Energy Methods for Free Boundary Problems

"The book contains some new unpublished results. It will appeal to researchers in partial differential equations, nonlinear analysis, and continuum mechanics." -Zentralblatt Math "... very important and useful monograph ... The book is addressed to pure or applied mathematicians at universities and in industry and also to graduate level students. All results are presented in a systematic and self-contained manner, precise proofs being usually sufficiently detailed. The topic makes the book appealing for any specialist interested in applications of energy methods for free boundary problems in nonlinear PDEs and fluid mechanics." -Mathematical Reviews "The book...is a well-done selection of the most powerful energy methods - some of them very recent - used in the study of free boundary problems in nonlinear partial differential equations. Clearly and concisely written and containing a plenty of examples and applications, this monograph is of interset for all those mathematicians, or applied mathematicians, working in Partial Diffferential Equations, Nonlinear Analysis and Fluid Mechanics, physicists and engineers." ---Analele Stiintifice ale Auniversitatii ,,al.I. cuza din Iasi

• 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.