Skip to main content

Part of the book series: The IMA Volumes in Mathematics and Its Applications ((IMA,volume 29))

Abstract

The subject of nonlinear hyperbolic waves is surveyed, with an emphasis on the discussion of a number of open problems.

Supported in part by the Applied Mathematical Sciences Program of the DOE, grant DE-FG02-88ER25053

Supported in part by the NSF Grant DMS-8619856

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Aavatsmark, To Appear, “Capillary Energy and Entropy Condition for the Buckley-Leverett Equation” Contemporary Mathematics.

    Google Scholar 

  2. M. Artola and A. Majda, 1987, “Nonlinear Development of Instabilities in Supersonic Vortex Sheets” Physica D 28, pp. 253–281.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Artola and A. Majda, 1989, “Nonlinear Kink Modes for Supersonic Vortex Sheets,” Phys. Fluids.

    Google Scholar 

  4. J. B. Bell, J. A. Trangenstein, and G. R. Shubin, 1986, “Conservation Laws of Mixed Type Describing Three-Phase Flow in Porous Media” SIAM J. Appl. Math. 46, pp. 1000–1017.

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Caginalp, 1986, “An Analysis of a Phase Field Model of a Free Boundary” Archive for Rational Mechanics and Analysis 92, pp. 205–245.

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Caginalp, 1986, “The Role of Microscopic Anisotropy in the Macroscopic Behavior of a Phase Field Boundary” Ann. Phys. 172, pp. 136–146.

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Caginalp, To Appear, Phase Field Models: Some Conjectures on Theorems for their Sharp Interface Limits

    Google Scholar 

  8. G. Caginalp, To Appear, Stefan and Hele-Shaw Type Models as Asymptotic Limits of the Phase Field Equations

    Google Scholar 

  9. G. Caginalp, To Appear, “The Dynamics of a Conserved Phase Field System: Stephan-like, Hele-Shaw and Cahn-Hilliard Models as Asymptotic Limits” IMA J. Applied Math.

    Google Scholar 

  10. Tung Chang and Ling Hsiao, 1988, The Riemann problem and Interaction of Waves in Gas Dynamics (John Wiley, New York).

    Google Scholar 

  11. Guiqiang Chen, 1987, “Overtaking of Shocks of the same kind in the Isentorpic Steady Supersonic Plane Flow” Acta Math. Sinica 7, pp. 311–327.

    MATH  Google Scholar 

  12. I-Liang Chern and T.-P. Liu, 1987, “Convergence to Diffusion Waves of Solutions for Viscous Conservation Laws” Comm. in Math. Phys. 110, pp. 503–517.

    Article  MathSciNet  MATH  Google Scholar 

  13. F. Furtado, 1989, “Stability of Nonlinear Waves for Conservation Laws” New York University Thesis.

    Google Scholar 

  14. F. Furtado, Eli Isaacson, D. Marchesin, and B. Plohr, To Appear, Stability of Riemann Solutions in the Large

    Google Scholar 

  15. X. Garaizar, 1989, “The Small Anisotropy Formulation of Elastic Deformation” Acta Applicandae Mathematica 14, pp. 259–268.

    Article  MathSciNet  MATH  Google Scholar 

  16. X. Garaizar, 1989, Private Communication

    Google Scholar 

  17. C. Gardner, J. Glimm, O. McBryan, R. Menikoff, D. H. Sharp, and Q. Zhang, 1988, “The Dynamics of Bubble Growth for Rayleigh-Taylor Unstable Interfaces,” Phys. of Fluids 31, pp. 447–465.

    Article  MATH  Google Scholar 

  18. H. Gilquin, 1989, “Glimm’s scheme and conservation laws of mixed type” SIAM Jour. Sci. Stat. Computing 10, pp. 133–153.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Glimm, C. Klingenberg, O. McBryan, B. Plohr, D. Sharp, and S. Yaniv, 1985, “Front Tracking and Two Dimensional Riemann Problems” Advances in Appl. Math. 6, pp. 259–290.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Glimm and D.H. Sharp, 1986, “An S Matrix Theory for Classical Nonlinear Physics” Foundations of Physics 16, pp. 125–141.

    Article  MathSciNet  Google Scholar 

  21. J. Glimm and David H. Sharp, 1987, “Numerical Analysis and the Scientifíc Method” IBM J. Research and Development 31, pp. 169–177.

    Article  MathSciNet  Google Scholar 

  22. J. Glimm, 1988, “The Interactions of Nonlinear Hyperbolic Waves,” Comm. Pure Appl. Math. 41, pp. 569–590.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. Glimm, Jan 1988, “The Continuous Structure of Discontinuities” in Proceedings of Nice Conference.

    Google Scholar 

  24. J. Glimm and X.L. Li, 1988, “On the Validation of the Sharp-Wheeler Bubble Merger Model from Experimental and Computational Data” Phys. of Fluids 31, pp. 2077–2085.

    Article  Google Scholar 

  25. J. Glimm, X. L. Li, R. Menikoff, D. H. Sharp, and Q. Zhang, To appear, A Numerical Study of Bubble Interactions in Rayleigh-Taylor Instability for Compressible Fluids

    Google Scholar 

  26. J. Glimm, To appear, “Scientific Computing: von Neumann’s vision, today’s realities and the promise of the future” in The Legacy of John von Neumann, ed. J. Impagliazzo (Amer. Math. Soc, Providence).

    Google Scholar 

  27. J. Goodman and X. Xin, To Appear, Viscous Limits for Piecewise Smooth Solutions to Systems of Conservation Laws

    Google Scholar 

  28. J. W. Grove and R. Menikoff, 1988, “The Anomalous Reflection of a Shock Wave through a Material Interface” in preparation.

    Google Scholar 

  29. L. F. Henderson, 1988, “On the Refraction of Longitudinal Waves in Compressible Media,” LLNL Report UCRL-53853.

    Google Scholar 

  30. D. Hoff and T.-P. Liu, To Appear, “The Inviscid Limit for the Navier-Stokes equations of Compressible, Isentropic now with shock data” Indiana J. Math..

    Google Scholar 

  31. H. Holden, 1987, “On the Riemann Problem for a Prototype of a Mixed Type Conservation Law” Comm. Pure Appl. Math. 40, pp. 229–264.

    Article  MathSciNet  MATH  Google Scholar 

  32. H. Holden and L. Holden, To Appear, “On the Riemann problem for a Prototype of a Mixed Type Conservation Law II” Contemporary Mathematics.

    Google Scholar 

  33. Ling Hsiao and Tung Chang, 1980 Acta Appl. Math. Sinica 4, pp. 343–375.

    Google Scholar 

  34. P.-T. Kan, 1989, “On the Cauchy Problem of a 2 × 2 System of Nonstrictly Hyperbolic Conservation Laws,” NYU Thesis.

    Google Scholar 

  35. B. Keyfitz, To Appear, “Criterion for Certain Wave Structures in Systems that Change Type” Contemporary Mathematics.

    Google Scholar 

  36. T.-P. Liu, 1985, “Nonlinear stability of shock waves for viscous conservation laws,” Memoir, AMS:328, pp. 1–108.

    Google Scholar 

  37. T.-P. Liu, 1987, “Hyperbolic Conservation Laws with Relaxation” Comm Math Phys 108, pp. 153–175.

    Article  MathSciNet  MATH  Google Scholar 

  38. T.-P. Liu and X. Xin, To Appear, Stability of Viscous Shock Wave Asociated with a System of Nonstrictly Hyperbolic Conservation Laws

    Google Scholar 

  39. A. Majda and V. Roytburd, To Appear, “Numerical Study of the Mechanisms for Initiation of Reacting Shock Waves,” Siam J. Sci Stat Comp.

    Google Scholar 

  40. R. Menikoff and B. Plohr, 1989, “Riemann Problem for Fluid Flow of Real Materials” Rev. Mod. Phys. 61, pp. 75–130.

    Article  MathSciNet  MATH  Google Scholar 

  41. R. Menikoff, 1989, Private Communication

    Google Scholar 

  42. R. von Mises, 1958, Mathematical Theory of Compressible Fluid Flow (Academic Press, New York).

    MATH  Google Scholar 

  43. R. L. Rabie, G. R. Fowles, and W. Fickett, 1979, “The Polymorphic Detonation,” Phys. of Fluids 22, pp. 422–435.

    Article  Google Scholar 

  44. K. I. Read, 1984, “Experimental Investigation of Turbulent Mixing by Rayleigh-Taylor Instability” Physica 12D, pp. 45–48.

    Google Scholar 

  45. D. H. Sharp and J. A. Wheeler, 1961, “Late Stage of Rayleigh-Taylor Instability” Institute for Defense Analyses.

    Google Scholar 

  46. M. Shearer, 1987, “Loss of Strict Hyperbolicity in the Buckley-Leverett Equations of Three Phase Flow in a Porous Medium.” in Numerical Simulation in Oil Recovery, ed. M. Wheeler (Springer Verlag, New York).

    Google Scholar 

  47. Z. Tang and T. C. T. Ting, 1987, “Wave Curves for the Riemann Problem of Plane Waves in Simple Isotropic Elastic Solids” Int. J. Eng. Science 25, pp. 1343–1381.

    Article  MathSciNet  MATH  Google Scholar 

  48. P. Woodward, 1985, “Simulation of the Kelvin-Helmholtz Instability of a Supersonic Slipsuface with a Piecewise Parabolic Method” Proc. INRIA Workshop on Numerical Methods for Euler Equations, p. 114.

    Google Scholar 

  49. J. A. Zufiria, “Vortex-in-Cell Simulation of Bubble Competition in Rayleigh-Taylor Instability,” Preprint, 1988.

    Google Scholar 

  50. J. A. Zufiria, 1988, “Bubble Competition in Rayleigh-Taylor Instability” Phys. of Fluids 31, pp. 440–446.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag New York, Inc.

About this paper

Cite this paper

Glimm, J. (1991). Nonlinear Waves: Overview and Problems. In: Glimm, J., Majda, A.J. (eds) Multidimensional Hyperbolic Problems and Computations. The IMA Volumes in Mathematics and Its Applications, vol 29. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9121-0_8

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-9121-0_8

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-9123-4

  • Online ISBN: 978-1-4613-9121-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics