The starting point is our new formulations 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.
I will then describe a main application: the study of stable shock formation, without symmetry assumptions, in more than one spatial dimension. I will emphasize the role that nonlinear geometric optics plays in the analysis and highlight how the new formulations allow for its implementation. Finally, I will describe some important open problems, and I will connect the results to the broader goal of obtaining a rigorous mathematical theory that models the long-time behavior of solutions in the presence of shock singularities.