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

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Abstract

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 $\alpha>\alpha_c$.