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A Regularity Result For Quasilinear Parabolic Systems
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Book Synopsis Linear and Quasilinear Parabolic Systems: Sobolev Space Theory by : David Hoff
Download or read book Linear and Quasilinear Parabolic Systems: Sobolev Space Theory written by David Hoff and published by American Mathematical Soc.. This book was released on 2020-11-18 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph presents a systematic theory of weak solutions in Hilbert-Sobolev spaces of initial-boundary value problems for parabolic systems of partial differential equations with general essential and natural boundary conditions and minimal hypotheses on coefficients. Applications to quasilinear systems are given, including local existence for large data, global existence near an attractor, the Leray and Hopf theorems for the Navier-Stokes equations and results concerning invariant regions. Supplementary material is provided, including a self-contained treatment of the calculus of Sobolev functions on the boundaries of Lipschitz domains and a thorough discussion of measurability considerations for elements of Bochner-Sobolev spaces. This book will be particularly useful both for researchers requiring accessible and broadly applicable formulations of standard results as well as for students preparing for research in applied analysis. Readers should be familiar with the basic facts of measure theory and functional analysis, including weak derivatives and Sobolev spaces. Prior work in partial differential equations is helpful but not required.
Book Synopsis Regularity Problem for Quasilinear Elliptic and Parabolic Systems by : Alexander Koshelev
Download or read book Regularity Problem for Quasilinear Elliptic and Parabolic Systems written by Alexander Koshelev and published by Springer. This book was released on 2006-11-14 with total page 277 pages. Available in PDF, EPUB and Kindle. Book excerpt: The smoothness of solutions for quasilinear systems is one of the most important problems in modern mathematical physics. This book deals with regular or strong solutions for general quasilinear second-order elliptic and parabolic systems. Applications in solid mechanics, hydrodynamics, elasticity and plasticity are described. The results presented are based on two main ideas: the universal iterative method, and explicit, sometimes sharp, coercivity estimates in weighted spaces. Readers are assumed to have a standard background in analysis and PDEs.
Book Synopsis Linear and Quasilinear Parabolic Problems by : Herbert Amann
Download or read book Linear and Quasilinear Parabolic Problems written by Herbert Amann and published by Birkhäuser. This book was released on 2012-12-06 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this treatise we present the semigroup approach to quasilinear evolution equa of parabolic type that has been developed over the last ten years, approxi tions mately. It emphasizes the dynamic viewpoint and is sufficiently general and flexible to encompass a great variety of concrete systems of partial differential equations occurring in science, some of those being of rather 'nonstandard' type. In partic ular, to date it is the only general method that applies to noncoercive systems. Although we are interested in nonlinear problems, our method is based on the theory of linear holomorphic semigroups. This distinguishes it from the theory of nonlinear contraction semigroups whose basis is a nonlinear version of the Hille Yosida theorem: the Crandall-Liggett theorem. The latter theory is well-known and well-documented in the literature. Even though it is a powerful technique having found many applications, it is limited in its scope by the fact that, in concrete applications, it is closely tied to the maximum principle. Thus the theory of nonlinear contraction semigroups does not apply to systems, in general, since they do not allow for a maximum principle. For these reasons we do not include that theory.
Book Synopsis The Regularity of General Parabolic Systems with Degenerate Diffusion by : Verena Bögelein
Download or read book The Regularity of General Parabolic Systems with Degenerate Diffusion written by Verena Bögelein and published by American Mathematical Soc.. This book was released on 2013-01-28 with total page 155 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of the paper is twofold. On one hand the authors want to present a new technique called $p$-caloric approximation, which is a proper generalization of the classical compactness methods first developed by DeGiorgi with his Harmonic Approximation Lemma. This last result, initially introduced in the setting of Geometric Measure Theory to prove the regularity of minimal surfaces, is nowadays a classical tool to prove linearization and regularity results for vectorial problems. Here the authors develop a very far reaching version of this general principle devised to linearize general degenerate parabolic systems. The use of this result in turn allows the authors to achieve the subsequent and main aim of the paper, that is, the implementation of a partial regularity theory for parabolic systems with degenerate diffusion of the type $\partial_t u - \mathrm{div} a(Du)=0$, without necessarily assuming a quasi-diagonal structure, i.e. a structure prescribing that the gradient non-linearities depend only on the the explicit scalar quantity.
Book Synopsis Linear and Quasilinear Parabolic Problems by : Herbert Amann
Download or read book Linear and Quasilinear Parabolic Problems written by Herbert Amann and published by Springer Science & Business Media. This book was released on 1995-03-27 with total page 688 pages. Available in PDF, EPUB and Kindle. Book excerpt: This treatise gives an exposition of the functional analytical approach to quasilinear parabolic evolution equations, developed to a large extent by the author during the last 10 years. This approach is based on the theory of linear nonautonomous parabolic evolution equations and on interpolation-extrapolation techniques. It is the only general method that applies to noncoercive quasilinear parabolic systems under nonlinear boundary conditions. The present first volume is devoted to a detailed study of nonautonomous linear parabolic evolution equations in general Banach spaces. It contains a careful exposition of the constant domain case, leading to some improvements of the classical Sobolevskii-Tanabe results. It also includes recent results for equations possessing constant interpolation spaces. In addition, systematic presentations of the theory of maximal regularity in spaces of continuous and Hölder continuous functions, and in Lebesgue spaces, are given. It includes related recent theorems in the field of harmonic analysis in Banach spaces and on operators possessing bounded imaginary powers. Lastly, there is a complete presentation of the technique of interpolation-extrapolation spaces and of evolution equations in those spaces, containing many new results.
Book Synopsis Regularity Results for Nonlinear Elliptic Systems and Applications by : Alain Bensoussan
Download or read book Regularity Results for Nonlinear Elliptic Systems and Applications written by Alain Bensoussan and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book collects many helpful techniques for obtaining regularity results for solutions of nonlinear systems of partial differential equations. These are applied in various cases to provide useful examples and relevant results, particularly in such fields as fluid mechanics, solid mechanics, semiconductor theory and game theory.
Book Synopsis Moving Interfaces and Quasilinear Parabolic Evolution Equations by : Jan Prüss
Download or read book Moving Interfaces and Quasilinear Parabolic Evolution Equations written by Jan Prüss and published by Birkhäuser. This book was released on 2016-07-25 with total page 618 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this monograph, the authors develop a comprehensive approach for the mathematical analysis of a wide array of problems involving moving interfaces. It includes an in-depth study of abstract quasilinear parabolic evolution equations, elliptic and parabolic boundary value problems, transmission problems, one- and two-phase Stokes problems, and the equations of incompressible viscous one- and two-phase fluid flows. The theory of maximal regularity, an essential element, is also fully developed. The authors present a modern approach based on powerful tools in classical analysis, functional analysis, and vector-valued harmonic analysis. The theory is applied to problems in two-phase fluid dynamics and phase transitions, one-phase generalized Newtonian fluids, nematic liquid crystal flows, Maxwell-Stefan diffusion, and a variety of geometric evolution equations. The book also includes a discussion of the underlying physical and thermodynamic principles governing the equations of fluid flows and phase transitions, and an exposition of the geometry of moving hypersurfaces.
Book Synopsis Elliptic Regularity Theory by : Lisa Beck
Download or read book Elliptic Regularity Theory written by Lisa Beck and published by Springer. This book was released on 2016-04-08 with total page 214 pages. Available in PDF, EPUB and Kindle. Book excerpt: These lecture notes provide a self-contained introduction to regularity theory for elliptic equations and systems in divergence form. After a short review of some classical results on everywhere regularity for scalar-valued weak solutions, the presentation focuses on vector-valued weak solutions to a system of several coupled equations. In the vectorial case, weak solutions may have discontinuities and so are expected, in general, to be regular only outside of a set of measure zero. Several methods are presented concerning the proof of such partial regularity results, and optimal regularity is discussed. Finally, a short overview is given on the current state of the art concerning the size of the singular set on which discontinuities may occur. The notes are intended for graduate and postgraduate students with a solid background in functional analysis and some familiarity with partial differential equations; they will also be of interest to researchers working on related topics.
Book Synopsis Parabolic Problems by : Joachim Escher
Download or read book Parabolic Problems written by Joachim Escher and published by Springer Science & Business Media. This book was released on 2011-07-20 with total page 712 pages. Available in PDF, EPUB and Kindle. Book excerpt: The volume originates from the 'Conference on Nonlinear Parabolic Problems' held in celebration of Herbert Amann's 70th birthday at the Banach Center in Bedlewo, Poland. It features a collection of peer-reviewed research papers by recognized experts highlighting recent advances in fields of Herbert Amann's interest such as nonlinear evolution equations, fluid dynamics, quasi-linear parabolic equations and systems, functional analysis, and more.
Book Synopsis Analytic Semigroups and Optimal Regularity in Parabolic Problems by : Alessandra Lunardi
Download or read book Analytic Semigroups and Optimal Regularity in Parabolic Problems written by Alessandra Lunardi and published by Springer Science & Business Media. This book was released on 2012-12-13 with total page 437 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book shows how the abstract methods of analytic semigroups and evolution equations in Banach spaces can be fruitfully applied to the study of parabolic problems. Particular attention is paid to optimal regularity results in linear equations. Furthermore, these results are used to study several other problems, especially fully nonlinear ones. Owing to the new unified approach chosen, known theorems are presented from a novel perspective and new results are derived. The book is self-contained. It is addressed to PhD students and researchers interested in abstract evolution equations and in parabolic partial differential equations and systems. It gives a comprehensive overview on the present state of the art in the field, teaching at the same time how to exploit its basic techniques. - - - This very interesting book provides a systematic treatment of the basic theory of analytic semigroups and abstract parabolic equations in general Banach spaces, and how this theory may be used in the study of parabolic partial differential equations; it takes into account the developments of the theory during the last fifteen years. (...) For instance, optimal regularity results are a typical feature of abstract parabolic equations; they are comprehensively studied in this book, and yield new and old regularity results for parabolic partial differential equations and systems. (Mathematical Reviews) Motivated by applications to fully nonlinear problems the approach is focused on classical solutions with continuous or Hölder continuous derivatives. (Zentralblatt MATH)
Book Synopsis Parabolic Systems with Polynomial Growth and Regularity by : Frank Duzaar
Download or read book Parabolic Systems with Polynomial Growth and Regularity written by Frank Duzaar and published by American Mathematical Soc.. This book was released on 2011 with total page 135 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors establish a series of optimal regularity results for solutions to general non-linear parabolic systems $ u_t- \mathrm{div} \ a(x,t,u,Du)+H=0,$ under the main assumption of polynomial growth at rate $p$ i.e. $ a(x,t,u,Du) \leq L(1+ Du ^{p-1}), p \geq 2.$ They give a unified treatment of various interconnected aspects of the regularity theory: optimal partial regularity results for the spatial gradient of solutions, the first estimates on the (parabolic) Hausdorff dimension of the related singular set, and the first Calderon-Zygmund estimates for non-homogeneous problems are achieved here.
Book Synopsis Nonlinear Partial Differential Equations and Related Topics by : Arina A. Arkhipova
Download or read book Nonlinear Partial Differential Equations and Related Topics written by Arina A. Arkhipova and published by American Mathematical Soc.. This book was released on 2010 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: "St. Petersburg PDE seminar, special session dedicated to N.N. Uraltseva's [75th] anniversary, June 2009"--P. [vi].
Book Synopsis Cross Diffusion Systems by : Dung Le
Download or read book Cross Diffusion Systems written by Dung Le and published by Walter de Gruyter GmbH & Co KG. This book was released on 2022-10-24 with total page 214 pages. Available in PDF, EPUB and Kindle. Book excerpt: The introduction of cross diffusivity opens many questions in the theory of reactiondiffusion systems. This book will be the first to investigate such problems presenting new findings for researchers interested in studying parabolic and elliptic systems where classical methods are not applicable. In addition, The Gagliardo-Nirenberg inequality involving BMO norms is improved and new techniques are covered that will be of interest. This book also provides many open problems suitable for interested Ph.D students.
Download or read book Nematics written by Jean-Michel Coron and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 426 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume (>Ie) NEMATICS Mathematical and Physical aspects constitutes the proceedings of a workshop which was held at l'Universite de Paris Sud (Orsay) in May 1990. This meeting was an Advanced Research Workshop sponsored by NATO. We gratefully acknowledge the help and support of the NATO Science Committee. Additional support has been provided by the Ministere des affaires etrangeres (Paris) and by the Direction des Recherches et Etudes Techniques (Paris). Also logistic support has been provided by the Association des Numericiens d'Orsay. (*) These proceedings are published in the framework of the "Contrat DRET W 90/316/ AOOO". v Contents (*) FOREWORD v INTRODUCTION 1. M. CORON, 1. M. GHIDAGLIA, F. HELEIN xi AN ENERGY-DECREASING ALGORITHM FOR HARMONIC MAPS F. ALOUGES 1 A COHOMOLOGICAL CRITERION FOR DENSITY OF SMOOTH MAPS IN SOBOLEV SPACES BETWEEN TWO MANIFOLDS F. BETHUEL, 1. M. CORON, F. DEMENGEL, F. HELEIN 15 ON THE MATHEMATICAL MODELING OF TEXTURES IN POLYMERIC LIQUID CRYSTALS M. C. CAmERER 25 A RESULT ON THE GLOBAL EXISTENCE FOR HEAT FLOWS OF HARMONIC MAPS FROM D2 INTO S2 K. C. CHANG, W. Y. DING 37 BLOW-UP ANALYSIS FOR HEAT FLOW OF HARMONIC MAPS Y. CHEN 49 T AYLOR-COUETTE INSTABILITY IN NEMATIC LIQUID CRYSTALS P. E. ClADIS 65 ON A CLASS OF SOLUTIONS IN THE THEORY OF NEMATIC PHASES B. D. COLEMAN, 1. T. JENKINS 93 RHEOLOGY OF THERMOTROPIC NEMATIC LIQUID CRYSTALLINE POLYMERS M. M. DENN, 1. A.
Book Synopsis Regularity of Landau Lifshitz Equations by : Seungsuk Seo
Download or read book Regularity of Landau Lifshitz Equations written by Seungsuk Seo and published by . This book was released on 2003 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author :Emmanuele DiBenedetto Publisher :Springer Science & Business Media ISBN 13 :1461415845 Total Pages :287 pages Book Rating :4.4/5 (614 download)
Book Synopsis Harnack's Inequality for Degenerate and Singular Parabolic Equations by : Emmanuele DiBenedetto
Download or read book Harnack's Inequality for Degenerate and Singular Parabolic Equations written by Emmanuele DiBenedetto and published by Springer Science & Business Media. This book was released on 2011-11-13 with total page 287 pages. Available in PDF, EPUB and Kindle. Book excerpt: Degenerate and singular parabolic equations have been the subject of extensive research for the last 25 years. Despite important achievements, the issue of the Harnack inequality for non-negative solutions to these equations, both of p-Laplacian and porous medium type, while raised by several authors, has remained basically open. Recently considerable progress has been made on this issue, to the point that, except for the singular sub-critical range, both for the p-laplacian and the porous medium equations, the theory is reasonably complete. It seemed therefore timely to trace a comprehensive overview, that would highlight the main issues and also the problems that still remain open. The authors give a comprehensive treatment of the Harnack inequality for non-negative solutions to p-laplace and porous medium type equations, both in the degenerate (p/i”2 or im/i”1) and in the singular range (1“ip/i2 or 0“im/i
Book Synopsis Handbook of Differential Equations: Evolutionary Equations by : C.M. Dafermos
Download or read book Handbook of Differential Equations: Evolutionary Equations written by C.M. Dafermos and published by Elsevier. This book was released on 2004-08-24 with total page 579 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains several introductory texts concerning the main directions in the theory of evolutionary partial differential equations. The main objective is to present clear, rigorous,and in depth surveys on the most important aspects of the present theory. The table of contents includes: W.Arendt: Semigroups and evolution equations: Calculus, regularity and kernel estimatesA.Bressan: The front tracking method for systems of conservation lawsE.DiBenedetto, J.M.Urbano,V.Vespri: Current issues on singular and degenerate evolution equations;L.Hsiao, S.Jiang: Nonlinear hyperbolic-parabolic coupled systemsA.Lunardi: Nonlinear parabolic equations and systemsD.Serre:L1-stability of nonlinear waves in scalar conservation laws B.Perthame:Kinetic formulations of parabolic and hyperbolic PDE's: from theory to numerics
