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

Abstract

We investigate the formation of singularities in the incompressible Navier-Stokes equations in $d\ge 2$ 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 $\alpha \ge {\alpha }_c=1+d/2$. Moreover, initial singularities become unstable for $\alpha >{\alpha }_c$.

5 pages

References in corpus (3)

• Fractional Laplacian in Bounded Domains
• Amnestically induced persistence in random walks
• Turbulence and the Navier-Stokes equations

Cited by in corpus (1)

• A Solvability criterion for Navier-Stokes equations in high dimensions