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ArticlesHydrodynamics at the smallest scales: a solvability criterion for Navier–Stokes equations in high dimensions
Published:28 January 2011https://doi.org/10.1098/rsta.2010.0257
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 αl(n)=(2+n)/4 for Navier–Stokes equations with dissipation −(−Δ)α in 
, for all n≥2. We review the proof of strong global solvability when α≥αl(n), given smooth initial data. If the corresponding Euler equations for n>2 were to allow uncontrolled growth of the enstrophy 
, then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier–Stokes equations for α<αl(n). The energy is critical under scale transformations only for α=αl(n).
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