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Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves
Abstract
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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Cites
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Cited by
Finite-volume schemes for shallow-water equations
Kurganov, A. (2018) · Acta Numerica · doi:10.1017/s0962492918000028 · Cited by 108
Metadata
- Authors
- LeFloch, & Novotny
- Venue
- Applied Mechanics Reviews
- Publisher
- ASME International
- Published
- 2003-07-01
- Volume
- 56
- Issue
- 4
- Pages
- B53-B54
- ISSN
- 0003-6900 (ISSN-L)
- 2379-0407
- OpenAlex
- W1981825355
- Cited by
- 275
- References
- 24
- FWCI
- 20.28
- Type
- journal article
- Language
- en
- Open access
- closed
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