective. Thus, this model family can be trusted as a reference for processes on scales larger than their small-scale limits of validity of about 25 km horizontally, 5-10 km vertically, and 10 min in time. This motivates this project's first workpackage: Rigorous justification of reduced models for large-scale atmospheric flows relative to the HPEs.
While the established mathematical theory for the HPEs is impressive, a number of pertinent questions remain, such as the uniqueness of weak solutions of the viscous model and the weak solution theory in the inviscid case. Recent progress on the 3D Navier-Stokes system suggests that global weak solutions of the viscous HPE system may not be unique, and recent results on the Onsager-type rigidity question shed additional and very different light on the issue. In a second work package we build on these works and initiate a systematic study of weak solutions of the HPEs: Weak solutions of the hydrostatic primitive equations.
For flow models addressing smaller scales, one likely faces the same challenges as recently posed for the Euler and Navier-Stokes equations: The concept of weak solutions, one of the backbones of PDE theory, has been questioned by new results from convex integration (CI): There are weak solutions of the Euler and Navier-Stokes equations that are compactly supported in space and time. This, of course, is not physically meaningful as it would imply the existence of a perpetuum mobile. Important challenges to applying CI in the geophysical setting are (i) to take care of the strong gravitational and rotational effects and (ii) the extreme vertical-to-horizontal aspect ratio of atmosphere-ocean flows. These aspects give rise to an array of interesting questions regarding smaller scale geophysical flows to be addressed in the third workpackage of this project: Convex integration theory for non-hydrostatic (smaller-scale) reduced atmospheric flow models.
Prof. Dr. Matthias Hieber
Technische Universität Darmstadt
Prof. Dr.-Ing. Rupert Klein
Freie Universität Berlin
Prof. Dr. László Székelyhidi
Max-Planck-Institut für Mathematik in den Naturwissenschaften