*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.

Smith Colloquium Fall 2019