On the global existence for the axisymmetric Euler equations
Journées équations aux dérivées partielles (2008), article no. 4, 17 p.

This paper deals with the global well-posedness of the 3D 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 .

DOI : 10.5802/jedp.48
Abidi, Hammadi 1 ; Hmidi, Taoufik 1 ; Keraani, Sahbi 1

1 IRMAR, Université de Rennes 1, Campus de Beaulieu, 35 042 Rennes cedex. France
@article{JEDP_2008____A4_0,
     author = {Abidi, Hammadi and Hmidi, Taoufik and Keraani, Sahbi},
     title = {On the global existence for the axisymmetric {Euler} equations},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {4},
     pages = {1--17},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2008},
     doi = {10.5802/jedp.48},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jedp.48/}
}
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
PY  - 2008
SP  - 1
EP  - 17
PB  - Groupement de recherche 2434 du CNRS
UR  - http://archive.numdam.org/articles/10.5802/jedp.48/
DO  - 10.5802/jedp.48
LA  - en
ID  - JEDP_2008____A4_0
ER  - 
%0 Journal Article
%A Abidi, Hammadi
%A Hmidi, Taoufik
%A Keraani, Sahbi
%T On the global existence for the axisymmetric Euler equations
%J Journées équations aux dérivées partielles
%D 2008
%P 1-17
%I Groupement de recherche 2434 du CNRS
%U http://archive.numdam.org/articles/10.5802/jedp.48/
%R 10.5802/jedp.48
%G en
%F JEDP_2008____A4_0
Abidi, Hammadi; Hmidi, Taoufik; Keraani, Sahbi. On the global existence for the axisymmetric Euler equations. Journées équations aux dérivées partielles (2008), article  no. 4, 17 p. doi : 10.5802/jedp.48. http://archive.numdam.org/articles/10.5802/jedp.48/

[1] J. T. Beale, T. Kato, A. Majda, Remarks on the Breakdown of Smooth Solutions for the 3D Euler Equations, Comm. Math. Phys. 94 (1984) 61-66. | MR | Zbl

[2] J. Bergh, J. Löfström, Interpolation spaces. An introduction, Springer-Verlag, 1976. | MR | Zbl

[3] J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. de l’École Norm. Sup. 14 (1981) 209-246. | Numdam | MR | Zbl

[4] D. Chae, Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptot. Anal. 38 (2004), no. 3-4, 339-358. | MR | Zbl

[5] J.-Y. Chemin, Perfect incompressible Fluids, Clarendon press, Oxford, 1998. | MR | Zbl

[6] P. Constantin, C. Fefferman, A. Majda, J. Geometric constraints on potentially singular solutions for the 3D Euler equations, Comm. Partial Diff. Eqs. 21 (1996), no. 3-4, 559-571. | MR | Zbl

[7] R. Danchin, Axisymmetric incompressible flows with bounded vorticity, Russian Math. Surveys 62 (2007), no 3, 73-94. | MR | Zbl

[8] T. Hmidi, S. Keraani, Incompressible viscous flows in borderline Besov spaces, Arch. Ration. Mech. Anal. 189 (2008), no. 2, 283-300. | MR | Zbl

[9] T. Kato, Nonstationary flows of viscous and ideal fluids in 3 , J. Functional analysis, 9 (1972), 296-305. | MR | Zbl

[10] R. O’Neil, Convolution operators and L(p,q) spaces, Duke Math. J. 30 (1963), 129-142. | MR | Zbl

[11] H. C. Pak, Y. J. Park, Existence of solution for the Euler equations in a critical Besov space B ,1 1 ( n ), Comm. Partial Diff. Eqs, 29 (2004) 1149-1166. | MR | Zbl

[12] J. Peetre, New thoughts on Besov spaces, Duke University Mathematical Series 1, Durham N. C. 1976. | MR | Zbl

[13] X. Saint Raymond, Remarks on axisymmetric solutions of the incompressible Euler system, Comm. Partial Differential Equations 19 (1994), no. 1-2, 321-334. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[16] M. Vishik, Hydrodynamics in Besov Spaces, Arch. Rational Mech. Anal 145, 197-214, 1998. | MR | Zbl

Cité par Sources :