Kinetic Formulation of Conservation Laws

Hardcover | July 15, 2002

byBenoit Perthame

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This book gives a general presentation of the mathematical and numerical connections kinetic theory and conservation laws based on several earlier works with P. L. Lions and E. Tadmor, as well as on more recent developments. The kinetic formalism approach allows the reader to consider PartialDifferential Equations, such as some nonlinear conservation laws, as linear kinetic (or semi-kinetic) equations acting on a nonlinear quantity. It also aids the reader with using Fourier transform, regularisation, and moments methods to provide new approaches for proving uniqueness, regularizingeffects, and a priori bounds. Special care has been given to introduce basic tools, including the classical Boltzmann formalism to derive compressible fluid dynamics, the study of oscillatons through the kinetic defect measure, and an elementary construction of solutions to scalar conservation laws. More advanced materialcontains regularizing effects through averaging lemmas, existence of global large solutions to isentropic gas dynamics, and a new uniqueness proof for scalar conservation laws. Sections are also devoted to the derivation of numerical approaches, the 'kinetic schemes', and the analysis of theirtheoretical properties.

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From the Publisher

This book gives a general presentation of the mathematical and numerical connections kinetic theory and conservation laws based on several earlier works with P. L. Lions and E. Tadmor, as well as on more recent developments. The kinetic formalism approach allows the reader to consider PartialDifferential Equations, such as some nonline...

Prof. Benoit Perthame ENS/DMA 45, rue d'Ulm; F-75230 Paris Cedex 05 FRANCE 33-1-44322036 phone 33-1-44322080 fax benoit.perthame@ens.fr French, born France, June 23 1959

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Format:HardcoverDimensions:210 pages, 9.21 × 6.14 × 0.64 inPublished:July 15, 2002Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0198509138

ISBN - 13:9780198509134

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Table of Contents

Foreword1. A brief overview of the kinetic approach2. The function chi, entropies and representation of nonlinear functions3. Kinetic formulation of multidimensional scalar conservation laws4. Uniqueness of solutions to scalar conservation laws and consequences5. Compactness, cancellation of oscillations and averaging lemmas6. Kinetic schemes for SCL7. Isentropic gas dynamics8. Kinetic schemes for gas dynamicsAppendices

Editorial Reviews

"It has been known within the community that Benoit Perthame is writing such a book. The book is anticipated from the community and expected to fill a gap in the literature."A Tzavaras, Department of Mathematics, University of Wisconsin-Madison