Quasilinear hyperbolic PDE systems arise in many branches of mathematics and physics. A fundamental issue permeating their study is that, aside from equations with exceptional structure, initially smooth solutions are often expected to form shocks in finite time. Roughly, a shock is a singularity such that the solution remains bounded but its derivatives blow up. Although many shock formation results have been proved in one spatial dimension, there are very few constructive proofs of shock formation in more than one spatial dimension. In this colloquium, I will provide an overview of recent progress on the formation of shocks in two and three spatial dimensions. I will start by describing prior contributions from many researchers including B. Riemann, P. Lax, F.John, S. Alinhac, and especially D. Christodoulou, whose remarkable 2007 monograph yielded a sharp description of shock formation in vorticity-free small-data solutions to the relativistic Euler equations in three spatial dimensions.

I will then describe some of my recent work, some of it joint with G. Holzegel, S. Klainerman, J. Luk, and W. Wong, in which we extended Christodoulou's framework and proved similar results for general classes of equations and new types of initial conditions. I will especially focus on my work with J. Luk on the compressible Euler equations, in which we obtained the first constructive result on the long-time behavior of the vorticity up to the first shock: for an open set of initial data, generic first derivatives of the velocity blow up but the vorticity remains bounded! The proof relies on a new formulation of the equations exhibiting surprisingly good ``null structures,'' reminiscent of the type found in equations that admit global solutions. Remarkably, the good structures are a key ingredient in proving that a singularity forms. Throughout the talk, I will highlight some of the main ideas behind the analysis including the critical role played by geometric decompositions adapted to characteristic hypersurfaces.

### Penn Mathematics Colloquium

Wednesday, January 25, 2017 - 3:30pm

#### Jared Speck

MIT