Linear And Quasi Linear Evolution Equations In Hilbert Spaces

Author : Pascal Cherrier,Albert Milani
Publisher : American Mathematical Society
Page : 400 pages
File Size : 54,8 Mb
Release : 2022-07-14
Category : Mathematics
ISBN : 9781470471446

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Linear and Quasi-linear Evolution Equations in Hilbert Spaces by Pascal Cherrier,Albert Milani Pdf

This book considers evolution equations of hyperbolic and parabolic type. These equations are studied from a common point of view, using elementary methods, such as that of energy estimates, which prove to be quite versatile. The authors emphasize the Cauchy problem and present a unified theory for the treatment of these equations. In particular, they provide local and global existence results, as well as strong well-posedness and asymptotic behavior results for the Cauchy problem for quasi-linear equations. Solutions of linear equations are constructed explicitly, using the Galerkin method; the linear theory is then applied to quasi-linear equations, by means of a linearization and fixed-point technique. The authors also compare hyperbolic and parabolic problems, both in terms of singular perturbations, on compact time intervals, and asymptotically, in terms of the diffusion phenomenon, with new results on decay estimates for strong solutions of homogeneous quasi-linear equations of each type. This textbook presents a valuable introduction to topics in the theory of evolution equations, suitable for advanced graduate students. The exposition is largely self-contained. The initial chapter reviews the essential material from functional analysis. New ideas are introduced along with their context. Proofs are detailed and carefully presented. The book concludes with a chapter on applications of the theory to Maxwell's equations and von Karman's equations.

Author : Horst Reinhard Beyer
Publisher : Springer
Page : 291 pages
File Size : 54,6 Mb
Release : 2007-04-10
Category : Mathematics
ISBN : 9783540711292

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Beyond Partial Differential Equations by Horst Reinhard Beyer Pdf

This book introduces the treatment of linear and nonlinear (quasi-linear) abstract evolution equations by methods from the theory of strongly continuous semigroups. The theoretical part is accessible to graduate students with basic knowledge in functional analysis, with only some examples requiring more specialized knowledge from the spectral theory of linear, self-adjoint operators in Hilbert spaces. Emphasis is placed on equations of the hyperbolic type which are less often treated in the literature.

Author : Behzad Djafari Rouhani
Publisher : CRC Press
Page : 450 pages
File Size : 53,8 Mb
Release : 2019-05-20
Category : Mathematics
ISBN : 9781482228199

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Nonlinear Evolution and Difference Equations of Monotone Type in Hilbert Spaces by Behzad Djafari Rouhani Pdf

This book is devoted to the study of non-linear evolution and difference equations of first or second order governed by maximal monotone operator. This class of abstract evolution equations contains ordinary differential equations, as well as the unification of some important partial differential equations including heat equation, wave equation, Schrodinger equation, etc. The book contains a collection of the authors' work and applications in this field, as well as those of other authors.

Author : Oliver Caps
Publisher : Springer Science & Business Media
Page : 310 pages
File Size : 45,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783322800398

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Evolution Equations in Scales of Banach Spaces by Oliver Caps Pdf

The book provides a new functional-analytic approach to evolution equations by considering the abstract Cauchy problem in a scale of Banach spaces. Conditions are proved characterizing well-posedness of the linear, time-dependent Cauchy problem in scales of Banach spaces and implying local existence, uniqueness, and regularity of solutions of the quasilinear Cauchy problem. Many applications illustrate the generality of the approach. In particular, using the Fefferman-Phong inequality unifying results on parabolic and hyperbolic equations generalizing classical ones and a unified treatment of Navier-Stokes and Euler equations is described. Assuming only basic knowledge in analysis and functional analysis the book provides all mathematical tools and is aimed for students, graduates, researchers, and lecturers.

Author : Giuseppe Da Prato,Jerzy Zabczyk
Publisher : Cambridge University Press
Page : 397 pages
File Size : 50,8 Mb
Release : 2002-07-25
Category : Mathematics
ISBN : 9781139433433

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Second Order Partial Differential Equations in Hilbert Spaces by Giuseppe Da Prato,Jerzy Zabczyk Pdf

State of the art treatment of a subject which has applications in mathematical physics, biology and finance. Includes discussion of applications to control theory. There are numerous notes and references that point to further reading. Coverage of some essential background material helps to make the book self contained.

Author : Toka Diagana
Publisher : Springer
Page : 189 pages
File Size : 40,8 Mb
Release : 2018-10-23
Category : Mathematics
ISBN : 9783030004491

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Semilinear Evolution Equations and Their Applications by Toka Diagana Pdf

This book, which is a continuation of Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, presents recent trends and developments upon fractional, first, and second order semilinear difference and differential equations, including degenerate ones. Various stability, uniqueness, and existence results are established using various tools from nonlinear functional analysis and operator theory (such as semigroup methods). Various applications to partial differential equations and the dynamic of populations are amply discussed. This self-contained volume is primarily intended for advanced undergraduate and graduate students, post-graduates and researchers, but may also be of interest to non-mathematicians such as physicists and theoretically oriented engineers. It can also be used as a graduate text on evolution equations and difference equations and their applications to partial differential equations and practical problems arising in population dynamics. For completeness, detailed preliminary background on Banach and Hilbert spaces, operator theory, semigroups of operators, and almost periodic functions and their spectral theory are included as well.

Author : Nina Nikolaevna Uraltseva
Publisher : American Mathematical Soc.
Page : 240 pages
File Size : 44,9 Mb
Release : 1995-05-19
Category : Mathematics
ISBN : 0821895958

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Nonlinear Evolution Equations by Nina Nikolaevna Uraltseva Pdf

This collection focuses on nonlinear problems in partial differential equations. Most of the papers are based on lectures presented at the seminar on partial differential equations and mathematical physics at St. Petersburg University. Among the topics explored are the existence and properties of solutions of various classes of nonlinear evolution equations, nonlinear imbedding theorems, bifurcations of solutions, and equations of mathematical physics (Navier-Stokes type equations and the nonlinear Schrodinger equation). The book will be useful to researchers and graduate students working in partial differential equations and mathematical physics.

Author : C.M. Dafermos,Eduard Feireisl
Publisher : Elsevier
Page : 579 pages
File Size : 49,7 Mb
Release : 2004-08-24
Category : Mathematics
ISBN : 9780080521824

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Handbook of Differential Equations: Evolutionary Equations by C.M. Dafermos,Eduard Feireisl Pdf

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 estimates A.Bressan: The front tracking method for systems of conservation laws E.DiBenedetto, J.M.Urbano,V.Vespri: Current issues on singular and degenerate evolution equations; L.Hsiao, S.Jiang: Nonlinear hyperbolic-parabolic coupled systems A.Lunardi: Nonlinear parabolic equations and systems D.Serre:L1-stability of nonlinear waves in scalar conservation laws B.Perthame:Kinetic formulations of parabolic and hyperbolic PDE’s: from theory to numerics

Author : Jan Prüss,Gieri Simonett
Publisher : Birkhäuser
Page : 609 pages
File Size : 53,5 Mb
Release : 2016-07-25
Category : Mathematics
ISBN : 9783319276984

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Moving Interfaces and Quasilinear Parabolic Evolution Equations by Jan Prüss,Gieri Simonett Pdf

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.

Author : Thierry Cazenave,Alain Haraux
Publisher : Oxford University Press
Page : 204 pages
File Size : 55,8 Mb
Release : 1998
Category : Computers
ISBN : 019850277X

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An Introduction to Semilinear Evolution Equations by Thierry Cazenave,Alain Haraux Pdf

This book presents in a self-contained form the typical basic properties of solutions to semilinear evolutionary partial differential equations, with special emphasis on global properties. It has a didactic ambition and will be useful for an applied readership as well as theoretical researchers.

Author : J. Kral
Publisher : Springer Science & Business Media
Page : 138 pages
File Size : 48,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461344254

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Nonlinear Evolution Equations and Potential Theory by J. Kral Pdf

Preface.- Gottfried Anger: Direct and inverse problems in potential theory.- Viorel Barbu: Regularity results for sane differential equations associated with maximal monotone operators in Hilbert spaces.- Haim Brezis: Classes d'interpolation associées à un opérateur monotone et applications.- Siegfried Dnümmel: On inverse problems for k-dimensional potentials.- Jozef Ka?ur: Application of Rothe's method to nonlinear parabolic boundary value problems.- Josef Král: Potentials and removability of singularities.- Vladimir Lovicar: Theorem of Fréchet and asymptotically almost periodid solutions of.

Author : Herbert Amann
Publisher : Birkhäuser
Page : 366 pages
File Size : 43,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034892216

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Linear and Quasilinear Parabolic Problems by Herbert Amann Pdf

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.

Author : Ralph Edwin Showalter
Publisher : Unknown
Page : 212 pages
File Size : 55,5 Mb
Release : 1977
Category : Mathematics
ISBN : MINN:31951000515763V

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Hilbert Space Methods for Partial Differential Equations by Ralph Edwin Showalter Pdf

Author : Kazufumi Ito,Franz Kappel
Publisher : World Scientific
Page : 520 pages
File Size : 40,8 Mb
Release : 2002-05-24
Category : Mathematics
ISBN : 9789814488389

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Evolution Equations And Approximations by Kazufumi Ito,Franz Kappel Pdf

This book presents an approximation theory for a general class of nonlinear evolution equations in Banach spaces and the semigroup theory, including the linear (Hille–Yosida), nonlinear (Crandall–Liggett) and time-dependent (Crandall–Pazy) theorems. The implicit finite difference method of Euler is shown to generate a sequence convergent to the unique integral solution of evolution equations of the maximal monotone type. Moreover, the Chernoff theory provides a sufficient condition for consistent and stable time integration of time-dependent nonlinear equations. The Trotter–Kato theorem and the Lie–Trotter type product formula give a mathematical framework for the convergence analysis of numerical approximations of solutions to a general class of partial differential equations. This book contains examples demonstrating the applicability of the generation as well as the approximation theory. In addition, the Kobayashi–Oharu approach of locally quasi-dissipative operators is discussed for homogeneous as well as nonhomogeneous equations. Applications to the delay differential equations, Navier–Stokes equation and scalar conservation equation are given. Contents: Dissipative and Maximal Monotone OperatorsLinear SemigroupsAnalytic SemigroupsApproximation of C0-SemigroupsNonlinear Semigroups of ContractionsLocally Quasi-Dissipative Evolution EquationsThe Crandall–Pazy ClassVariational Formulations and Gelfand TriplesApplications to Concrete SystemsApproximation of Solutions for Evolution EquationsSemilinear Evolution EquationsAppendices:Some InequalitiesConvergence of Steklov MeansSome Technical Results Needed in Section 9.2 Readership: Researchers in the fields of analysis & differential equations and approximation theory. Keywords:Evolution Equations;Approximations;Euler;Trotter-Kato;Lie-Trotter;Quasi-Dissipative Operators;K and Y Kobayashi;S OharuReviews:“Ito and Kappel offer a unified presentation of the general approach for well-posedness results using abstract evolution equations, drawing from and modifying the work of K and Y Kobayashi and S Oharu … their work is not a textbook, but they explain how instructors can use various sections, or combinations of them, as a foundation for a range of courses.”Book News, Inc.

Author : Herbert Amann
Publisher : Springer Science & Business Media
Page : 688 pages
File Size : 41,6 Mb
Release : 1995-03-27
Category : Language Arts & Disciplines
ISBN : 3764351144

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Linear and Quasilinear Parabolic Problems by Herbert Amann Pdf

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.