On the global existence for the axisymmetric Euler equations

Hammadi Abidi[1]; Taoufik Hmidi[1]; Sahbi Keraani[1]

  • [1] IRMAR, Université de Rennes 1, Campus de Beaulieu, 35 042 Rennes cedex. France

Journées Équations aux dérivées partielles (2008)

  • Volume: 347, Issue: 1, page 1-17
  • ISSN: 0752-0360

Abstract

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This paper deals with the global well-posedness of the 3 D axisymmetric Euler equations for initial data lying in critical Besov spaces B p , 1 1 + 3 p . In this case the BKM criterion is not known to be valid and to circumvent this difficulty we use a new decomposition of the vorticity .

How to cite

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Abidi, Hammadi, Hmidi, Taoufik, and Keraani, Sahbi. "On the global existence for the axisymmetric Euler equations." Journées Équations aux dérivées partielles 347.1 (2008): 1-17. <http://eudml.org/doc/10636>.

@article{Abidi2008,
abstract = {This paper deals with the global well-posedness of the $3$D axisymmetric Euler equations for initial data lying in critical Besov spaces $B_\{p,1\}^\{1+\frac\{3\}\{p\}\}$. In this case the BKM criterion is not known to be valid and to circumvent this difficulty we use a new decomposition of the vorticity .},
affiliation = {IRMAR, Université de Rennes 1, Campus de Beaulieu, 35 042 Rennes cedex. France; IRMAR, Université de Rennes 1, Campus de Beaulieu, 35 042 Rennes cedex. France; IRMAR, Université de Rennes 1, Campus de Beaulieu, 35 042 Rennes cedex. France},
author = {Abidi, Hammadi, Hmidi, Taoufik, Keraani, Sahbi},
journal = {Journées Équations aux dérivées partielles},
language = {eng},
month = {6},
number = {1},
pages = {1-17},
publisher = {Groupement de recherche 2434 du CNRS},
title = {On the global existence for the axisymmetric Euler equations},
url = {http://eudml.org/doc/10636},
volume = {347},
year = {2008},
}

TY - JOUR
AU - Abidi, Hammadi
AU - Hmidi, Taoufik
AU - Keraani, Sahbi
TI - On the global existence for the axisymmetric Euler equations
JO - Journées Équations aux dérivées partielles
DA - 2008/6//
PB - Groupement de recherche 2434 du CNRS
VL - 347
IS - 1
SP - 1
EP - 17
AB - This paper deals with the global well-posedness of the $3$D axisymmetric Euler equations for initial data lying in critical Besov spaces $B_{p,1}^{1+\frac{3}{p}}$. In this case the BKM criterion is not known to be valid and to circumvent this difficulty we use a new decomposition of the vorticity .
LA - eng
UR - http://eudml.org/doc/10636
ER -

References

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  11. H. C. Pak, Y. J. Park, Existence of solution for the Euler equations in a critical Besov space B ∞ , 1 1 ( ℝ n ) , Comm. Partial Diff. Eqs, 29 (2004) 1149-1166. Zbl1091.76006MR2097579
  12. J. Peetre, New thoughts on Besov spaces, Duke University Mathematical Series 1, Durham N. C. 1976. Zbl0356.46038MR461123
  13. X. Saint Raymond, Remarks on axisymmetric solutions of the incompressible Euler system, Comm. Partial Differential Equations 19 (1994), no. 1-2, 321-334. Zbl0795.35063MR1257007
  14. T. Shirota, T. Yanagisawa, Note on global existence for axially symmetric solutions of the Euler system, Proc. Japan Acad. Ser. A Math. Sci. 70 (1994), no. 10, 299–304. Zbl0831.35141MR1313183
  15. M. R. Ukhovskii, V. I. Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, Prikl. Mat. Meh. 32 (1968), no. 1, 59-69. Zbl0172.53405MR239293
  16. M. Vishik, Hydrodynamics in Besov Spaces, Arch. Rational Mech. Anal 145, 197-214, 1998. Zbl0926.35123MR1664597

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