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Full Description
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.
Contents
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.
