Hyperbolic Systems of Conservation Laws

1. Front Matter

   Pages i-x

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2. Fundamental Concepts and Examples

   1. Fundamental Concepts and Examples

      • Philippe G. LeFloch

      Pages 1-26

3. Scalar Conservation Laws

   1. Front Matter

      Pages 27-27

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   2. The Riemann Problem

      • Philippe G. LeFloch

      Pages 29-50

   3. Diffusive-Dispersive Traveling Waves

      • Philippe G. LeFloch

      Pages 51-80

   4. Existence Theory for the Cauchy Problem

      • Philippe G. LeFloch

      Pages 81-117

   5. Continuous Dependence of Solutions

      • Philippe G. LeFloch

      Pages 118-136

4. Systems of Conservation Laws

   1. Front Matter

      Pages 137-137

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   2. The Riemann Problem

      • Philippe G. LeFloch

      Pages 139-166

   3. Classical Entropy Solutions of the Cauchy Problem

      • Philippe G. LeFloch

      Pages 167-187

   4. Nonclassical Entropy Solutions of the Cauchy Problem

      • Philippe G. LeFloch

      Pages 188-211

   5. Continuous Dependence of Solutions

      • Philippe G. LeFloch

      Pages 212-240

   6. Uniqueness of Entropy Solutions

      • Philippe G. LeFloch

      Pages 241-258

5. Back Matter

   Pages 259-294

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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.

• Partial differential equations
• Phase
• Potential
• Wave
• compressible fluids
• hyperbolic equation
• linearity
• partial differential equation
• shock waves

• Centre de Mathématiques Appliquées & Centre National de la Recherche Scientifique, Ecole Polytechnique, Palaiseau, France

  Philippe G. LeFloch

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