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A sufficient condition for a finite-time $L_2 $ singularity of the 3d Euler Equations

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He, Xinyu. (2005) A sufficient condition for a finite-time $L_2 $ singularity of the 3d Euler Equations. Mathematical Proceedings of the Cambridge Philosophical Society, Vol.139 (No.3). pp. 555-561. ISSN 0305-0041

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Official URL: http://dx.doi.org/10.1017/S0305004105008777

Abstract

A sufficient condition is derived for a finite-time $L_2 $ singularity of the 3d incompressible Euler equations, making appropriate assumptions on eigenvalues of the Hessian of pressure. Under this condition $ \ \lim_{ t \uparrow T_*} \sup \|\frac{ D \omega} { Dt}\|_{L_2(\Omega)} = \infty $, where $\Omega \subset \mathbb{R}$ moves with the fluid. In particular, $| \omega | $, $| \S_{ij} | $, and $| \P_{ij} | $ all become unbounded at one point $(x_1, T_1) $, $T_1 $ being the first blow-up time in $L_2 $.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Fluid dynamics -- Mathematical models, Fluid mechanics, Eigenvectors
Journal or Publication Title:	Mathematical Proceedings of the Cambridge Philosophical Society
Publisher:	Cambridge University Press
ISSN:	0305-0041
Date:	November 2005
Volume:	Vol.139
Number:	No.3
Page Range:	pp. 555-561
Identification Number:	10.1017/S0305004105008777
Status:	Peer Reviewed
Access rights to Published version:	Open Access
References:	[1] J. T. Beale, T. Kato and A. J. Majda. Remarks on the breakdown of smooth solutions for the 3d Euler equations. Comm. Math. Phys. 94 (1984), 61–66. [2] P. Constantin. Geometric and analytical studies in turbulence. Appl. Math. Sci. 100 (1994). [3] P. Constantin, C. Fefferman and A. J. Majda. Geometric constraints on potentially singular solutions for the 3d Euler equations. Comm. Partial Differential Equations 21 (1996), 559–571. [4] J. D. Gibbon. A quaternionic structure in the 3d Euler and ideal MHD equations. Physica D 166 (2002), 17–28. [5] J. D. Gibbon, B. Galanti and R. M. Kerr. Stretching and compression of vorticity in the 3d Euler equations. In Turbulence Structure and Vortex Dynamics (ed. J. C. R. Hunt & J. C. Vassilicos), (Cambridge University Press, 2000). [6] X. He. An invariant for the 3d Euler equations. Applied Mathematics Letters 12 (1999), 55–58. [7] R. M. Kerr. Evidence for a singularity of the 3d, incompressible Euler equations. Phys. Fluids A 5 (1993), 1725–1746. [8] A. J. Majda. Vorticity and the mathematical theory of incompressible fluid flow. Comm. Pure Appl. Math. 39 (1986), S187–S220. [9] H. K. Moffatt. The interaction of skewed vortex pairs: a model for blow-up of the Navier–Stokes equations. J. Fluid Mech. 409 (2000), 51–68. [10] K. Ohkitani. Eigenvalue problems in 3d Euler flows. Phys. Fluids 5 (1993), 2570–2572. [11] R. B. Pelz. Symmetry and the hydrodynamic blow-up problem. J. Fluid Mech. 444 (2001), 299–320 (see also: Discrete groups, symmetric flows and hydrodynamic blowup, in Tubes, Sheets and Singularities in Fluid Dynamics (Eds. K. Bajer and H. K. Moffatt), IUTAM Symposium Series, (Kluwer, 2002). [12] G. Ponce. Remarks on a paper by J. T. Beale, T. Kato and A. J. Majda. Comm. Math. Phys. 98 (1985), 349–353.
URI:	http://wrap.warwick.ac.uk/id/eprint/737