Hydrodynamics at the smallest scales: a solvability criterion for Navier–Stokes equations in high dimensions

Abstract

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 a L (n) = (2 + n)/4 for Navier-Stokes equations with dissipation −(−D) a in R n , for all n ≥ 2. We review the proof of strong global solvability when a ≥ a L (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 L 2 , then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier-Stokes equations for a < a L (n). The energy is critical under scale transformations only for a = a L (n).

Nonlinear dynamics in meso and nano scales: fundamental aspects and applications

“…In two works of this issue, it is shown that the characteristic sizes in which nonlinearities may arise in fluid and plasma physics can be in the nano and meso scales. In the son-father collaboration, Viswanathan & Viswanathan [78] tackle the extremely hard problem of stability and solvability of the Navier-Stokes equation. They argue that the difficulty found in settling the question has its roots in the interplay between the nonlinearity and the scale-invariant aspects (which can cascade to very small scales) in hydrodynamics.…”