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Energy methods for free boundary problems. Applications to nonlinear PDEs and fluid mechanics. (English) Zbl 0988.35002

Progress in Nonlinear Differential Equations and their Applications. 48. Basel: Birkhäuser. xi, 329 p. (2002).
This book is the result of collaboration among the authors over the last fifteen years. Their collaboration centered on using different energy methods to get conditions on the structure of a PDE or system of PDEs which yields the formation of a free boundary; in other words, this means that the support of a solution is localized in the space-time domain. Energy methods are of special interest in those situations in which traditional methods based on comparison principles have failed, or even when the comparison principle holds, it may be extremely difficult to construct suitable sub- or supersolutions. The main idea of the energy methods consists in deriving and studying suitable ordinary differential inequalities for various types of energy. In typical situations, these inequalities follow from the conservation and balance laws of continuum mechanics. In the simplest situations, the energy functions defined through a formal procedure coincide with the kinetic and potential energy. The first three chapters begin with a systematic explanation of the energy methods in applications to nonlinear PDEs. Each chapter concludes with a section devoted to bibliographical comments and open problems. A thorough exposition of the application of the methods is developed for problems in fluid mechanics in Chapter 4, where bibliographical comments are incorporated within the sections. Some useful facts from the theory of Sobolev spaces are given in the Appendix. The book contains some new unpublished results. It will appeal to researchers in partial differential equations, nonlinear analysis, and continuum mechanics.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
35R35 Free boundary problems for PDEs
80A20 Heat and mass transfer, heat flow (MSC2010)
80A22 Stefan problems, phase changes, etc.
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