Constantine M. Dafermos

Hyperbolic Conservation Laws in Continuum Physics

• Gebundenes Buch

Produktbeschreibung

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 the rudiments of analysis and the qualitative theory of partial differential equations, whereas prior exposure to continuum mechanics is not required.

As with its earlier editions, the book is addressed to a diverse group of readers: younger researchers, newcomers to the area, looking for a broad overview of the field; specialists in the analysis of hyperbolic conservation laws aspiring to fathom its genetic relation to mathematical physics; experts in continuum mechanics, as a vehicle for acquiring the necessary analytical tools; and numerical analysts as a reference source to the general theory. This new edition contains an account of recent results on the Euler equations, pertaining to the breakdown of classical solutions and the construction of very weak, measure-valued, solutions as well as of milder, continuous but wildly oscillating turbulent solutions. Furthermore, the presentation of a number of topics in the previous editions has been revised, expanded and brought up to date. The bibliography has also been expanded, now comprising twenty-five hundred titles.

From the reviews of earlier editions:

Written from a unique and unifying perspective, this treatise provides the mathematical community with a wonderfully entire and exhaustive portrait of the field . Heinrich Freistühler, Jahresber Dtsch Math-Ver.

A monumental book encompassing all aspects of the mathematical theory of hyperbolic conservation laws, widely recognized as the 'Bible' on the subject. Phillippe G. LeFloch, Math. Reviews.

Produktdetails

• Grundlehren der mathematischen Wissenschaften 325
• Verlag: Springer, Berlin; Springer Berlin Heidelberg; Springe
• Artikelnr. des Verlages: 89586539
• 5. Aufl. 2025.
• Seitenzahl: 1034
• Erscheinungstermin: Dezember 2025
• Englisch
• Abmessung: 235mm x 155mm
• ISBN-13: 9783662724446
• ISBN-10: 3662724448
• Artikelnr.: 75480709

Autorenporträt

Professor Dafermos received a Diploma in Civil Engineering from the National Technical University of Athens (1964) and a Ph.D. in Mechanics from the Johns Hopkins University (1967). He has taught at Cornell University (1968-1971) and at Brown University (1971-2022), where he is currently the Alumni-Alumnae University Professor Emeritus of Applied Mathematics. He is a Member of the U.S. National Academy of Sciences, a Fellow of the American Academy of Arts and Sciences, a Foreign Member of the Accademia Nazionale dei Lincei, a Correspondent Member of the Academy of Athens and an Honorary Professor of Academia Sinica; also a Fellow of the American Mathematical Society and of the Society of Industrial and Applied Mathematics. He has been awarded the Wiener Prize, by the American Mathematical Society, the W.T. and Idalia Prize, by the Society of Industrial and Applied Mathematics, the Cataldo and Angiola Agostinelli Prize, by the Accademia Nazionale dei Lincei, the Prize of the Society of the Interaction of Mechanics and Mathematics, and the Galileo Medal of the city of Padua.

Inhaltsangabe

I Balance Laws.
II Introduction to Continuum Physics.
III Hyperbolic Systems of Balance Laws.
IV The Cauchy Problem.
V Entropy and the Stability of Classical Solutions.
VI The General Theory for Scalar Conservation Laws.
VII Hyperbolic Systems of Balance Laws in One
Space Dimension.
VIII Admissible Shocks.
IX Admissible Wave Fans and the Riemann Problem.
X Generalized Characteristics.
 XI Scalar Conservation Laws in One Space Dimension.
XII Genuinely Nonlinear Systems of Two Conservation Laws.
XIII The Random Choice Method.
XIV The Front Tracking Method and Standard Riemann Semigroups.
XV Construction of BV Solutions by the Vanishing Viscosity Method.
XVI BV Solutions for Systems of Balance Laws.
XVII Compensated Compactness.
XVIII Steady and Self
similar Solutions in Multi
Space Dimensions.
XIX Euler Equations.
Bibliography.
Author Index.
Subject Index.

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I Balance Laws. - II Introduction to Continuum Physics. - III Hyperbolic Systems of Balance Laws. - IV The Cauchy Problem. - V Entropy and the Stability of Classical Solutions. - VI The L1 Theory for Scalar Conservation Laws. - VII Hyperbolic Systems of Balance Laws in One-Space Dimension. - VIII Admissible Shocks. - IX Admissible Wave Fans and the Riemann Problem. - X Generalized Characteristics. - XI Scalar Conservation Laws in One Space Dimension. - XII Genuinely Nonlinear Systems of Two Conservation Laws. - XIII The Random Choice Method. - XIV The Front Tracking Method and Standard Riemann Semigroups. - XV Construction of BV Solutions by the Vanishing Viscosity Method. - XVI BV Solutions for Systems of Balance Laws. - XVII Compensated Compactness. - XVIII Steady and Self-similar Solutions in Multi-Space Dimensions. - Bibliography. - Author Index. - Subject Index.

Rezensionen

"This book is a contribution to the fields of mathematical analysis and partial differential equations and their broad interactions with various branches of applied mathematics and continuum physics, whose value cannot be overstated. ... a useful resource for a variety of courses as well as an almost encyclopedic reference point for advanced researchers. ... It is a great achievement that will influence the PDEs and mathematical analysis community for years to come." (Marta Lewicka, Mathematical Reviews, November, 2017)

From the reviews of the second edition:

"The second edition of the famous book Grundlehren der Mathematischen Wissenschaften 325 is devoted to the mathematical theory of hyperbolic conservation and balance laws. 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 of conservations laws. ... the original text has been reorganized so as to streamline the exposition, enrich the collection of examples, and improve the notation. ... The bibliography has been considerably expanded ... ." (Evgeniy Panov, Zentralblatt MATH, Vol. 1078, 2006)

"This comprehensive book is about rigorous mathematical theory of balance and conservation laws ... . The statements of theorems are carefully and precisely written. The proofs are canonical and illuminating ... . This book is sure to convince every reader that working in this area is challenging, enlightening, and joyful. I heartily recommend this book to anyone who wants to learn about the foundations of the theory of balance and conservation laws and their generic relations to continuum physics ... ." (Katarina Jegdic, SIAM Review, Vol. 48 (3), 2006)

From the reviews of the third edition:

"Covers the existence, uniqueness, continuous dependence, and qualitative behavior of entropy solutions for, successively, scalar conservation laws, systems of two conservation laws, and general strictly hyperbolic systems. ... This book is now widely recognized as the 'Bible' on hyperbolic conservation laws, and has contributed to the training of many students while providing a huge amount of invaluable information (including a 100 page bibliography) for researchers working in this field or related areas." (Philippe G. LeFloch, Mathematical Reviews, Issue 2011 i)

"The third edition of the famous book by C. M. Dafermos ... devoted to the mathematical theory of hyperbolic conservation and balance laws. This masterly written book is, surely,the most entire exposition in the subject of conservation laws." (Evgeniy Panov, Zentralblatt MATH, Vol. 1196, 2010)