Skip to main content

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

Abstract

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.

[12345678910]

References

  1. 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/S0002-9947-06-04028-1

    MATH  MathSciNet  Article  Google Scholar 

  2. 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/s00033-002-8154-7

    MATH  MathSciNet  Article  Google Scholar 

  3. 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-H

    MATH  MathSciNet  Article  Google Scholar 

  4. 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/s00205-003-0257-6

    MATH  MathSciNet  Article  Google Scholar 

  5. 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-120020497

    MATH  MathSciNet  Article  Google Scholar 

  6. 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/s00205-003-0304-3

    MATH  MathSciNet  Article  Google Scholar 

  7. 7.

    Fang DY, Xu J: Existence and asymptotic behavior ofsolutions to the multidimensional compressible Euler equations with damping. http://arxiv.org/abs/math.AP/0703621

  8. 8.

    Simon J: Compact sets in the space. Annali di Matematica Pura ed Applicata 1987,146(1):65–96.

    MATH  Article  Google Scholar 

  9. 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. 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.

    MATH  MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jiang Xu.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Xu, J., Fang, D. A Note on the Relaxation-Time Limit of the Isothermal Euler Equations. Bound Value Probl 2007, 056945 (2007). https://doi.org/10.1155/2007/56945

Download citation

Keywords

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