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Introduction

  • Thomas H. Otway
Chapter
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.

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References

  1. 1.
    Barros-Neto, J., Gelfand, I.M.: Fundamental solutions for the Tricomi operator. Duke Math. J. 98, 465–483 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Barros-Neto, J., Gelfand, I.M.: Fundamental solutions for the Tricomi operator II. Duke Math. J. 111, 561–584 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Barros-Neto, J., Gelfand, I.M.: Fundamental solutions for the Tricomi operator III, Duke Math. J. 128, 119–140 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bers, L.: Mathematical Aspects of Subsonic and Transonic Gas Dynamics. Wiley, New York (1958)zbMATHGoogle Scholar
  5. 5.
    Čanić, S., Keyfitz, B.: A smooth solution for a Keldysh type equation. Commun. Partial Diff. Equations 21, 319–340 (1996)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chapman, C.J.: High Speed Flow. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  7. 7.
    Chen, G-Q., Slemrod, M., Wang, D.: Isometric immersions and compensated compactness. Commun. Math. Phys. 294, 411–437 (2010)Google Scholar
  8. 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. 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. 10.
    Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Interscience, New York (1948)zbMATHGoogle Scholar
  11. 11.
    Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Mechanics. Springer, Berlin (2005)Google Scholar
  12. 12.
    Fabes, E., Kenig, C., Serapioni, R.: The local regularity of solutions of degenerate elliptic equations. Commun. Partial Diff. Equations 7, 77–116 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 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. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 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. 16.
    Gu, C.: On partial differential equations of mixed type in n independent variables. Commun. Pure Appl. Math. 34, 333–345 (1981)CrossRefzbMATHGoogle Scholar
  17. 17.
    Han, Q.: Local solutions to a class of Monge-Ampère equations of mixed type. Duke Math. J. 136 421–473 (2007)zbMATHMathSciNetGoogle Scholar
  18. 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. 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)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Hersh, R.: How to classify differential polynomials. Amer. Math. Monthly 80, 641–654 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Jang, J., Masmoudi, N.: Well-posedness for compressible Euler equations with physical vacuum singularity. Commun. Pure Appl. Math. 62, 1327–1385 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 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. 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)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Khuri, M.A.: Boundary value problems for mixed type equations and applications. J. Nonlinear Anal. Ser. A: TMA (to appear); arXiv:1106.4000v1Google Scholar
  25. 25.
    Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)Google Scholar
  26. 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)CrossRefzbMATHGoogle Scholar
  27. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 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. 29.
    Magnanini, R., Talenti, G.: On complex-valued solutions to a 2D eikonal equation. Part one: qualitative properties. Contemp. Math. 283, 203–229 (1999)MathSciNetGoogle Scholar
  30. 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. 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)zbMATHMathSciNetGoogle Scholar
  32. 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)zbMATHMathSciNetGoogle Scholar
  33. 33.
    Magnanini, R., Talenti, G.: On complex-valued 2D eikonals. IV. Continuation past a caustic. Milan J. Math. 77, 1–66 (2009)zbMATHMathSciNetGoogle Scholar
  34. 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. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966)zbMATHGoogle Scholar
  37. 37.
    Otway, T.H.: Hodge equations with change of type. Ann. Mat. Pura Appl. 181, 437–452 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Otway, T.H.: Harmonic fields on the extended projective disc and a problem in optics. J. Math. Phys. 46, 113501 (2005)CrossRefMathSciNetGoogle Scholar
  39. 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. 40.
    Otway, T.H.: Energy inequalities for a model of wave propagation in cold plasma. Publ. Mat. 52, 195–234 (2008)zbMATHMathSciNetGoogle Scholar
  41. 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. 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)zbMATHGoogle Scholar
  43. 43.
    Payne, K.R.: Interior regularity of the Dirichlet problem for the Tricomi equation. J. Mat. Anal. Appl. 199, 271–292 (1996)CrossRefzbMATHGoogle Scholar
  44. 44.
    Payne, K.R.: Solvability theorems for linear equations of Tricomi type. J. Mat. Anal. Appl. 215, 262–273 (1997)CrossRefzbMATHGoogle Scholar
  45. 45.
    Rassias, J.M.: Lecture notes on Mixed Type Partial Differential Equations. World Scientific, Teaneck (1990)CrossRefzbMATHGoogle Scholar
  46. 46.
    Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Springer, Berlin (1983)zbMATHGoogle Scholar
  47. 47.
    Stix, T.H.: The Theory of Plasma Waves. McGraw-Hill, New York (1962)zbMATHGoogle Scholar
  48. 48.
    Stoker, J.J.: Water Waves. Interscience, New York (1987)Google Scholar
  49. 49.
    Stokes, G.G.: Mathematical and Physical Papers, vol. 5, Appendix. Cambridge University Press, Cambridge (1880–1905)Google Scholar
  50. 50.
    Swanson, D.G.: Plasma Waves. Institute of Physics, Bristol (2003)Google Scholar
  51. 51.
    Taylor, M.E.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)zbMATHGoogle Scholar
  52. 52.
    Zheng, Y.: Systems of Conservation Laws: Two-Dimensional Riemann Problems. Birkhauser, Boston (2001)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Thomas H. Otway
    • 1
  1. 1.Department of Mathematical SciencesYeshiva UniversityNew YorkUSA

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