T. M. Viswanathan | Searcharxiv

arXiv2009

A Solvability criterion for Navier-Stokes equations in high dimensions

We define the Ladyzhenskaya-Lions exponent $α_{\rm {\tiny \sc l}} (n)=({2+n})/4$ for Navier-Stokes equations with dissipation $-(-Δ)^α$ in ${\Bbb R}^n$, for all $n\geq 2$. We review the proof of strong global solvability when $α\geq α_{\rm {\tiny \sc l}} (n)$, given smooth initial data. If the corresponding Euler equations for $n>2$ were to allow uncontrolled growth of the enstrophy ${1\over 2} \|\nabla u \|^2_{L^2}$, then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier-Stokes equations for $α<α_{\rm {\tiny \sc l}} (n)$. The energy is critical under scale transformations only for $α=α_{\rm {\tiny \sc l}} (n)$.

math.AP↗

arXiv2008

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

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

physics.flu-dyn↗