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

G. M. Viswanathan; T. M. Viswanathan

doi:doi:10.1209/0295-5075/84/50006

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. Viswanathan^{1, 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

Gandhi.Viswanathan@pq.cnpq.br
viswanathan.tenkasi@gmail.com

received 2 August 2008; accepted in final form 27 October 2008; published December 2008
published online 9 December 2008

Abstract
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 $\alpha =\alpha _{c}$ .

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

© EPLA 2008