Skip to main content

Introduction

  • 829 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 2043)

Abstract

In the introductory chapters to most plasma physics texts (e.g., [43,45]), an idealized model is presented in which the plasma ion and electron temperatures – rather than being millions of degrees – are set to absolute zero. This is done to reduce the mathematics to its simplest possible form, but even in this case a rigorous description of the plasma is problematic: the classical Dirichlet problem, which is the physically natural boundary value problem in this context, turns out to be ill-posed on typical domains.

Keywords

  • Dirichlet Problem
  • Hyperbolic Equation
  • Isometric Embedding
  • Beltrami Equation
  • Inhomogeneous Boundary Condition

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barros-Neto, J., Gelfand, I.M.: Fundamental solutions for the Tricomi operator. Duke Math. J. 98, 465–483 (1999)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. Barros-Neto, J., Gelfand, I.M.: Fundamental solutions for the Tricomi operator II. Duke Math. J. 111, 561–584 (2002)

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. Barros-Neto, J., Gelfand, I.M.: Fundamental solutions for the Tricomi operator III, Duke Math. J. 128, 119–140 (2005)

    CrossRef  MATH  MathSciNet  Google Scholar 

  4. Bers, L.: Mathematical Aspects of Subsonic and Transonic Gas Dynamics. Wiley, New York (1958)

    MATH  Google Scholar 

  5. Čanić, S., Keyfitz, B.: A smooth solution for a Keldysh type equation. Commun. Partial Diff. Equations 21, 319–340 (1996)

    CrossRef  MATH  Google Scholar 

  6. Chapman, C.J.: High Speed Flow. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  7. Chen, G-Q., Slemrod, M., Wang, D.: Isometric immersions and compensated compactness. Commun. Math. Phys. 294, 411–437 (2010)

    Google Scholar 

  8. Cibrario, M.: Sulla riduzione a forma canonica delle equazioni lineari alle derivate parziali di secondo ordine di tipo misto. Rendiconti del R. Insituto Lombardo 65 (1932)

    Google Scholar 

  9. Cinquini-Cibrario, M.: Review of Rassias, John M. (GR-UATH) Lecture notes on mixed type partial differential equations, World Scientific Publishing Co., Teaneck, 1990. In: Mathematical Reviews, MR1082555 (91m:35162)

    Google Scholar 

  10. Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Interscience, New York (1948)

    MATH  Google Scholar 

  11. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Mechanics. Springer, Berlin (2005)

    Google Scholar 

  12. Fabes, E., Kenig, C., Serapioni, R.: The local regularity of solutions of degenerate elliptic equations. Commun. Partial Diff. Equations 7, 77–116 (1982)

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. Frankl’, F.L.: Problems of Chaplygin for mixed sub- and supersonic flows [in Russian]. Izv. Akad. Nauk SSSR, ser. mat. 9(2), 121–143 (1945)

    Google Scholar 

  14. Gramchev, T.V.: An application of the analytic microlocal analysis to a class of differential operators of mixed type. Math. Nachr. 121, 41–51 (1985)

    CrossRef  MATH  MathSciNet  Google Scholar 

  15. Groothuizen, R.J.P.: Mixed Elliptic-Hyperbolic Partial Differential Operators: A Case Study in Fourier Integral Operators. CWI Tract, vol. 16, Centrum voor Wiskunde en Informatica, Amsterdam (1985)

    Google Scholar 

  16. Gu, C.: On partial differential equations of mixed type in n independent variables. Commun. Pure Appl. Math. 34, 333–345 (1981)

    CrossRef  MATH  Google Scholar 

  17. Han, Q.: Local solutions to a class of Monge-Ampère equations of mixed type. Duke Math. J. 136 421–473 (2007)

    MATH  MathSciNet  Google Scholar 

  18. Han, Q., Hong, J.-X.: Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Mathematical Surveys and Monographs, vol. 130, American Mathematical Society, Providence (2006)

    Google Scholar 

  19. Han, Q., Khuri, M.: On the local isometric embedding in R 3 of surfaces with Gaussian curvature of mixed sign. Commun. Anal. Geom. 18, 649–704 (2010)

    MATH  MathSciNet  Google Scholar 

  20. Hersh, R.: How to classify differential polynomials. Amer. Math. Monthly 80, 641–654 (1973)

    CrossRef  MATH  MathSciNet  Google Scholar 

  21. Jang, J., Masmoudi, N.: Well-posedness for compressible Euler equations with physical vacuum singularity. Commun. Pure Appl. Math. 62, 1327–1385 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  22. Keldysh, M.V.: On certain classes of elliptic equations with singularity on the boundary of the domain [in Russian]. Dokl. Akad. Nauk SSSR 77, 181–183 (1951)

    Google Scholar 

  23. Khuri, M.A.: The local isometric embedding in R 3 of two-dimensional Riemannian manifolds with Gaussian curvature changing sign to finite order on a curve. J. Differential Geom. 76, 249–291 (2007)

    MATH  MathSciNet  Google Scholar 

  24. Khuri, M.A.: Boundary value problems for mixed type equations and applications. J. Nonlinear Anal. Ser. A: TMA (to appear); arXiv:1106.4000v1

    Google Scholar 

  25. Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)

    Google Scholar 

  26. Lin, C.S.: The local isometric embedding in R 3 of a 2-dimensional Riemanian manifolds with Gaussian curvature changing sign cleanly. Commun. Pure Appl. Math. 39, 867–887 (1986)

    CrossRef  MATH  Google Scholar 

  27. Lupo, D., Morawetz, C.S., Payne, K.R.: On closed boundary value problems for equations of mixed elliptic-hyperbolic type. Commun. Pure Appl. Math. 60, 1319–1348 (2007)

    CrossRef  MATH  MathSciNet  Google Scholar 

  28. Lupo, D., Morawetz, C.S., Payne, K.R.: Erratum: “On closed boundary value problems for equations of mixed elliptic-hyperbolic type,” [Commun. Pure Appl. Math. 60, 1319–1348 (2007)]. Commun. Pure Appl. Math. 61, 594 (2008)

    Google Scholar 

  29. Magnanini, R., Talenti, G.: On complex-valued solutions to a 2D eikonal equation. Part one: qualitative properties. Contemp. Math. 283, 203–229 (1999)

    MathSciNet  Google Scholar 

  30. Magnanini, R., Talenti, G.: Approaching a partial differential equation of mixed elliptic-hyperbolic type. In: Anikonov, Yu.E., Bukhageim, A.L., Kabanikhin, S.I., Romanov, V.G. (eds.) Ill-posed and Inverse Problems, pp. 263–276. VSP, Utrecht (2002)

    Google Scholar 

  31. Magnanini, R., Talenti, G.: On complex-valued solutions to a two-dimensional eikonal equation. II. Existence theorems. SIAM J. Math. Anal. 34, 805–835 (2003)

    MATH  MathSciNet  Google Scholar 

  32. Magnanini, R., Talenti, G.: On complex-valued solutions to a 2D eikonal equation. III. Analysis of a Bäcklund transformation. Appl. Anal. 85, 249–276 (2006)

    MATH  MathSciNet  Google Scholar 

  33. Magnanini, R., Talenti, G.: On complex-valued 2D eikonals. IV. Continuation past a caustic. Milan J. Math. 77, 1–66 (2009)

    MATH  MathSciNet  Google Scholar 

  34. Morawetz, C.S.: Note on a maximum principle and a uniqueness theorem for an elliptic-hyperbolic equation. Proc. R. Soc. London, Ser. A 236, 141–144 (1956)

    Google Scholar 

  35. Morawetz, C.S., Stevens, D.C., Weitzner, H.: A numerical experiment on a second-order partial differential equation of mixed type. Commun. Pure Appl. Math. 44, 1091–1106 (1991)

    CrossRef  MATH  MathSciNet  Google Scholar 

  36. Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966)

    MATH  Google Scholar 

  37. Otway, T.H.: Hodge equations with change of type. Ann. Mat. Pura Appl. 181, 437–452 (2002)

    CrossRef  MATH  MathSciNet  Google Scholar 

  38. Otway, T.H.: Harmonic fields on the extended projective disc and a problem in optics. J. Math. Phys. 46, 113501 (2005)

    CrossRef  MathSciNet  Google Scholar 

  39. Otway, T.H.: Erratum: Harmonic fields on the extended projective disc and a problem in optics [J. Math. Phys. 46, 113501 (2005)]. J. Math. Phys. 48, 079901 (2007)

    Google Scholar 

  40. Otway, T.H.: Energy inequalities for a model of wave propagation in cold plasma. Publ. Mat. 52, 195–234 (2008)

    MATH  MathSciNet  Google Scholar 

  41. Parker, P.E.: Geometry of bicharacteristics. In: Advances in Differential Geometry and General Relativity, Contemp. Math., vol. 359, pp. 31–40. American Mathematical Society, Providence (2004)

    Google Scholar 

  42. Payne, K.R.: Propagation of singularities for solutions to the Dirichlet problem for equations of Tricomi type, Rend. Sem. Mat. Univ. Pol. Torino 54, 115–137 (1996)

    MATH  Google Scholar 

  43. Payne, K.R.: Interior regularity of the Dirichlet problem for the Tricomi equation. J. Mat. Anal. Appl. 199, 271–292 (1996)

    CrossRef  MATH  Google Scholar 

  44. Payne, K.R.: Solvability theorems for linear equations of Tricomi type. J. Mat. Anal. Appl. 215, 262–273 (1997)

    CrossRef  MATH  Google Scholar 

  45. Rassias, J.M.: Lecture notes on Mixed Type Partial Differential Equations. World Scientific, Teaneck (1990)

    CrossRef  MATH  Google Scholar 

  46. Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Springer, Berlin (1983)

    MATH  Google Scholar 

  47. Stix, T.H.: The Theory of Plasma Waves. McGraw-Hill, New York (1962)

    MATH  Google Scholar 

  48. Stoker, J.J.: Water Waves. Interscience, New York (1987)

    Google Scholar 

  49. Stokes, G.G.: Mathematical and Physical Papers, vol. 5, Appendix. Cambridge University Press, Cambridge (1880–1905)

    Google Scholar 

  50. Swanson, D.G.: Plasma Waves. Institute of Physics, Bristol (2003)

    Google Scholar 

  51. Taylor, M.E.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)

    MATH  Google Scholar 

  52. Zheng, Y.: Systems of Conservation Laws: Two-Dimensional Riemann Problems. Birkhauser, Boston (2001)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Otway, T.H. (2012). Introduction. In: The Dirichlet Problem for Elliptic-Hyperbolic Equations of Keldysh Type. Lecture Notes in Mathematics(), vol 2043. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24415-5_1

Download citation