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: 22 October 2007


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.


Authors’ Affiliations

Department of Mathematics, Nanjing University of Aeronautics and Astronautics
Department of Mathematics, Zhejiang University


  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


© J. Xu and D. Fang 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.