Spontaneous symmetry breaking and finite-time singularities in d-dimensional incompressible flows with fractional dissipationG. M. Viswanathan1, 2 and T. M. Viswanathan3
1 Consortium of the Americas for Interdisciplinary Science, University of New Mexico 800 Yale Blvd. NE, Albuquerque, NM 87131, USA
2 Instituto de Física, Universidade Federal de Alagoas - Maceió - AL, CEP 57072-970, Brazil
3 Universidade Federal de Alagoas - Maceió - AL, CEP 57072-970, Brazil
received 2 August 2008; accepted in final form 27 October 2008; published December 2008
published online 9 December 2008
We investigate the formation of singularities in incompressible flows governed by Navier-Stokes equations in d 2 dimensions with a fractional Laplacian . We derive analytically a sufficient but not necessary condition for the solutions to remain always smooth and show that finite-time singularities cannot form for . Moreover, initial singularities become unstable for . The scale invariance symmetry intrinsic to the Navier-Stokes system becomes spontaneously broken, except at the critical point .
05.40.Fb - Random walks and Levy flights.
47.10.ad - Navier-Stokes equations.
05.70.Jk - Critical point phenomena.
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