Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves / Edition 1

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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.

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Table of Contents

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.- 1. Entropy conditions.- 2. Classical Riemann solver.- 3. Entropy dissipation function.- 4. Nonclassical Riemann solver for concave-convex flux.- 5. Nonclassical Riemann solver for convex-concave flux.- III. Diffusive-dispersive traveling waves.- 1. Diffusive traveling waves.- 2. Kinetic functions for the cubic flux.- 3. Kinetic functions for general flux.- 4. Traveling waves for a given speed.- 5. Traveling waves for a given diffusion-dispersion ratio.- IV. Existence theory for the Cauchy problem.- 1. Classical entropy solutions for convex flux.- 2. Classical entropy solutions for general flux.- 3. Nonclassical entropy solutions.- 4. Refined estimates.- V. Continuous dependence of solutions.- 1. A class of linear hyperbolic equations.- 2. L1 continuous dependence estimate.- 3. Sharp version of the continuous dependence estimate.- 4. Generalizations.- 2. Systems of Conservation Laws.- VI. The Riemann problem.- 1. Shock and rarefaction waves.- 2. Classical Riemann solver.- 3. Entropy dissipation and wave sets.- 4. Kinetic relation and nonclassical Riemann solver.- VII. Classical entropy solutions of the Cauchy problem.- 1. Glimm interaction estimates.- 2. Existence theory.- 3. Uniform estimates.- 4. Pointwise regularity properties.- VIII. Nonclassical entropy solutions of the Cauchy problem.- 1. A generalized total variation functional.- 2. A generalized weighted interaction potential.- 3. Existence theory.- 4. Pointwise regularity properties.- IX. Continuous dependence of solutions.- 1. A class of linear hyperbolic systems.- 2. L1 continuous dependence estimate.- 3. Sharp version of the continuous dependence estimate.- 4. Generalizations.- X. Uniqueness of entropy solutions.- 1. Admissible entropy solutions.- 2. Tangency property.- 3. Uniqueness theory.- 4. Applications.

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