Abstract: I will discuss my works (some joint) on the compressible Euler equations with non-trivial vorticity and entropy. We derived a new formulation of the equations exhibiting miraculous geoanalytic 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 solutions to the relativistic Euler equations 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. 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.
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