The Formation of Shock Waves in the Presence of Vorticity

In his 2007 monograph, D. Christodoulou proved a breakthrough result giving a complete description of the formation of shock waves, starting from small, regular initial conditions, in solutions to the relativistic Euler equations. In 2014, Christodoulou–Miao extended the result to the nonrelativistic compressible Euler equations. In both works, the assumptions on the initial conditions caused the shock to form in the acoustic wave zone, far from the region where vorticity is present. Consequently, Christodoulou and Miao were able to use the potential formulation of the Euler equations to study the shock formation. In this talk, I will describe my recent joint work with J. Luk, in which we prove a similar shock formation result for the compressible Euler equations, but allowing for small, non-zero vorticity in a neighborhood of the shock. To control the vorticity up to the shock, we rely on a coalition of new geometric and analytic insights that complement the ones used by Christodoulou–Miao. In particular, since the potential formulation is not available, we rely on our new formulation of the compressible Euler equations. The new formulation exhibits remarkable nonlinear null structures, reminiscent of the type found in geometric field theories. By exploiting these structures, we are able to prove that the vorticity remains uniformly bounded up to the shock and that it does not interfere with the singularity formation mechanisms. Thus, our work yields the first constructive description of the behavior of vorticity in a neighborhood of a singularity formed from compression. From a pure PDE perspective, our work provides the first constructive proof of stable blowup without symmetry assumptions for a quasilinear hyperbolic system featuring multiple speeds of propagation: the speed of sound and the speed associated to vorticity propagation.