Hydrodynamics at the smallest scales: A solvability criterion for Navier-Stokes equations in high dimensions

摘要

Strong global solvability is difficult to prove for high-dimensional hydrodynamic systems because of the complex interplay between nonlinearity and scale invariance. We define the Ladyzhenskaya-Lions exponent α1(n)=(2 + n)/4 for Navier-Stokes equations with dissipation -(-Δ)a in Rn, for all n = 2. We review the proof of strong global solvability when α ≥ α1(n), given smooth initial data. If the corresponding Euler equations for n ≥ 2 were to allow uncontrolled growth of the enstrophy (1/2)||Vu||2 L2, then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier-Stokes equations for α < α1(n). The energy is critical under scale transformations only for α=α1(n).