The study of the physical evolution of a compressible fluid beyond the point where surfaces in spacetime appear where the derivatives of the physical variables blow up leads to the shock development problem. This is a free boundary problem for a quasilinear hyperbolic system of partial differential equations, with nonlinear conditions at the free boundary, the shock hypersurface, and with initial conditions on a transversal characteristic hypersurface which are singular at its past boundary, which is at the same time
the past boundary of the shock hypersurface. My recent work solves this problem by introducing two new geometric methods and one new analytic method. The first geometric method transforms the problem into a problem of two coupled differential systems on a manifold, the first of which is fully nonlinear. What is achieved by this transformation is a complete regularization of the problem. The second geometric method is concerned with the derivation of energy identities for free boundary problems and features vectorfields in the role of variation fields. The analytic method is designed to overcome the difficulties arising from the severely singular integrals appearing as error integrals in the energy identities. Speaker(s): Demetrios Christodoulou (ETH Zurich, Switzerland)