|Location:||MSRI: Simons Auditorium|
Speaker: Steve Shkoller
Title: A Short Introduction to Free-boundary Problems for Incompressible and Compressible Fluid Dynamics
The motion of interfaces and free-boundaries (moving hypersurfaces) in perfect fluids are governed by the Euler equations of hydrodynamics. I will describe the analysis for the class of problems in which the speed of the moving free-boundary is given by the normal component of the fluid velocity, which itself is found from the solution of the Euler equations (usually in three space dimensions).
These lectures will introduce the Lagrangian version of the Euler equations, the geometry of the inverse deformation of the flow map, nonlinear a priori estimates, and new technical tools to construct solutions for this class of problems. If time permits, I will explain the connections with the Hele-Shaw equations, as well as the MHD equations of magneto-hydrodynamics.