Applied Math Colloquium: Dr. Marcelo Disconzi

Title: Rough sound waves in the three-dimensional compressible Euler system.

Abstract: We study the three-dimensional compressible Euler equations and prove that under low regularity assumptions on the initial data, it is possible to avoid, at least for short times, the formation of shocks. More precisely, our main result is that the time of classical existence can be controlled under low regularity assumptions on the part of the initial data associated with propagation of sound waves in the fluid. Such low regularity assumptions are in fact optimal. We treat the equations in full generality; in particular, the fluid is allowed to have vorticity and entropy. Compared to previous works on low regularity, the main new feature of our result is that the quasilinear PDE system under study exhibits multiple speeds of propagation. In fact, this is the first result of its kind for a system with multiple characteristic speeds. An interesting feature of our proof is the use of techniques that originated in the study of the vacuum Einstein equations in general relativity. This is joint work with C. Luo, G. Mazzone, and J. Speck (arXiv:1909.02550 [math.AP]).