Fluid Dynamics

Summer School

June 17.-22. 2000

The purpose of the Summer School is twofold: on the one hand participants will learn about the mathematical subject, which is fluid dynamics. On the other hand, the School is addressed to young mathematicians in the process of becoming independent researchers, so the program is designed to give a maximum of insight into what mathematical research is like nowadays. Thus we will focus on (but not be entirely restricted to) a number of recent papers/preprints, and some written with the participation of very young researchers.

This also means that we will not devellop every detail of the theory, and we will have to take some of the results and implications quoted in the papers as granted without full explanation.

Fluid dynamics is the study of various PDE which govern the behaviour of fluids under various assumptions. For example, Euler's equation describes an incompressible fluid without viscosity, whereas the Navier-Stokes equation governs the incompressible fluid with viscosity. For both equations (say in 3D), it is a fundamental question whether smooth solutions to these equations exist for all time given smooth initial data. In the case of the Navier Stokes equation, this is one of the seven Millenium problems posed by the Clay Institute. All PDE in this area are nonlinear. The papers by Cordoba/Fefferman, Beale-Kato-Majda, Constantin-Fefferman-Majda deal with possible scenarios for a blowup of the solutions to Navier-Stokes or Euler equation. The paper by Koch-Tataru proves existence of solutions to Navier Stokes under smallness assumption on the initial data, which guarantee that the viscosity term dominates and thus Navier-Stokes equation behaves like the heat equation. The Korteweg de Vries equation models water waves in long shallow channels. This equation has been much studied in the past 30 years, since it has remarkable structures (solitons, inverse scattering, connection to Schroedinger equations). The article by Kenig-Ponce-Vega proves existence of solutions under mild assumptions on the initial data. The paper by Sijue Wu proves existence of solutions for the equation that describes the surface of 2D water. It uses the Riemann mapping in a remarkable way.