The Shock Development Problem

This is a very lengthy monograph on the development of shocks in fluid mechanics. The author is a mathematical physicist who is probably best known for his proof of the nonlinear stability of Minkowski spacetime (from special relativity) within the context of general relativity. 

 

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.

 

Starting with smooth initial data this system of equations leads to a formation of a surface where the derivatives of the physical quantities with respect to the standard rectangular coordinates blows up. This is the shock.

 

Shock development, as the author describes it, is an initial boundary layer problem with a free boundary and singular initial conditions. The free boundary is the shock hypersurface. The boundary conditions are jump conditions relative to a prior solution. That prior solution is described in a previous monograph by the author that focuses on the maximal classical development of smooth initial data. The author develops geometric and analytic methods that are new to the fluid mechanics world. He creates an acoustical structure with a Lorentzian metric defined on the spacetime manifold of the fluid. In his analysis, the acoustical structure interacts with the background spacetime structure.

 

The author then reformulates the basic equations as a system of two coupled first-order systems; one of these he calls the characteristic system, which is fully linear. The other is a wave system, which is quasilinear. In the end, he gets a complete regularization of the problem. Along the way, he uses his geometric formulation to characterize the free boundary, and he develops analytic methods to handle the singular integrals that arise in certain energy identities.

 

Other than the general area of fluid dynamics no specific applications are identified. The reader is left to wonder why a fairly complicated relativistic treatment is important here. Perhaps the author has astrophysical applications in mind. 

 

The book has only a basic bibliography. It has an index, but a very weak one - a significant handicap for a book of this length. There are a few figures, but readers could certainly benefit from seeing more of them.

 

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.

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Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films and material science. He did his PhD work in dynamical systems and celestial mechanics.