A Solvability criterion for Navier-Stokes equations in high dimensions

Abstract

We define the Ladyzhenskaya-Lions exponent $α_{\rm {\tiny \sc l}} (n)=({2+n})/4$ for Navier-Stokes equations with dissipation $-(-\Delta {)}^{\alpha }$ in ${\mathbb{R}}^n$, for all $n\ge 2$. 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 $\frac{1}{2}\Vert \nabla u{\Vert }_{L^2}^2$, 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 $α=α_{\rm {\tiny \sc l}} (n)$.

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References in corpus (1)

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