7R42. Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics. Progress in Nonlinear Differential Equations and Their Applications, Vol 48. - SN Antontsev (Dept de Matematica, Univ de Beira Interior, Covilha, 6201-001, Portugal), JI Diaz (Dept de Matematica Aplicada, Univ Complutenese, Madrid, 28040, Spain), S Shmarev (Dept de Matematicas, Univ de Oviedo, Oviedo, 33007, Spain). Birkhauser Boston, Cambridge MA. 2002. 329 pp. ISBN 0-8176-4123-8. \$79.95.

Reviewed by AJ Kassab (Dept of Mech, Mat, and Aerospace Eng, Col of Eng, Univ of Central Florida, Orlando FL 32816-2450).

This is a research monograph reporting on collaborative research carried out by the authors over the past 15 years in the field of free boundary problem. Specifically, the authors use energy methods as the basic tool to investigate the underlying mathematical structure of the partial differential equations (PDE) governing various field problems whose solutions lead to the formation of a free boundary. Given the definition of an energy norm, the technique results in inequalities involving ordinary differential equations whose study allows determination of the existence of a free boundary and provides a theoretical framework for the study of the solutions for the PDEs of interest. This book mainly considers application of energy methods to quasi-nonlinear equations of fluid mechanics.

The first three chapters lay the formal mathematical and theoretical framework of energy methods applied to nonlinear PDEs. The first chapter introduces energy methods to study nonlinear stationary problems giving rise to localized solutions and hence free boundaries. Second-order elliptic equations, systems of elliptic equations, and higher-order elliptic equations as well as isotropic and anisotropic diffusion problems are considered. The second chapter addresses stabilization to a stationary solution for nonlinear evolution problems. Here again, the authors progress from parabolic equations to anisotropic parabolic equations, to systems of equations of combined type and higher order parabolic equations. The third chapter addresses space and time localization of weak solutions for nonlinear degenerate parabolic equations as well as systems of these equations. Application of energy methods to various fluid mechanics problems is provided in the fourth chapter. The chapter begins with a brief review of theoretical fluid mechanics, followed by a series of applications ranging from free boundary gas dynamics problems such as free jets and collisions of co-axial jets to filtration of immiscible fluids in porous media which finds application in recovery of oil from a reservoir and to flows of non-homogeneous non-Newtonian fluids. The authors also treat a class of problems which require the simultaneous study of surface channel flow as well as underground water flow. The final two topics studied involves the drift diffusion model for free electrons that forms quasi-linear degenerate systems of PDEs and finds application in semi-conductor theory and the so-called blow up regimes in the solution of coupled PDEs for heat conduction and electrostatic potential resulting from the model of heat diffusion problems in a conductor in the presence of Joule heating.

The book often follows mathematical exposition of statement of lemmas and theorems, followed by proofs and corollaries as well as relevant remarks. Each chapter terminates with a bibliographic review and a set of open problems that are essentially proposals for a program of research in topics not fully investigated in the literature or by the authors. There are some illustrations, all of which are of high quality. There are no illustrative computational results. There is an appendix which contains some basic definitions of function spaces and a brief review of elementary inequalities and imbedding theorems. There is a copious number of $300+$ references which support the text and a detailed index.

The monograph is the 48th volume in a series entitled Progress in Nonlinear Differential Equations and Their Applications, edited by H Brezis of the Mathematics Department of both Rutgers University in New Jersey, and the Universite Pierre and Marie Curie in Paris. The series is described as lying at the interface of pure and applied mathematics, and indeed this book falls into such a category. Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics can serve as a reference on the subject of energy methods when they are treated as part of mathematics post-graduate courses on partial differential equations; It is recommended for acquisition by university libraries as a quality addition to their mathematics collections.