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Remarks on the Blow-up of the Euler Equations and the Related Equations

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Abstract

We consider the Euler system for inviscid incompressible fluid flows, and its perturbations in ℝn, n≥2. We prove global well-posedness of this perturbed Euler system in the Triebel-Lizorkin spaces for initial vorticity which is small in the critical Besov norms. Comparison type theorems about the blow-up of solutions are proved between the Euler system and its perturbations. We also study the possiblity of the self-similar type of blow-up of solutions to the equations.

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References

  1. Bahouri, H., Dehman, B.: Remarques sur l’apparition de singularités dans les écoulements Eulériens incompressibles à donnée initiale Höldérienne. J. Math. Pures Appl. 73, 335–346 (1994)

    MathSciNet  MATH  Google Scholar 

  2. Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66 (1984)

    MathSciNet  MATH  Google Scholar 

  3. Chae, D.: On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces. Commun. Pure and Appl. Math. 55(5), 654–678 (2002)

    Article  MATH  Google Scholar 

  4. Chae, D.: On the Euler equations in the critical Triebel-Lizorkin spaces. Arch. Rat. Mech. Anal. 170(3), 185–210 (2003)

    Google Scholar 

  5. Chemin, J.Y.: Régularité des trajectoires des particules d’un fluide incompressible remplissant l’espace. J. Math. Pures Appl. 71(5), 407–417 (1992)

    MATH  Google Scholar 

  6. Chemin, J.Y.: Perfect incompressible fluids. Oxford: Clarendon Press, 1998

  7. Constantin, P.: Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations. Commun. Math. Phys. 104, 311–326 (1986)

    MathSciNet  MATH  Google Scholar 

  8. Constantin, P., Fefferman, C., Majda, A.: Geometric constraints on potential singularity formation in the 3-D Euler equations. Commun. P. D. E. 21(3-4), 559–571 (1996)

    Google Scholar 

  9. Constantin, P., Lax, P.D., Majda, A.: A simple one-dimensional model for the three-dimensional vorticity equations. Commun. Pure Appl. Math. 38, 715-724 (1985)

    MathSciNet  MATH  Google Scholar 

  10. Frazier, M., Torres, R., Weiss, G.: The boundedness of Caleron-Zygmund operators on the spaces \(\dot{F}^{\alpha,q}_{p}\). Revista Mathemática Ibero-Americana 4(1), 41–72 (1988)

    MATH  Google Scholar 

  11. Jawerth, B.: Some observations on Besov and Lizorkin-Triebel Spaces. Math. Sacand. 40, 94–104 (1977)

    MATH  Google Scholar 

  12. Kato, T.: Nonstationary flows of viscous and ideal fluids in ℝ3. J. Funct. Anal. 9, 296–305 (1972)

    MATH  Google Scholar 

  13. Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41, 891–907 (1988)

    MATH  Google Scholar 

  14. Kato, T., Ponce, G.: Well posedness of the Euler and Navier-Stokes equations in Lebesgue spaces L s p (ℝ2 ). Revista Mathemática Ibero-Americana, 2, 73–88 (1986)

    Google Scholar 

  15. Kozono, H., Taniuchi, Y.: Limiting case of the Sobolev inequality in BMO, with applications to the Euler equations. Commun. Math. Phys. 214, 191–200 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kozono, H., Ogawa, T., Taniuchi, T.: The critical Sobolev inequalities in Besov spaces and regularity criterion to some semilinear evolution equations. Math Z. 242(2), 251–278 (2002)

    Article  Google Scholar 

  17. Leray, J.: Essai sur le mouvement d’un fluide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)

    MATH  Google Scholar 

  18. Majda, A., Bertozzi, A.: Vorticity and Incompressible Flow. Cambridge: Cambridge Univ. Press, 2002

  19. Nečas, J., Ružička, M., Šverák, V.: On Leray’s self-similar solutions of the Navier-Stokes equations. Acta Math. 176, 283–294 (1996)

    MathSciNet  Google Scholar 

  20. Triebel, H.: Theory of Function Spaces. Basel-Boston: Birkhäuser, 1983

  21. Vishik, M.: Hydrodynamics in Besov spaces. Arch. Rational Mech. Anal. 145, 197–214 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Vishik, M.: Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type. Ann. Scient. Éc. Norm. Sup., 4e série. 32, 769–812 (1999)

    Google Scholar 

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  1. Department of Mathematics, Seoul National University, Seoul, 151-742, Korea

    Dongho Chae

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  1. Dongho Chae
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Correspondence to Dongho Chae.

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Chae, D. Remarks on the Blow-up of the Euler Equations and the Related Equations. Commun. Math. Phys. 245, 539–550 (2004). https://doi.org/10.1007/s00220-003-1018-y

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  • DOI: https://doi.org/10.1007/s00220-003-1018-y

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