The Shock Development Problem

TL;DR

This paper addresses the complex shock development problem in fluid mechanics, focusing on constructing a hypersurface in spacetime where physical variables exhibit discontinuities, using novel geometric and analytic methods to handle singular initial data.

Contribution

It introduces new geometric and analytic techniques to solve a restricted form of the shock development problem in multiple dimensions with singular initial conditions.

Findings

Constructed a spacetime hypersurface with discontinuities obeying conservation laws.

Developed methods to handle singular initial data at the surface of origin.

Extended solutions to any number of spatial dimensions.

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.