A Solvability criterion for Navier-Stokes equations in high dimensions

TL;DR

This paper establishes a critical exponent for the dissipation term in Navier-Stokes equations in high dimensions, determining when global solutions are guaranteed based on the level of dissipation.

Contribution

It introduces the Ladyzhenskaya-Lions exponent as a solvability criterion for high-dimensional Navier-Stokes equations with fractional dissipation.

Findings

Global solvability when dissipation exponent exceeds the critical value

Identification of the critical energy scale under scale transformations

Highlighting the gap in controlling solutions for lower dissipation exponents

Abstract

We define the Ladyzhenskaya-Lions exponent $\alpha_{\rm {\tiny \sc l}} (n)=({2+n})/4$ for Navier-Stokes equations with dissipation $-(-\Delta)^{\alpha}$ in ${\Bbb R}^n$, for all $n\geq 2$. We review the proof of strong global solvability when $\alpha\geq \alpha_{\rm {\tiny \sc l}} (n)$, 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 $\alpha<\alpha_{\rm {\tiny \sc l}} (n)$. The energy is critical under scale transformations only for $\alpha=\alpha_{\rm {\tiny \sc l}} (n)$.