Sur la solution à support compact de l’equation d’Euler compressible

  • Tetu Makino
  • Seiji Ukai
  • Shuichi Kawashima


The Cauchy problem for the compressible Euler equation is discussed with compactly supported initials. To establish the localexistence of classical solutions by the aid of the theory of quasilinear symmetric hyperbolic systems, a new symmetrization is introduced which works for initials having compact support or vanishing at infinity. It is further shown that as far as the classical solution is concerned, its support does not change, and that the life span is finite for any solution except for the trivial zero solution.

Key words

compressible Euler equation compactly supported solution quasi-linear symmetric hyperbolic system non-existence of global solution 


  1. [1]
    T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Rational Mech. Anal.,58 (1975), 181–205.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    S. Klainerman and A. Majda, Compressible and incompressible fluids. Comm. Pure Appl. Math.,35 (1982), 629–651.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves. SIAM Regional Conf. Lecture #11, Philadelphia, 1973.Google Scholar
  4. [4]
    A. Majda, Compressible fluid flow and systems of conservation laws in several space variables. Appl. Math. Sci.,53, Springer-Verlag, 1984.Google Scholar
  5. [5]
    T. Makino, On a local existence theorem for the evolution equation of gaseous stars. à paraître.Google Scholar
  6. [6]
    T. Sideris, Formation of singularities in solutions to nonlinear hyperbolic equations. Arch. Rational Mech. Anal.,86 (1984), 369–381.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    T. Sideris, Formation of singularities in three-dimensional compressible fluids. Comm. Math. Phys.,101 (1985), 475–485.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© JJAM Publishing Committee 1986

Authors and Affiliations

  • Tetu Makino
    • 1
  • Seiji Ukai
    • 2
  • Shuichi Kawashima
    • 3
  1. 1.Department of Liberal ArtsOsaka Industrial UniversityOsakaJapan
  2. 2.Department of Applied PhysicsOsaka City UniversityOsakaJapan
  3. 3.Department of MathematicsNara Women’s UniversityNaraJapan

Personalised recommendations