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Elliptic and Parabolic Problems : A Special Tribute to the Work of Haim Brezis

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The goal of these proceedings and of the meeting of Gaeta was to celebrate and honor the mathematical achievements of Haim Brezis. The prodigious in?uence of histalentandhispersonalityinthedomainofnonlinearanalysisisunanimously- claimed!This impactis visible inthe huge number ofhis formerstudents (dozens), students of former students (hundreds) and collaborators (hundreds). Thus the Gaeta meeting was, to some extent, the family reunion of part of this large c- munity sharing a joint interest in the ?eld of elliptic and parabolic equations and pushing it to a very high standard. Italyhasa longtraditionandtasteforanalysisandwecouldnot?ndabetter placeneitheramorecompletesupportfortherealisationofourproject.Wehaveto thank here the university of Cassino, Napoli, Roma la Sapienza , the GNAMPA- Istituto di Alta Matematica, CNR-IAC, MEMOMAT, RTN Fronts-Singularities, the commune of Gaeta. Additional founding came from the universities of M- house and Zur ¨ ich. Finally, we are grateful to Birkh¨ auser and Dr. Hemp?ing who allowed us to record the talks of this conference in a prestigious volume. The organizers Progress in Nonlinear Di?erential Equations and Their Applications, Vol. 63, 1-12 c 2005 Birkh¨ auser Verlag Basel/Switzerland One-Layer Free Boundary Problems with Two Free Boundaries Andrew Acker Abstract. We studythe uniquenessand successive approximation of solutions of a class of two-dimensional steady-state ?uid problems involving in?nite periodic ?ows between two periodic free boundaries, each characterized by a ?ow-speed condition related to Bernoulli's law.


Multi
Mathematical Foundation of Turbulent Viscous Flows : Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, SEptember 1-5, 2003
Authors: --- ---
ISBN: 9783540324546 Year: 2006 Publisher: Berlin Heidelberg Springer Berlin Heidelberg

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Five leading specialists reflect on different and complementary approaches to fundamental questions in the study of the Fluid Mechanics and Gas Dynamics equations. Constantin presents the Euler equations of ideal incompressible fluids and discusses the blow-up problem for the Navier-Stokes equations of viscous fluids, describing some of the major mathematical questions of turbulence theory. These questions are connected to the Caffarelli-Kohn-Nirenberg theory of singularities for the incompressible Navier-Stokes equations that is explained in Gallavotti's lectures. Kazhikhov introduces the theory of strong approximation of weak limits via the method of averaging, applied to Navier-Stokes equations. Y. Meyer focuses on several nonlinear evolution equations - in particular Navier-Stokes - and some related unexpected cancellation properties, either imposed on the initial condition, or satisfied by the solution itself, whenever it is localized in space or in time variable. Ukai presents the asymptotic analysis theory of fluid equations. He discusses the Cauchy-Kovalevskaya technique for the Boltzmann-Grad limit of the Newtonian equation, the multi-scale analysis, giving the compressible and incompressible limits of the Boltzmann equation, and the analysis of their initial layers.


Multi
Hyperbolic Problems and Regularity Questions
Authors: --- ---
ISBN: 9783764374518 Year: 2007 Publisher: Basel Birkhäuser Basel

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This book discusses new challenges in the quickly developing field of hyperbolic problems. Particular emphasis lies on the interaction between nonlinear partial differential equations, functional analysis and applied analysis as well as mechanics. The book originates from a recent conference focusing on hyperbolic problems and regularity questions. It is intended for researchers in functional analysis, PDE, fluid dynamics and differential geometry.

Singularly Perturbed Boundary-Value Problems
Authors: --- ---
ISBN: 9783764383312 Year: 2007 Publisher: Basel Birkhäuser Basel

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It is well known that many phenomena in biology, chemistry, engineering, physics canbedescribedbyboundaryvalueproblemsassociatedwithvarioustypes ofp- tial di?erential equations or systems. When we associate a mathematical model with a phenomenon, we generally try to capture what is essential, retaining the important quantities and omitting the negligible ones which involve small par- eters. The model that would be obtained by maintaining the small parameters is called the perturbed model, whereas the simpli?ed model (the one that does not includethesmallparameters)iscalledunperturbed(orreducedmodel). Ofcourse, the unperturbed model is to be preferred, because it is simpler. What matters is that it should describefaithfully enoughthe respectivephenomenon, which means that its solution must be close enough  to the solution of the corresponding perturbed model. This fact holds in the case of regular perturbations (which are de?ned later). On the other hand, in the case of singular perturbations, things get morecomplicated. If we refer to aninitial-boundary value problem,the solutionof theunperturbed problemdoes notsatisfy ingeneralallthe originalboundary c- ditions and/or initial conditions (because some of the derivatives may disappear byneglecting the small parameters). Thus, somediscrepancymay appear between the solution of the perturbed model and that of the corresponding reduced model. Therefore, to ?ll in this gap, in the asymptotic expansion of the solution of the perturbed problem with respect to the small parameter (considering, for the sake of simplicity, that we have a single parameter), we must introduce corrections (or boundary layer functions). Morethanhalfacenturyago,A. N.


Multi
The Method of Intrinsic Scaling : A Systematic Approach to Regularity for Degenerate and Singular PDEs
Authors: ---
ISBN: 9783540759324 Year: 2008 Publisher: Berlin Heidelberg Springer Berlin Heidelberg

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This set of lectures, which had its origin in a mini course delivered at the Summer Program of IMPA (Rio de Janeiro), is an introduction to intrinsic scaling, a powerful method in the analysis of degenerate and singular PDEs. In the first part, the theory is presented from scratch for the model case of the degenerate p-Laplace equation. This approach brings to light what is really essential in the method, leaving aside technical refinements needed to deal with more general equations, and is entirely self-contained. The second part deals with three applications of the theory to relevant models arising from flows in porous media and phase transitions. The aim is to convince the reader of the strength of the method as a systematic approach to regularity for this important class of equations.


Multi
Functional Analysis and Evolution Equations : The Günter Lumer Volume
Authors: --- --- --- --- --- et al.
ISBN: 9783764377946 Year: 2008 Publisher: Basel Birkhäuser Basel

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Günter Lumer was an outstanding mathematician whose work has great influence on the research community in mathematical analysis and evolution equations. He was at the origin of the breath-taking development the theory of semigroups saw after the pioneering book of Hille and Phillips of 1957. This volume contains invited contributions presenting the state of the art of these topics and reflecting the broad interests of Günter Lumer.


Digital
Partial Differential Equations in Action From Modelling to Theory
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ISBN: 9788847007529 Year: 2008 Publisher: Milano Springer-Verlag Italia, Milano

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Introduction to Nonlinear Dispersive Equations
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ISBN: 9780387848990 Year: 2009 Publisher: New York NY Springer New York

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The aim of this textbook is to introduce the theory of nonlinear dispersive equations to graduate students in a constructive way. The first three chapters are dedicated to preliminary material, such as Fourier transform, interpolation theory and Sobolev spaces. The authors then proceed to use the linear Schrodinger equation to describe properties enjoyed by general dispersive equations. This information is then used to treat local and global well-posedness for the semi-linear Schrodinger equations. The end of each chapter contains recent developments and open problems, as well as exercises.


Digital
Hyperbolic Partial Differential Equations
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ISBN: 9780387878232 Year: 2009 Publisher: New York, NY Springer-Verlag New York

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Principles of Partial Differential Equations
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ISBN: 9781441910967 Year: 2009 Publisher: New York NY Springer New York

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This concise book covers the classical tools of PDE theory used in today's science and engineering: characteristics, the wave propagation, the Fourier method, distributions, Sobolev spaces, fundamental solutions, and Green's functions. The approach is problem-oriented, giving the reader an opportunity to master solution techniques. The theoretical part is rigorous and with important details presented with care. Hints are provided to help the reader restore the arguments to their full rigor. Many examples from physics are intended to keep the book intuitive and to illustrate the applied nature of the subject. The book is useful for a higher-level undergraduate course and for self-study.

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