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Geometric constraints on potentially singular solutions for the \(3-D\) Euler equations. (English) Zbl 0853.35091
We discuss necessary and sufficient conditions for the formation of finite time singularities (blow up) in the incompressible three-dimensional Euler equations. The well-known result of J. T. Beale, T. Kato and A. Majda [Commun. Math. Phys. 94, 61-66 (1984; Zbl 0573.76029)] states that these equations have smooth solutions on the time interval \([0, T)\) if, and only if \[ \lim_{t\to T} \int^t_0 |\omega(\cdot, s)|_{L^\infty(dx)} ds< \infty, \] where \(\omega= \nabla \times u\) is the vorticity of the fluid and \(u\) is its divergence-free velocity. In this paper, we prove criteria in which the direction of vorticity \(\xi= \omega/|\omega|\) plays an important role.

35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
76B47 Vortex flows for incompressible inviscid fluids
Zbl 0573.76029
Full Text: DOI
[1] Kato T., Commun. Math. Phys. 94 pp 61– (1984) · Zbl 0573.76029
[2] Constantin P., Siam Review 36 pp 73– (1994) · Zbl 0803.35106
[3] P. Constantin, Geometric and analytic studies in turbulences, in Trends and Perspectives in Appl. Math., L. Sirovich ed., Appl. Math. Sciences 100, Springer-Verlag, 1994
[4] P. Constantin P., Tatra Mountain Math. Publ. 4 pp 25– (1994)
[5] P. Constantin P., Indiana Univ. Math.Journal 42 pp 775– (1993) · Zbl 0837.35113
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[8] Kerr R., Phys. Fluids A 5 pp 1725– (1993) · Zbl 0800.76083
[9] Majda A., SIAM Rev. pp 349– (1991) · Zbl 0850.76278
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