Skip to main content

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 258))

  • 3609 Accesses

Abstract

In this chapter we shall present a summary of what could be considered as some of the most significant results which have appeared since the appearance of the original edition of this book. Of course, limitations of space (and time !) forces us to give only statements of major results, and very brief outlines of some of their proofs. This chapter will be divided into four sections, the first two corresponding to Parts II and III of the text, and the last two related to Part IV of the text. The numbered references correspond to the new reference list given at the end of this chapter.

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 189.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover 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. Alexander, J., R. Gardner, and C.K.R.T. Jones A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math., 410 (1990), 167–212.

    MathSciNet  MATH  Google Scholar 

  2. Bates P. and C.K.R.T. Jones Invariant manifolds for semilinear partial differential equations. Dynamics Reported, 2 (1989), 1–38.

    MathSciNet  Google Scholar 

  3. Benci, V. A new approach to the Morse-Conley theory. In: Recent Advances in Hamiltonian Systems, edited by G.F. Dell’Antonio and B. D’Onofrio. World Scientific: Singapore, 1985, pp. 1–52.

    Google Scholar 

  4. Caginalp, G. and P. Fife Higher-order phase field models and detailed anisotropy. Phys. Rev. B., 34 (1986), 4940–4943.

    Article  MathSciNet  Google Scholar 

  5. Conley, C. and E. Zehnder The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold. Invent. Math., 73 (1983), 33–49.

    Article  MathSciNet  MATH  Google Scholar 

  6. Conley, C. and J. Smoller Bifurcation and stability of stationary solutions of the Fitz-Hugh-Nagumo equations. J. Diff. Eqs., 63 (1986), 389–405.

    Article  MathSciNet  MATH  Google Scholar 

  7. Conway, E., R. Gardner, and J. Smoller Stability and bifurcation of steady-state solutions for predator-prey equations. Adv. in Appl. Math., 3 (1982), 288–334.

    Article  MathSciNet  MATH  Google Scholar 

  8. Dancer, E.N. On non-radially symmetric bifurcation. J. London Math. Soc, 20 (1979), 287–292.

    Article  MathSciNet  MATH  Google Scholar 

  9. Ding Xiaxi, Chen Guiqiang, and Luo Peizhu Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics, I, II, III. Acta Math. Sinica (1985), 415–432,433-472; (1986), 75-120.

    Google Scholar 

  10. Ding Xiaxi, Chen Guiqiang, and Luo Peizhu Convergence of the generalized Lax-Friedrichs scheme and Godunov’s scheme for isentropic gas dynamics. Comm. Math. Phys., 121 (1989), 63–84.

    Article  MathSciNet  MATH  Google Scholar 

  11. DiPerna, R. Convergence of approximate solutions of conservation laws. Arch. Rat. Mech. Anal., 82 (1983), 27–70.

    Article  Google Scholar 

  12. DiPerna, R. Convergence of the viscosity method for isentropic gas dynamics. Comm. Math. Phys., 91 (1983), 1–30.

    Article  MathSciNet  Google Scholar 

  13. Evans, J.W. Nerve axon equations, I, II, III. Ind. U. Math. J., 21 (1972), 877–895; 22 (1972), 75-90, 577-594.

    Article  MATH  Google Scholar 

  14. Evans, J.W. Nerve axon equations, IV. Ind. U. Math. J., 24 (1975), 1169–1190.

    Article  MATH  Google Scholar 

  15. Fenichel, N. Persistence and smoothness of invariant manifolds for flows. Ind. U. Math. J., 22 (1972), 577–594.

    Article  Google Scholar 

  16. Floer, A. and E. Zehnder Fixed point results for sympletic maps related to the Arnold conjecture. Proc. of Workshop in Dynamical Systems and Bifurcation, Groningen (April 1984), pp. 16–19.

    Google Scholar 

  17. Floer, A. A refinement of the Conley index and an application to the stability of hyperbolic invariant sets. Ergodic Theory Dynamical Systems, 7 (1987), 93–103.

    Article  MathSciNet  MATH  Google Scholar 

  18. Floer, A. Proof of the Arnold conjecture for surfaces and generalizations for certain Kähler manifolds. Duke Math. J., 53 (1986), 1–32.

    Article  MathSciNet  MATH  Google Scholar 

  19. Franzosa, R. Index filtrations and connection metrics for partially ordered Morse decompositions. Trans. Amer. Math. Soc, 298 (1986), 193–213.

    Article  MathSciNet  MATH  Google Scholar 

  20. Franzosa, R. The connection matrix theory for Morse decompositions. Trans. Amer. Math. Soc, 311 (1989), 561–592.

    Article  MathSciNet  MATH  Google Scholar 

  21. Gardner, R. Existence of travelling wave solutions of predator-prey systems via the connection index. SIAM J. Appl. Math., 44 (1984), 56–79.

    Article  MathSciNet  MATH  Google Scholar 

  22. Gardner, R. On the detonation of a combustible gas. Trans. Amer. Math. Soc, 277 (1983), 431–468.

    Article  MathSciNet  MATH  Google Scholar 

  23. Gardner, R. and C.K.R.T. Jones Travelling waves of a perturbed diffusion equation arising in a phase field model. Ind. U. Math. J., 38 (1989), 1197–1222.

    Google Scholar 

  24. Gardner, R. and C.K.R.T. Jones A stability index for steady-state solutions of boundary-value problems for parabolic systems, J. Diff. Eqns., 91 (1991), 181–203.

    Article  MathSciNet  MATH  Google Scholar 

  25. Gardner, R. and C.K.R.T. Jones Stability of travelling wave solutions of diffusive-predator-prey systems. Trans. Amer. Math. Soc, 237 (1991), 465–524.

    Article  MathSciNet  Google Scholar 

  26. Gidas, B., W.M. Ni, and L. Nirenberg Symmetry of positive solutions of nonlinear elliptic equations in ℝn. Comm. Math. Phys., 68 (1979), 202–243.

    Article  MathSciNet  Google Scholar 

  27. Goodman, J. Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rat. Mech. Anal, 95 (1986), 325–344.

    Article  MATH  Google Scholar 

  28. Hastings, S. On the existence of heteroclinic and periodic orbits for the Fitz-Hugh-Nagumo equations. Quart. J. Appl. Math., Oxford Ser. 27 (1976), 123–134.

    Article  MathSciNet  MATH  Google Scholar 

  29. Helgason, S. Topics in Harmonic Analysis on Homogeneous Spaces. Boston: Birkhauser, 1981.

    MATH  Google Scholar 

  30. Hoff, D. and J. Smoller Solutions in the large for certain nonlinear parabolic systems. Ann. Inst. H. Poincaré, 2 (1985), 213–235.

    MathSciNet  MATH  Google Scholar 

  31. Husemoller, D. Fiber Bundles. Springer-Verlag: Berlin, 1966.

    Google Scholar 

  32. Hin, A.M. and O.A. Oleinik Behavior of the solution of the Cauchy problem for certain quasilinear equations for unbounded increase of time. Amer. Math. Soc. Transi, Ser. 2,42 (1964), 19–23.

    Google Scholar 

  33. Jones, C. Stability of travelling wave solutions for the Fitz-Hugh-Nagumo system. Trans. Amer. Math. Soc, 286 (1984), 431–469.

    Article  MathSciNet  MATH  Google Scholar 

  34. Jones, C. Some ideas in the proof that the Fitz-Hugh-Nagumo pulse is stable. In: Nonlinear Partial Differential Equations, edited by J. Smoller, Contemp. Math., No. 17. Amer. Math. Soc: Providence, 1983, pp. 287–292.

    Chapter  Google Scholar 

  35. Kawashima, S. and A. Matzumura Asymptotic stability of travelling wave solutions of systems of one-dimensional gas motion. Comm. Math. Phys., 101 (1985), 97–127.

    Article  MathSciNet  MATH  Google Scholar 

  36. Keyfitz, B.L. and H.C. Kranzer (eds.) Nonstrictly Hyperbolic Conservation Laws. Contemp. Math., No. 60, Amer. Math. Soc.: Providence, 1987.

    MATH  Google Scholar 

  37. Langer, R. Existence of homoclinic travelling wave solutions to the Fitz-Hugh-Nagumo equations. Ph.D. thesis, Northeastern University.

    Google Scholar 

  38. Liu, T.P. Nonlinear stability of shock waves for viscous conservation laws. Amer. Math. Soc. Memoir, No. 328. Amer. Math. Soc: Providence, 1985.

    Google Scholar 

  39. Liu, T.P. Shock waves for compressible Navier-Stokes equations are stable. Comm. Pure Appl. Math., 39 (1986), 565–594.

    Article  MathSciNet  MATH  Google Scholar 

  40. Liu, T.P. and Z. Xin Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations. Comm. Math. Phys., 118 (1986), 451–465.

    Article  MathSciNet  Google Scholar 

  41. Mischaikow, K. Conley’s connection matrix. In: Dynamics of Infinite Dimensional Dynamical Systems, edited by S.-N. Chow and J. Hale, NATO ASI, Series F: Computers and Systems Sciences, #37. Springer-Verlag: New York, 1987.

    Google Scholar 

  42. Mischaikow, K. Classification of travelling wave solutions of reaction-diffusion systems. Lefschetz Center for Dynamical Systems, Report #86-5, (1985).

    Google Scholar 

  43. Murat, F. Capacité par compensation. Ann. Scuola Norm. Pisa Sci. Fis. Mat., 5 (1978). 489–507.

    MathSciNet  MATH  Google Scholar 

  44. Nishiura, Y. and M. Mimura Layer oscillations in reaction-diffusion systems. SIAM J. Appl. Math., 49 (1989), 481–514.

    Article  MathSciNet  MATH  Google Scholar 

  45. Reineck, J. Connecting orbits in one-parameter families of flows. J. Ergodic Theory Dynamical Systems, 8 (1988), 359–374.

    Article  MathSciNet  Google Scholar 

  46. Rybakowski, K. On the homotopy index for infinite-dimensional semiflows. Trans. Amer. Math. Soc., 269 (1982), 351–382.

    Article  MathSciNet  MATH  Google Scholar 

  47. Rybakowski, K. The Morse index, repellor-attractor pairs, and the connection index for semiflows on noncompact spaces. J. Diff. Eqns., 47 (1983), 66–98.

    Article  MathSciNet  MATH  Google Scholar 

  48. Rybakowski, K. Trajectories joining critical points of nonlinear parabolic and hyperbolic partial differential equations. J. Diff. Eqns., 51 (1984), 182–212.

    Article  MathSciNet  MATH  Google Scholar 

  49. Rybakowski, K. and E. Zehnder A Morse equation in Conley’s index theory for semiflows on metric spaces. Ergodic Theory Dynamical Systems, 5 (1985), 123–143.

    Article  MathSciNet  MATH  Google Scholar 

  50. Salamon, D. Connected simple systems and the Conley index of isolated invariant sets. Trans. Amer. Math. Soc, 291 (1985), 1–41.

    Article  MathSciNet  MATH  Google Scholar 

  51. Smoller, J. and A. Wasserman On the monotonicity of the time map. J. Diff. Eqns., 77 (1989), 287–303.

    Article  MathSciNet  MATH  Google Scholar 

  52. Smoller, J. and A. Wasserman Symmetry, degeneracy, and universality in semilinear elliptic equations. J. Funct. Anal., 89 (1990), 364–409.

    Article  MathSciNet  MATH  Google Scholar 

  53. Smoller, J. and A. Wasserman Bifurcation and symmetry-breaking. Invent. Math., 100 (1990), 63–95.

    Article  MathSciNet  MATH  Google Scholar 

  54. Tartar, L. The compensated compactness method applied to systems of conservation laws. In: Systems of Non-Linear Partial Differential Equations, edited by J. Ball. Reidel: Dordrecht, 1983, pp. 263–288.

    Chapter  Google Scholar 

  55. Tartar, L. Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, IV, edited by R.J. Knops. Pitman: San Francisco, 1979, pp. 136–212.

    Google Scholar 

  56. Temple, B. Global solution of the Cauchy problem for a class of 2 x 2 nonstrictly hyperbolic conservation laws. Adv. in Appl. Math., 3 (1982), 335–375.

    Article  MathSciNet  MATH  Google Scholar 

  57. Temple, B. Nonlinear conservation laws with invariant submanifolds. Trans. Amer. Math. Soc, 280 (1983), 781–795.

    Article  MathSciNet  MATH  Google Scholar 

  58. Temple, B. Degenerate systems of conservation laws. In: Nonstrictly Hyperbolic Conservation Laws, edited by B. Keyfitz and H. Kranzer. Contemp. Math., No. 60, Amer. Math. Soc.: Providence, 1987, pp. 125–133.

    Chapter  Google Scholar 

  59. Temple, B. Stability of Godunov’s method for a class of 2 x 2 systems of conservation laws. Trans. Amer. Math. Soc, 288 (1985), 115–123.

    MathSciNet  MATH  Google Scholar 

  60. Temple, B. Systems of conservation laws with coinciding shock and rarefaction waves. Contemporary Math., 17 (1983), 143–151.

    Article  MATH  Google Scholar 

  61. Temple, B. Decay with a rate for noncompactly supported solutions of conservation laws. Trans. Amer. Math. Soc, 298 (1986), 43–82.

    Article  MathSciNet  MATH  Google Scholar 

  62. Young, L.C. Lectures on the Calculus of Variations and Optimal Control Theory, W.B. Saunders, Philadelphia, PA (1969).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer Science+Business Media New York

About this chapter

Cite this chapter

Smoller, J. (1994). Recent Results. In: Shock Waves and Reaction—Diffusion Equations. Grundlehren der mathematischen Wissenschaften, vol 258. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0873-0_25

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0873-0_25

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-6929-8

  • Online ISBN: 978-1-4612-0873-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics