Weekly Colloquium/Seminar Notices

In this talk we survey some recent results concerning non-uniqueness of weak solutions of inviscid equations of fluid mechanics, most prominently the Euler equations. Although non-uniqueness is well known and well understood in 1 space dimension (e.g. for Burgers), in higher dimensions unexpected phenomena appear, very much akin to the so-called h-principle in geometry. The latter is a term due to Gromov, an encompasses results such as the Nash-Kuiper isometric embedding theorem or the Smale sphere inversion, where the space of solutions to a certain system of PDEs is highly flexible. One still has to make sense of the analogous flexibility for fluid dynamics in the context of turbulence, but so far it seems that there is a close relationship to Kolmogorov's statistical theory.

Jupiter's quite stable belts and our ozone hole can both be modeled as Lagrangian Coherent Structures (LCS) of elliptic type in incompressible two-dimensional fluid flows. Elliptic LCS are material fluid curves that do not experience exponential stretching and folding over a finite-time interval. Elliptic LCS inhibit mixing. A vivid example of an elliptic LCS in a geophysical fluid flow system is that one at the perimeter of the austral stratospheric polar vortex. Such an elliptic LCS prevents ozone-depleted air from spreading toward lower latitudes, leading to the formation of the so-called ozone hole. I will also discuss a theory that provides support for their occurrence. It is related to the Kolmogorov--Arnold--Moser theory for time-quasiperiodic one-degree-of-freedom Hamiltonian systems for which the frequency mapping is degenerate in the Kolmogorov sense. This talk is the result of several years of collaboration with oceanographers at the University of Miami's Rosenstiel School of Marine and Atmospheric Science (Francisco Beron-Vera, Michael Brown, and Mara Olascoaga).

A Margulis spacetime is a complete affine 3-manifold M with nonsolvable fundamental group. Associated to every Margulis spacetime is a noncompact complete hyperbolic surface S. Surprisingly, every Margulis spacetime is orientable, even when S is nonorientable. We will talk about the case where S is a two-holed cross-surface.