Kinetic Formulation of Conservation Laws

Benoit Perthame (, Ecole Normale Superieure, Paris)

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English

Oxford University Press
01 January 2003

Differential calculus & equations; Calculus of variations; Applied mathematics; Fluid mechanics

Series: Oxford Lecture Series in Mathematics and its Applications

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 Partial Differential 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, regularizing effects, 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 material contains 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 their theoretical properties.

By: Benoit Perthame ( Ecole Normale Superieure Paris)
Imprint: Oxford University Press
Country of Publication: United Kingdom
Dimensions: Height: 242mm, Width: 161mm, Spine: 16mm
Weight: 431g
ISBN: 9780198509134
ISBN 10: 0198509138
Series: Oxford Lecture Series in Mathematics and its Applications
Pages: 212
Publication Date: 01 January 2003
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

Foreword 1: A brief overview of the kinetic approach 2: The function chi, entropies and representation of nonlinear functions 3: Kinetic formulation of multidimensional scalar conservation laws 4: Uniqueness of solutions to scalar conservation laws and consequences 5: Compactness, cancellation of oscillations and averaging lemmas 6: Kinetic schemes for SCL 7: Isentropic gas dynamics 8: Kinetic schemes for gas dynamics Appendices

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

Reviews for Kinetic Formulation of Conservation Laws

"""Kinetic Formulation has received wide attention and Perthame is the person most responsible for it."" T-P Liu, Department of Mathematics, Stanford, CA ""A well-known expert, who made some of the key breakthroughs in this area."" A Bressan, SISSA, Trieste, Italy ""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"