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The seeds of Continuum Physics were planted with the works of the natural philosophers of the eighteenth century, most notably Euler; by the mid-nineteenth century, the trees were fully grown and ready to yield fruit. It was in this envi ronment that the study of gas dynamics gave birth to the theory of quasilinear hyperbolic systems in divergence form, commonly called "hyperbolic conserva tion laws"; and these two subjects have been traveling hand-in-hand over the past one hundred and fifty years. This book aims at presenting the theory of hyper bolic conservation laws from the standpoint of its genetic relation to Continuum Physics. Even though research is still marching at a brisk pace, both fields have attained by now the degree of maturity that would warrant the writing of such an exposition. In the realm of Continuum Physics, material bodies are realized as continuous media, and so-called "extensive quantities", such as mass, momentum and energy, are monitored through the fields of their densities, which are related by balance laws and constitutive equations. A self-contained, though skeletal, introduction to this branch of classical physics is presented in Chapter II. The reader may flesh it out with the help of a specialized text on the subject.
Champs [Théorie des ] (Physique) --- Conservatiewetten (Fysica) --- Conservation [Lois de ] --- Conservation laws (Physics) --- Differentiaalvergelijkingen [Hyperbolische ] --- Differential equations [Hyperbolic] --- Equations différentielles hyperboliques --- Field theory (Physics) --- Lois de conservation (Physique) --- Veldtheorie (Fysica) --- Differential equations, Hyperbolic. --- Conservation laws (Physics). --- Field theory (Physics). --- Partial differential equations. --- Thermodynamics. --- Mechanics. --- Partial Differential Equations. --- Classical Mechanics. --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Chemistry, Physical and theoretical --- Mechanics --- Heat --- Heat-engines --- Partial differential equations
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This is a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws. It illustrates the essential role of continuum thermodynamics in providing motivation and direction for the development of the mathematical theory while also serving as the principal source of applications. The reader is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of analysis and the qualitative theory of partial differential equations, whereas prior exposure to continuum physics is not required. The target group of readers would consist of (a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics; (b) specialists in continuum mechanics who may need analytical tools; (c) experts in numerical analysis who wish to learn the underlying mathematical theory; and (d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conservation laws. New to the 3rd edition is an account of the early history of the subject, spanning the period between 1800 to 1957. Also new is a chapter recounting the recent solution of open problems of long standing in classical aerodynamics. Furthermore, the presentation of a number of topics in the previous edition has been revised and brought up to date, and the collection of applications has been substantially enriched. The bibliography, also expanded and updated, now comprises over fifteen hundred titles. From the reviews of the 2nd edition: "The author is known as one of the leading experts in the field. His masterly written book is, surely, the most complete exposition in the subject." Evgeniy Panov, Zentralblatt MATH "This book is sure to convince every reader that working in this area is challenging, enlightening, and joyful." Katarina Jegdic, SIAM Review.
Conservation laws (Physics). --- Differential equations, Hyperbolic. --- Field theory (Physics). --- Conservation laws (Physics) --- Differential equations, Hyperbolic --- Field theory (Physics) --- Mathematics --- Physics --- Nuclear Physics --- Calculus --- Physical Sciences & Mathematics --- Classical field theory --- Continuum physics --- Hyperbolic differential equations --- Mathematics. --- Partial differential equations. --- Mechanics. --- Thermodynamics. --- Continuum mechanics. --- Structural mechanics. --- Partial Differential Equations. --- Continuum Mechanics and Mechanics of Materials. --- Structural Mechanics. --- Continuum mechanics --- Differential equations, Partial --- Physical laws
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517.9 --- Differential equations --- -Equations, Differential --- Bessel functions --- Calculus --- Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Congresses --- Congresses. --- -517.91 Differential equations --- 517.9 Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- -Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- -517.9 Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- 517.91 Differential equations --- 517.91 --- Numerical solutions --- Numerical solutions&delete&
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The material collected in this volume discusses the present as well as expected future directions of development of the field with particular emphasis on applications. The seven survey articles present different topics in Evolutionary PDE's, written by leading experts.- Review of new results in the area - Continuation of previous volumes in the handbook series covering evolutionary PDEs - New content coverage of DE applications
Differential equations. --- Nonlinear optics. --- Optics, Nonlinear --- Optics --- Lasers --- 517.91 Differential equations --- Differential equations
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The material collected in this volume reflects the active present of this area of mathematics, ranging from the abstract theory of gradient flows to stochastic representations of non-linear parabolic PDE's.Articles will highlight the present as well as expected future directions of development of the field with particular emphasis on applications. The article by Ambrosio and Savaré discussesthe most recent development in the theory of gradient flow of probability measures. After an introduction reviewing the properties of the Wasserstein space and corresponding subdifferential calc
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The material collected in this volume discusses the present as well as expected future directions of development of the field with particular emphasis on applications. The seven survey articles present different topics in Evolutionary PDE's, written by leading experts.- Review of new results in the area- Continuation of previous volumes in the handbook series covering Evolutionary PDEs- Written by leading experts
Differential equations --- Mathematics. --- Math --- Science --- 517.91 Differential equations
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The aim of this Handbook is to acquaint the reader with the current status of the theory of evolutionary partial differential equations, and with some of its applications. Evolutionary partial differential equations made their first appearance in the 18th century, in the endeavor to understand the motion of fluids and other continuous media. The active research effort over the span of two centuries, combined with the wide variety of physical phenomena that had to be explained, has resulted in an enormous body of literature. Any attempt to produce a comprehensive survey would be futile. The aim
Differential equations --- Mathematics. --- Math --- Science --- 517.91 Differential equations
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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: No
Differential equations --- Evolution equations --- Evolutionary equations --- Equations, Evolution --- Equations of evolution --- 517.91 Differential equations
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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 PDEs: from theory to numerics. The material collected in this volume reflects the active present of this area of mathematics, ranging from the abstract theory of gradient flows to stochastic representations of non-linear parabolic PDE's. Articles will highlight the present as well as expected future directions of development of the field with particular emphasis on applications. The article by Ambrosio and Savař discusses the most recent development in the theory of gradient flow of probability measures. After an introduction reviewing the properties of the Wasserstein space and corresponding subdifferential calculus, applications are given to evolutionary partial differential equations. The contribution of Herrero provides a description of some mathematical approaches developed to account for quantitative as well as qualitative aspects of chemotaxis. Particular attention is paid to the limits of cell's capability to measure external cues on the one hand, and to provide an overall description of aggregation models for the slim mold Dictyostelium discoideum on the other. The chapter written by Masmoudi deals with a rather different topic - examples of singular limits in hydrodynamics. This is nowadays a well-studied issue given the amount of new results based on the development of the existence theory for rather general systems of equations in hydrodynamics. The paper by DeLellis addreses the most recent results for the transport equations with regard to possible applications in the theory of hyperbolic systems of conservation laws. Emphasis is put on the development of the theory in the case when the governing field is only a BV function. The chapter by Rein represents a comprehensive survey of results on the Poisson-Vlasov system in astrophysics. The question of global stability of steady states is addressed in detail. The contribution of Soner is devoted to different representations of non-linear parabolic equations in terms of Markov processes. After a brief introduction on the linear theory, a class of non-linear equations is investigated, with applications to stochastic control and differential games. The chapter written by Zuazua presents some of the recent progresses done on the problem of controllabilty of partial differential equations. The applications include the linear wave and heat equations, parabolic equations with coefficients of low regularity, and some fluid-structure interaction models. - Volume 1 focuses on the abstract theory of evolution - Volume 2 considers more concrete probelms relating to specific applications - Volume 3 reflects the active present of this area of mathematics, ranging from the abstract theory of gradient flows to stochastic representations of non-linear PDEs. The material collected in this volume discusses the present as well as expected future directions of development of the field with particular emphasis on applications. The seven survey articles present different topics in Evolutionary PDE's, written by leading experts. - Review of new results in the area - Continuation of previous volumes in the handbook series covering Evolutionary PDEs - Written by leading experts. The material collected in this volume discusses the present as well as expected future directions of development of the field with particular emphasis on applications. The seven survey articles present different topics in Evolutionary PDE's, written by leading experts. - Review of new results in the area - Continuation of previous volumes in the handbook series covering evolutionary PDEs - New content coverage of DE applications.
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