Local well-posedness for the incompressible Euler equations in the critical Besov spaces

Yong Zhou[1]

  • [1] Chinese University of Hong Kong, Institute of Mathematical Sciences and Department of Mathematics, Shatin, N.T. (Hong Kong), Xiamen University, Xiamen, Fujian (Chine)

Annales de l’institut Fourier (2004)

  • Volume: 54, Issue: 3, page 773-786
  • ISSN: 0373-0956

Abstract

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In this paper we establish the existence and uniqueness of the local solutions to the incompressible Euler equations in ℝ N , N ≥ 3 , with any given initial data belonging to the critical Besov spaces B p , 1 N / p + 1 . Moreover, a blowup criterion is given in terms of the vorticity field.

How to cite

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Zhou, Yong. "Local well-posedness for the incompressible Euler equations in the critical Besov spaces." Annales de l’institut Fourier 54.3 (2004): 773-786. <http://eudml.org/doc/116126>.

@article{Zhou2004,
abstract = {In this paper we establish the existence and uniqueness of the local solutions to the incompressible Euler equations in $\mathbb \{R\}$$^N$, $N \ge 3$, with any given initial data belonging to the critical Besov spaces $B_\{p,1\}^\{N/p+1\}$. Moreover, a blowup criterion is given in terms of the vorticity field.},
affiliation = {Chinese University of Hong Kong, Institute of Mathematical Sciences and Department of Mathematics, Shatin, N.T. (Hong Kong), Xiamen University, Xiamen, Fujian (Chine)},
author = {Zhou, Yong},
journal = {Annales de l’institut Fourier},
keywords = {well-posedness; Euler equations; Besov spaces},
language = {eng},
number = {3},
pages = {773-786},
publisher = {Association des Annales de l'Institut Fourier},
title = {Local well-posedness for the incompressible Euler equations in the critical Besov spaces},
url = {http://eudml.org/doc/116126},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Zhou, Yong
TI - Local well-posedness for the incompressible Euler equations in the critical Besov spaces
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 3
SP - 773
EP - 786
AB - In this paper we establish the existence and uniqueness of the local solutions to the incompressible Euler equations in $\mathbb {R}$$^N$, $N \ge 3$, with any given initial data belonging to the critical Besov spaces $B_{p,1}^{N/p+1}$. Moreover, a blowup criterion is given in terms of the vorticity field.
LA - eng
KW - well-posedness; Euler equations; Besov spaces
UR - http://eudml.org/doc/116126
ER -

References

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  1. J.-T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys 94 (1984), 61-66 Zbl0573.76029MR763762
  2. J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. Ecole Norm. Sup. (4) 14 (1981), 209-246 Zbl0495.35024MR631751
  3. D. Chae, On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces., Comm. Pure Appl. Math 55 (2002), 654-678 Zbl1025.35016MR1880646
  4. J.-Y. Chemin, Régularité de la trajectoire des particules d'un fluide parfait incompressible remplissant l'espace, J. Math. Pures Appl. (9) 71 (1992), 407-417 Zbl0833.35112MR1191582
  5. J.-Y. Chemin, Perfect incompressible fluids, 14 (1998), The Clarendon Press, Oxford University Press, New York Zbl0927.76002MR1688875
  6. R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math 141 (2000), 579-614 Zbl0958.35100MR1779621
  7. R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations 26 (2001), 1183-1233 Zbl1007.35071MR1855277
  8. M. Frazier, B. Jawerth, G. Weiss, Littlewood-Paley theory and the study of function spaces, 79 (1991), American Mathematical Society, Providence, RI Zbl0757.42006
  9. T. Kato, Nonstationary flows of viscous and ideal fluids in R 3 , J. Functional Analysis 9 (1972), 296-305 Zbl0229.76018MR481652
  10. T. Kato, G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math 41 (1988), 891-907 Zbl0671.35066MR951744
  11. H. Kozono, Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys. 214 (2000), 191-200 Zbl0985.46015MR1794270
  12. H. Kozono, T. Ogawa, Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z 242 (2002), 251-278 Zbl1055.35087MR1980623
  13. A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Frontiers of the mathematical sciences (New York, 1985) 39, suppl. (1986), S186-S220 Zbl0595.76021
  14. T. Runst, W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations., 3 (1996), Walter de Gruyter & Co., Berlin Zbl0873.35001MR1419319
  15. E.M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Monographs in Harmonic Analysis, III 43 (1993), Princeton University Press, Princeton, NJ Zbl0821.42001
  16. H. Triebel, Theory of function spaces. II., 84 (1992), Birkhäuser Verlag, Basel Zbl0763.46025
  17. M. Vishik, Hydrodynamics in Besov spaces., Arch. Ration. Mech. Anal 145 (1998), 197-214 Zbl0926.35123MR1664597
  18. M. Vishik, Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Ann. Sci. Ecole Norm. Sup. (4) 32 (1999), 769-812 Zbl0938.35128MR1717576

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