Energy Methods for Free Boundary Problems, Softcover reprint of the original 1st ed. 2002 Applications to Nonlinear PDEs and Fluid Mechanics Progress in Nonlinear Differential Equations and Their Applications Series, Vol. 48

For the past several decades, the study of free boundary problems has been a very active subject of research occurring in a variety of applied sciences. What these problems have in common is their formulation in terms of suitably posed initial and boundary value problems for nonlinear partial differential equations. Such problems arise, for example, in the mathematical treatment of the processes of heat conduction, filtration through porous media, flows of non-Newtonian fluids, boundary layers, chemical reactions, semiconductors, and so on. The growing interest in these problems is reflected by the series of meetings held under the title "Free Boundary Problems: Theory and Applications" (Ox­ ford 1974, Pavia 1979, Durham 1978, Montecatini 1981, Maubuisson 1984, Irsee 1987, Montreal 1990, Toledo 1993, Zakopane 1995, Crete 1997, Chiba 1999). From the proceedings of these meetings, we can learn about the different kinds of mathematical areas that fall within the scope of free boundary problems. It is worth mentioning that the European Science Foundation supported a vast research project on free boundary problems from 1993 until 1999. The recent creation of the specialized journal Interfaces and Free Boundaries: Modeling, Analysis and Computation gives us an idea of the vitality of the subject and its present state of development. This book is a result of collaboration among the authors over the last 15 years.

1 Localized Solutions of Nonlinear Stationary Problems.- 1 Introduction.- 2 Second-order elliptic equations.- 3 The weighted diffusion/absorption balance.- 4 Anisotropic equations: Diffusion/absorption balance.- 5 Systems of second-order elliptic equations.- 6 Higher-order elliptic equations.- 7 Bibliographical notes and open problems.- 2 Stabilization in a Finite Time to a Stationary State.- 1 Introduction.- 2. Second-order parabolic equations.- 3 The weighted diffusion-absorption balance.- 4 The Cauchy problem.- 5 Equations with nonpower and isotropic nonlinearities.- 6 Systems of equations of combined type.- 7 Higher-order parabolic equations and other applications.- 8 Bibliographical notes and open problems.- 3 Space and Time Localization in Nonlinear Evolution Problems.- 1 Introduction.- 2 General second-order equations.- 3 The waiting time property.- 4 Shrinking of supports and formation of a dead core.- 5 Equations with nonhomogeneous absorption terms.- 6 Equations with anisotropic nonlinearities.- 7 Systems of parabolic equations.- 8 Higher-order parabolic equations.- 9 Bibliographical notes.- 4 Applications to Problems in Fluid Mechanics.- 1 Introduction.- 2 The balance laws of fluid mechanics.- 3 Stationary problems of gas dynamics with free boundaries.- 4 Two-phase filtration of immiscible incompressible fluids.- 5 Flows of gas with density-dependent viscosity.- 6 Viscous-elastic media.- 7 Flows of nonhomogeneous non-Newtonian fluid.- 8 Boundary layers in dilatant fluid.- 9 Boussinesq system involving nonlinear thermal diffusion.- 10 Simultaneous motion in the surface channel and the underground water.- 11 Solute transport through a porous medium with micro and macro structures.- 12 A quasilinear degenerate system arising in semiconductor theory.- 13 Blowup in solutions of the thermistor problem.- The function spaces.- Elementary inequalities.- 2.1 Algebraic inequalities.- 2.2 Integral inequalities.- 3 Embedding theorems.- 3.1 Interpolation inequalities.- 3.2 Anisotropic function spaces.- References.