  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.
