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Explore a 50-minute lecture by Vlad Vicol examining shock formation and maximal hyperbolic development in multi-dimensional gas dynamics. Delve into the Cauchy problem for multi-dimensional compressible Euler equations evolving from compressive and generic smooth initial data. Learn about the construction of unique solutions that are as smooth as the initial data within the maximal globally hyperbolic development (MGHD) of given Cauchy data. Understand how the future temporal boundary of this spacetime region forms a singular hypersurface comprising three distinct sets: a co-dimension-2 surface of "first singularities" called the pre-shock, a downstream co-dimension-1 surface with gradient catastrophes, and an upstream co-dimension-1 Cauchy horizon surface. Discover the new geometric framework developed for describing acoustic characteristic surfaces, combined with differentiated Riemann-type variables that blend velocity/sound speed gradients with fast acoustic characteristic surface curvature. This lecture presents joint work with Steve Shkoller from the University of California at Davis.
