The Shock Development Problem

Abstract

The subject of this work is the shock development problem in fluid mechanics. A shock originates from an acoustically spacelike surface in spacetime at which the 1st derivatives of the physical variables blow up. The solution requires the construction of a hypersurface in spacetime which is acoustically timelike as viewed from its future, acoustically spacelike as viewed from its past, the shock hypersurface, across which the physical variables suffer discontinuities obeying jump conditions in accordance with the integral form of the particle and energy-momentum conservation laws. Mathematically, this is a free boundary problem, with nonlinear conditions at the free boundary, for a 1st order quasilinear hyperbolic system of p.d.e., with characteristic initial data which are singular at the past boundary of the initial characteristic hypersurface, that boundary being the surface of origin. This work solves, in any number of spatial dimensions, a restricted form of the problem which retains the essential difficulties due to the singular nature of the surface of origin. The solution is accomplished through the introduction of new geometric and analytic methods.