Skip to content


  • Research Article
  • Open Access

A Note on the Relaxation-Time Limit of the Isothermal Euler Equations

Boundary Value Problems20072007:056945

  • Received: 3 July 2007
  • Accepted: 30 August 2007
  • Published:


This work is concerned with the relaxation-time limit of the multidimensional isothermal Euler equations with relaxation. We show that Coulombel-Goudon's results (2007) can hold in the weaker and more general Sobolev space of fractional order. The method of proof used is the Littlewood-Paley decomposition.


  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • Sobolev Space


Authors’ Affiliations

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China
Department of Mathematics, Zhejiang University, Hangzhou, 310027, China


  1. Coulombel J-F, Goudon T: The strong relaxation limit of the multidimensional isothermal Euler equations. Transactions of the American Mathematical Society 2007,359(2):637–648. 10.1090/S0002-9947-06-04028-1MATHMathSciNetView ArticleGoogle Scholar
  2. Junca S, Rascle M: Strong relaxation of the isothermal Euler system to the heat equation. Zeitschrift für Angewandte Mathematik und Physik 2002,53(2):239–264. 10.1007/s00033-002-8154-7MATHMathSciNetView ArticleGoogle Scholar
  3. Marcati P, Milani A: The one-dimensional Darcy's law as the limit of a compressible Euler flow. Journal of Differential Equations 1990,84(1):129–147. 10.1016/0022-0396(90)90130-HMATHMathSciNetView ArticleGoogle Scholar
  4. Hanouzet B, Natalini R: Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Archive for Rational Mechanics and Analysis 2003,169(2):89–117. 10.1007/s00205-003-0257-6MATHMathSciNetView ArticleGoogle Scholar
  5. Sideris TC, Thomases B, Wang D: Long time behavior of solutions to the 3D compressible Euler with damping. Communications in Partial Differential Equations 2003,28(3–4):795–816. 10.1081/PDE-120020497MATHMathSciNetView ArticleGoogle Scholar
  6. Yong W-A: Entropy and global existence for hyperbolic balance laws. Archive for Rational Mechanics and Analysis 2004,172(2):247–266. 10.1007/s00205-003-0304-3MATHMathSciNetView ArticleGoogle Scholar
  7. Fang DY, Xu J: Existence and asymptotic behavior ofsolutions to the multidimensional compressible Euler equations with damping.
  8. Simon J: Compact sets in the space. Annali di Matematica Pura ed Applicata 1987,146(1):65–96.MATHView ArticleGoogle Scholar
  9. Chemin J-Y: Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and Its Applications. Volume 14. The Clarendon Press, Oxford University Press, New York, NY, USA; 1998:x+187.Google Scholar
  10. Shizuta Y, Kawashima S: Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Mathematical Journal 1985,14(2):249–275.MATHMathSciNetView ArticleGoogle Scholar