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Communications in Mathematical Physics

, Volume 101, Issue 4, pp 475–485 | Cite as

Formation of singularities in three-dimensional compressible fluids

  • Thomas C. Sideris
Article

Abstract

Presented are several results on the formation of singularities in solutions to the three-dimensional Euler equations for a polytropic, ideal fluid under various assumptions on the initial data. In particular, it is shown that a localized fluid which is initially compressed and outgoing, on average, will develop singularities regardless of the size of the initial disturbance.

Keywords

Neural Network Statistical Physic Complex System Initial Data Nonlinear Dynamics 
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References

  1. 1.
    Courant, R., Friedrichs, K. O.: Supersonic flow and shock waves. New York: Interscience 1948Google Scholar
  2. 2.
    John, F.: Formation of singularities in one-dimensional nonlinear wave propagation. Commun. Pure Appl. Math.27, 377–405 (1974)Google Scholar
  3. 3.
    John, F.: Blow-up of radial solutions ofu tt=C 2(u tu in three space dimensions. To appearGoogle Scholar
  4. 4.
    Kato, T.: The Cauchy problem for quasilinear symmetric systems. Arch. Ration. Mech. Anal.58, 181–205 (1975)Google Scholar
  5. 5.
    Klainerman, S. Majda, A.: Formation of singularities for wave equations including the nonlinear vibrating string. Commun. Pure Appl. Math.33, 241–263 (1980)Google Scholar
  6. 6.
    Klainerman, S. Majda, A.: Compressible and incompressible fluids. Commun. Pure Appl. Math.35, 629–651 (1982)Google Scholar
  7. 7.
    Lax, P.: Development of singularities of solutions of nonlinear hyperbolic partial differential equations. J. Math. Phys.5, 611–613 (1964)Google Scholar
  8. 8.
    Liu, T.-P.: The development of singularities in the nonlinear waves for quasi-linear hyperbolic partial differential equations. J. Differ. Equations33, 92–111 (1979)Google Scholar
  9. 9.
    Sideris, T.: Global behavior of solutions to nonlinear wave equations in three space dimensions. Commun. Partial Differ. Equations8, 12, 1291–1323 (1983)Google Scholar
  10. 10.
    Sideris, T.: Formation of singularities of solutions to nonlinear hyperbolic equations. Arch. Ration. Mech. Anal.86, 4, 369–381 (1984)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Thomas C. Sideris
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA

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