Wolfgang Pauli Institute (WPI) Vienna |
||
|---|---|---|
| ||
|
Topics: 10h15 coffee & cake 10h25 Norbert J. Mauser: introduction A) 10h30 – 11h20 : Rupert KLEIN (FU Berlin) "Asymptotic analysis: What is it and what is it good for?" Abstract: famous mathematicians, some of them practitioners themselves, have associated "asymptotics" with the "dark arts", "inventions of the devil", "a toolbox of tricks for special cases", and the like. In 40 years of experience with it I have come to a broader perspective, and that is the topic of this lecture. Most and foremost, I view Mathematics as the "science of structure", and I will argue why this leads me to consider asymptotics to be quite a systematic mathematical endeavour. I will highlight that asymptotics is found all over the place, from the applied mathematics of practical engineering to the award-winning work of a recent Fields medalist, and that the common denominator is the search for structure and understanding. Seen this way, the multi-faceted nature of asymptotics comes out as a necessity rather than as a reason for mockery. An asymptotic theory of tropical storms that is under development in my group will provide examples to go with my elaborations. B) 11h20 – 12h10 : Edriss TITI (U.Cambridge, Texas A&M, Weizmann) “Mathematical Analysis of Geophysical Models” Abstract: We present recent results concerning the global regularity of certain geophysical models. In particular, the three-dimensional Planetary Geostrophic and the Primitive Equations (PE) of oceanic and atmospheric dynamics with various anisotropic viscosity and turbulence mixing diffusion. However, in the non-viscous (inviscid) case it can be shown that, with or without rotation, the PE are linearly and nonlinearly ill-posed in the context of Sobolev spaces, and that there is a one-parameter family of initial data for which the corresponding smooth solutions of the primitive equations develop finite-time singularities (blowup). However, the PE will be shown to be well-posed in the space of real analytic functions, and we will discuss the effect of rotation on prolonging the life-span of analytic solutions. Capitalizing on the above results, we provide rigorous justification of the derivation of the viscous PE of planetary scale oceanic dynamics from the three-dimensional Navier-Stokes equations, for vanishing small values of the aspect ratio of the depth to horizontal width. Specifically, we can show that the Navier-Stokes equations, scaled appropriately by the small aspect ratio parameter of the physical domain, converge strongly to the primitive equations, globally and uniformly in time, and that the convergence rate is of the same order as the aspect ratio parameter. | ||
|
| © WPI 2001-2004. | www.wpi.ac.at |