Volume 84, Number 5, December 2008
Article Number 50006
Number of page(s) 6
Section General
Published online 09 December 2008
EPL, 84 (2008) 50006
DOI: 10.1209/0295-5075/84/50006

Spontaneous symmetry breaking and finite-time singularities in d-dimensional incompressible flows with fractional dissipation

G. 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 $\geqslant$ 2 dimensions with a fractional Laplacian $\vert\nabla \vert^{\alpha}$. 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 $\alpha \geqslant \alpha _{c}=1+d/2$. Moreover, initial singularities become unstable for $\alpha > \alpha _{c}$. The scale invariance symmetry intrinsic to the Navier-Stokes system becomes spontaneously broken, except at the critical point $\alpha =\alpha _{c}$.

05.40.Fb - Random walks and Levy flights. - Navier-Stokes equations.
05.70.Jk - Critical point phenomena.

© EPLA 2008