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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.
Table des matières
I. Fundamental concepts and examples.- 1. Hyperbolicity, genuine nonlinearity, and entropies.- 2. Shock formation and weak solutions.- 3. Singular limits and the entropy inequality.- 4. Examples of diffusive-dispersive models.- 5. Kinetic relations and traveling waves.- 1. Scalar Conservation Laws.- II. The Riemann problem.- III. Diffusive-dispersive traveling waves.- IV. Existence theory for the Cauchy problem.- V. Continuous dependence of solutions.- 2. Systems of Conservation Laws.- VI. The Riemann problem.- VII. Classical entropy solutions of the Cauchy problem.- VIII. Nonclassical entropy solutions of the Cauchy problem.- IX. Continuous dependence of solutions.- X. Uniqueness of entropy solutions.
Résumé
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.
