Georgia Institute of Technology College of Sciences

Compressible Euler equations describe the motion of compressible inviscid fluid. Physically, it states the basic conservation laws of mass, momentum, and energy. As one of the most important examples of nonlinear hyperbolic conservation laws, it is well-known that singularity will form in the solutions of Compressible Euler equations even with small smooth initial data. This talk will discuss some classical results in this direction, including some most recent results for the problem with large initial data.

Nonlinear dispersive and wave equations constitute an area of PDE that has witnessed tremendous activity over the past thirty years. Such equations mostly orginate from physics; examples include nonlinear Schroedinger, wave, Klein-Gordon, and water wave equations, as well as Einstein's equations in general relativity. The rapid developments in this theory were, to a large extent, driven by several successful interactions with other areas of mathematics, mainly harmonic analysis, but also geometry, mathematical physics, probability, and even analytic number theory (we will touch on this in another talk). This led to many elegant tools and rather beautiful mathematical arguments. We will try to give a panoramic, yet very selective, survey of this rich topic focusing on intuition rather than technicalities. In this second talk, we continue discussing some aspects of nonlinear dispersive equations posed on Euclidean spaces.