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 .

@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},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2008},
     doi = {10.5802/jedp.48},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/JEDP_2008____A4_0/

[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 763762 | Zbl 0573.76029

[2] J. Bergh, J. Löfström, Interpolation spaces. An introduction, Springer-Verlag, 1976. | MR 482275 | Zbl 0344.46071

[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 631751 | Zbl 0495.35024

[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 2072064 | Zbl 1068.35097

[5] J.-Y. Chemin, Perfect incompressible Fluids, Clarendon press, Oxford, 1998. | MR 1688875 | Zbl 0927.76002

[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 1387460 | Zbl 0853.35091

[7] R. Danchin, Axisymmetric incompressible flows with bounded vorticity, Russian Math. Surveys 62 (2007), no 3, 73-94. | MR 2355419 | Zbl 1139.76011

[8] T. Hmidi, S. Keraani, Incompressible viscous flows in borderline Besov spaces, Arch. Ration. Mech. Anal. 189 (2008), no. 2, 283-300. | MR 2413097 | Zbl 1147.76014

[9] T. Kato, Nonstationary flows of viscous and ideal fluids in 3 , J. Functional analysis, 9 (1972), 296-305. | MR 481652 | Zbl 0229.76018

[10] R. O’Neil, Convolution operators and L(p,q) spaces, Duke Math. J. 30 (1963), 129-142. | MR 146673 | Zbl 0178.47701

[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 2097579 | Zbl 1091.76006

[12] J. Peetre, New thoughts on Besov spaces, Duke University Mathematical Series 1, Durham N. C. 1976. | MR 461123 | Zbl 0356.46038

[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 1257007 | Zbl 0795.35063

[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 1313183 | Zbl 0831.35141

[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 239293 | Zbl 0172.53405

[16] M. Vishik, Hydrodynamics in Besov Spaces, Arch. Rational Mech. Anal 145, 197-214, 1998. | MR 1664597 | Zbl 0926.35123