The Shock Development Problem (EMS Monographs in Mathematics) by Demetrios Christodoulou (PDF)

Description

This monograph addresses the problem of the development of shocks in the context of the Eulerian equations of the mechanics of compressible fluids. The mathematical problem is that of an initial-boundary value problem for a nonlinear hyperbolic system of partial differential equations with a free boundary and singular initial conditions. The free boundary is the shock hypersurface and the boundary conditions are jump conditions relative to a prior solution, conditions following from the integral form of the mass, momentum and energy conservation laws. The prior solution is provided by the author’s previous work which studies the maximal classical development of smooth initial data. New geometric and analytic methods are introduced to solve the problem. Geometry enters as the acoustical structure, a Lorentzian metric structure defined on the spacetime manifold by the fluid. This acoustical structure interacts with the background spacetime structure. Reformulating the equations as two coupled first order systems, the characteristic system, which is fully nonlinear, and the wave system, which is quasilinear, a complete regularization of the problem is achieved. Geometric methods also arise from the need to treat the free boundary. These methods involve the concepts of bivariational stress and of variation fields. The main new analytic method arises from the need to handle the singular integrals appearing in the energy identities. Shocks are an ubiquitous phenomenon and also occur in magnetohydrodynamics, nonlinear elasticity, and the electrodynamics of nonlinear media. The methods developed in this monograph are likely to be found relevant in these fields as well.

User’s Reviews

Editorial Reviews: Review The shock development problem is presented here within the framework of Euler’s equations for a compressible, perfect fluid combined with the laws of thermodynamics. These equations are based on the conservation laws of mass, momentum, and energy. The result is a quasi-linear hyperbolic first-order system with physical variables that include fluid velocity and two positive thermodynamic quantities. This book requires a substantial background with partial differential equations, experience with fluid mechanics, and at least some acquaintance with Lorenzian geometry. It is clearly aimed at those with advanced expertise in the field. –Bill Satzer, MAA Reviews

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