Hyperbolic Systems Of Conservation Laws: The Theory Of Classical And Nonclassical Shock Waves (lectures In Mathematics. Eth Zürich)

This set of lecture notes was written for a Nachdiplom-Vorlesungen course given at the Forschungsinstitut fUr Mathematik (FIM), ETH Zurich, during the Fall Semester 2000. I would like to thank the faculty of the Mathematics Department, and especially Rolf Jeltsch and Michael Struwe, for giving me such a great opportunity to deliver the lectures in a very stimulating environment. Part of this material was also taught earlier as an advanced graduate course at the Ecole Poly technique (Palaiseau) during the years 1995-99, at the Instituto Superior Tecnico (Lisbon) in the Spring 1998, and at the University of Wisconsin (Madison) in the Fall 1998. This project started in the Summer 1995 when I gave a series of lectures at the Tata Institute of Fundamental Research (Bangalore). One main objective in this course is to provide a self-contained presentation of the well-posedness theory for nonlinear hyperbolic systems of first-order partial differential equations in divergence form, also called hyperbolic systems of con­ servation laws. Such equations arise in many areas of continuum physics when fundamental balance laws are formulated (for the mass, momentum, total energy . . . of a fluid or solid material) and small-scale mechanisms can be neglected (which are induced by viscosity, capillarity, heat conduction, Hall effect . . . ). Solutions to hyper­ bolic conservation laws exhibit singularities (shock waves), which appear in finite time even from smooth initial data. This set of lecture notes was written for a Nachdiplom-Vorlesungen course given at the Forschungsinstitut fUr Mathematik (FIM), ETH Zurich, during the Fall Semester 2000. I would like to thank the faculty of the Mathematics Department, and especially Rolf Jeltsch and Michael Struwe, for giving me such a great opportunity to deliver the lectures in a very stimulating environment. Part of this material was also taught earlier as an advanced graduate course at the Ecole Poly technique (Palaiseau) during the years 1995-99, at the Instituto Superior Tecnico (Lisbon) in the Spring 1998, and at the University of Wisconsin (Madison) in the Fall 1998. This project started in the Summer 1995 when I gave a series of lectures at the Tata Institute of Fundamental Research (Bangalore). One main objective in this course is to provide a self-contained presentation of the well-posedness theory for nonlinear hyperbolic systems of first-order partial differential equations in divergence form, also called hyperbolic systems of conƯ servation laws. Such equations arise in many areas of continuum physics when fundamental balance laws are formulated (for the mass, momentum, total energy ... of a fluid or solid material) and small-scale mechanisms can be neglected (which are induced by viscosity, capillarity, heat conduction, Hall effect ...). Solutions to hyperƯ bolic conservation laws exhibit singularities (shock waves), which appear in finite time even from smooth initial data

this Book Is A Self-contained Exposition Of The Well-posedness Theory For Nonlinear Hyperbolic Systems Of Conservation Laws, Recently Completed By The Author Together With His Collaborators. The Text Covers The Existence, Uniqueness, And Continuous Dependence Of Classical (compressive) Entropy Solutions. It Also Introduces The Reader To The Developing Theory Of Nonclassical (undercompressive) Entropy Solutions. The Study Of Nonclassical Shock Waves Is Based On The Concept Of A Kinetic Relation Introduced By The Author For General Hyperbolic Systems And Derived From Singular Limits Of Hyperbolic Conservation Laws With Balanced Diffusion And Dispersion Terms. The Systems Of Partial Differential Equations Under Consideration Arise In Many Areas Of Continuum Physics. No Familiarity With The Subject Is Assumed, So The Book Should Be Particularly Suitable For Graduate Students And Researchers Interested In Recent Developments About Nonlinear Partial Differential Equations And The Mathematical Aspects Of Shock Waves And Propagating Phase Boundaries.

Front Matter....Pages i-x Fundamental Concepts and Examples....Pages 1-26 Front Matter....Pages 27-27 The Riemann Problem....Pages 29-50 Diffusive-Dispersive Traveling Waves....Pages 51-80 Existence Theory for the Cauchy Problem....Pages 81-117 Continuous Dependence of Solutions....Pages 118-136 Front Matter....Pages 137-137 The Riemann Problem....Pages 139-166 Classical Entropy Solutions of the Cauchy Problem....Pages 167-187 Nonclassical Entropy Solutions of the Cauchy Problem....Pages 188-211 Continuous Dependence of Solutions....Pages 212-240 Uniqueness of Entropy Solutions....Pages 241-258 Back Matter....Pages 259-294 This book examines the well-posedness theory for nonlinear hyperbolic systems of conservation laws, recently completed by the author together with his collaborators. It covers the existence, uniqueness, and continuous dependence of classical entropy solutions. It also introduces the reader to the developing theory of nonclassical (undercompressive) entropy solutions. The systems of partial differential equations under consideration arise in many areas of continuum physics.