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

Details Spontaneous symmetry breaking and finite time singularities in $d$-dimensional incompressible flow with fractional dissipation Spontaneous symmetry breaking and finite time singularities in $d$-dimensional incompressible flow with fractional dissipation

2008-07-10

arxiv.org/abs/0807.1563v1

Physics

Fluid Dynamics

5 pages

Scientific paper

10.1209/0295-5075/84/50006

We investigate the formation of singularities in the incompressible Navier-Stokes equations in $d\geq 2$ dimensions with a fractional Laplacian $|\nabla |^\alpha$. We derive analytically a sufficient but not necessary condition for solutions to remain always smooth and show that finite time singularities cannot form for $\alpha\geq \alpha_c= 1+d/2$. Moreover, initial singularities become unstable for $\alpha>\alpha_c$.

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