In the field known as "the mathematical theory of shock waves," very exciting and unexpected developments have occurred in the last few years. Joel Smoller and Blake Temple have established classes of shock wave solutions to the Einstein Euler equations of general relativity; indeed, the mathematical and physical con sequences of these examples constitute a whole new area of research. The stability theory of "viscous" shock waves has received a new, geometric perspective due to the work of Kevin Zumbrun and collaborators, which offers a spectral approach to systems. Due to the intersection of point and essential spectrum, such an ap proach had for a long time seemed out of reach. The stability problem for "in viscid" shock waves has been given a novel, clear and concise treatment by Guy Metivier and coworkers through the use of paradifferential calculus. The L 1 semi group theory for systems of conservation laws, itself still a recent development, has been considerably condensed by the introduction of new distance functionals through Tai-Ping Liu and collaborators; these functionals compare solutions to different data by direct reference to their wave structure. The fundamental prop erties of systems with relaxation have found a systematic description through the papers of Wen-An Yong; for shock waves, this means a first general theorem on the existence of corresponding profiles. The five articles of this book reflect the above developments.
Erscheinungsdatum: 24.10.2012

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by Tai-Ping Liu, Guy Métivier, Joel Smoller, Blake Temple, Wen-An Yong, Kevin Zumbrun; edited by Heinrich Freistühler, Anders Szepessy

Tai-Ping Liu; Guy Métivier; Joel Smoller; Black Temple; Wen-An Yong; Kevin Zumbrun; Heinrich Freistühler; Anders Szepessy; et al

Heinrich Freistühler, Anders Szepessy, J. Smoller

Freistühler, Heinrich

Springer science + Business media, LLC

Springer; Brand: Birkhäuser

Birkhäuser Basel

Progress in Nonlinear Differential Equations and Their Applications -- 47, Progress in nonlinear differential equations and their applications -- 47., Boston, MA, Massachusetts, 2001

Progress in nonlinear differential equations and their applications, New York, cop. 2001

Softcover reprint of the original 1st ed. 2001, 2012

United States, United States of America

Springer Nature, Boston, MA, 2012

Oct 24, 2012

lg949493

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Source title: Advances in the Theory of Shock Waves (Progress in Nonlinear Differential Equations and Their Applications)

In the field known as "the mathematical theory of shock waves," very exciting and unexpected developments have occurred in the last few years. Joel Smoller and Blake Temple have established classes of shock wave solutions to the EinsteinƯ Euler equations of general relativity; indeed, the mathematical and physical conƯ sequences of these examples constitute a whole new area of research. The stability theory of "viscous" shock waves has received a new, geometric perspective due to the work of Kevin Zumbrun and collaborators, which offers a spectral approach to systems. Due to the intersection of point and essential spectrum, such an apƯ proach had for a long time seemed out of reach. The stability problem for "inƯ viscid" shock waves has been given a novel, clear and concise treatment by Guy Metivier and coworkers through the use of paradifferential calculus. The L 1 semiƯ group theory for systems of conservation laws, itself still a recent development, has been considerably condensed by the introduction of new distance functionals through Tai-Ping Liu and collaborators; these functionals compare solutions to different data by direct reference to their wave structure. The fundamental propƯ erties of systems with relaxation have found a systematic description through the papers of Wen-An Yong; for shock waves, this means a first general theorem on the existence of corresponding profiles. The five articles of this book reflect the above developments

This volume provides a comprehensive treatment of central themes in the modern mathematical theory of shock waves. Authored by leading scientists, the work covers: * the uniqueness of weak solutions to hyperbolic systems of conservation laws in one space variable (Tai-Ping Liu) * the multidimensional stability problem for shock fronts (Guy M,tivier) * shock wave solutions of the Einstein-Euler equations of general relativity (Joel Smoller and Blake Temple) * fundamental properties of hyperbolic systems with relaxation (Wen-An Yong) * the multidimensional stability problem for planar viscous shock waves (Kevin Zumbrun) The five articles, each self-contained and interrelated, combine the rigor of mathematical analysis with careful attention to the physical origins and applications of the field. A timely reference text for professional researchers in shock wave theory, the book also provides a basis for graduate seminars and courses for students of mathematics, physics, and theoretical engineering

This volume provides a comprehensive treatment of central themes in the modern mathematical theory of shock waves. Authored by leading scientists, the work covers: * the uniqueness of weak solutions to hyperbolic systems of conservation laws in one space variable (Tai-Ping Liu) * the multidimensional stability problem for shock fronts (Guy M, tivier) * shock wave solutions of the Einstein-Euler equations of general relativity (Joel Smoller and Blake Temple) * fundamental properties of hyperbolic systems with relaxation (Wen-An Yong) * the multidimensional stability problem for planar viscous shock waves (Kevin Zumbrun) The five articles, each self-contained and interrelated, combine the rigor of mathematical analysis with careful attention to the physical origins and applications of the field. A timely reference text for professional researchers in shock wave theory, the book also provides a basis for graduate seminars and courses for students of mathematics, physics, and theoretical engineering

Front Matter....Pages i-viii
Well-Posedness Theory for Hyperbolic Systems of Conservation Laws....Pages 1-24
Stability of Multidimensional Shocks....Pages 25-103
Shock Wave Solutions of the Einstein Equations: A General Theory with Examples....Pages 105-258
Basic Aspects of Hyperbolic Relaxation Systems....Pages 259-305
Multidimensional Stability of Planar Viscous Shock Waves....Pages 307-516
Back Matter....Pages 517-520

2013-08-01