Title: Smooth Solutions for the Euler Equations
Abstract: I will describe recent progress in our attempts to prove the existence of globally smooth shock-free solutions of the Euler equations of gas dynamics. Beginning with the physical phenomena that delay shock formation, we examine the structure of solutions in which shocks never form. I will describe the `simplest' solutions which are periodic in both space and time, and more recent examples which have higher regularity. We treat the existence of periodic solutions as a perturbation problem and reformulate it using a Nash-Moser iteration. The final step is expressed as an inversion of a projected operator. Much of this is joint work with Blake Temple.