Jared Speck, Vanderbilt University

Abstract: I will discuss my ongoing program on shock formation in solutions to the 3D compressible Euler equations with non-trivial vorticity and entropy. The program is based on a new formulation of the equations exhibiting miraculous geo-analytic structures, including I) A sharp decomposition of the flow into geometric “wave parts” and “transport-div-curl parts;” II) Null form source terms; III) Structures that allow one to propagate one additional degree of differentiability (compared to standard estimates) for the entropy and vorticity; and IV) Additional structures that allow one to localize the analysis in spacetime. We were inspired to search for such a formulation by Christodoulou’s groundbreaking 2007 monograph on shock formation for relativistic Euler solutions in irrotational and isentropic regions. I will then describe how the new formulation can be used to derive sharp results about the dynamics, with a special focus on stable shock formation without symmetry. In this context, I will discuss my recent work on the structure of the shock singularity at the time of first blowup as well as my forthcoming work on the structure of the maximal development. I will emphasize the role that nonlinear geometric optics plays in the framework and highlight how the new formulation allows for its implementation. Finally, I will connect the new formulation to the broader goal of obtaining a rigorous mathematical theory that models the long-time behavior of solutions that can develop shocks. Various aspects of this program are joint with L. Abbrescia, J. Luk, and M. Disconzi.