title: 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 α(L)(n)=(2+n)/4 for Navier-Stokes equations with dissipation -(-Δ)(α) in R(n), for all n≥2. We review the proof of strong global solvability when α≥α(L)(n), given smooth initial data. If the corresponding Euler equations for n>2 were to allow uncontrolled growth of the enstrophy (1/2)∥∇u∥(L²)(2), then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier-Stokes equations for α<α(L)(n). The energy is critical under scale transformations only for α=α(L)(n).
authors.affiliation: Universidade Federal de Alagoas, Maceió-AL, CEP 57072-970, Brazil.
authors.forename: T M
authors.initials: TM
authors.lastname: Viswanathan
authors.forename: G M
authors.initials: GM
authors.lastname: Viswanathan
mesh_terms: D001703:Biophysics / Q000379:methods*; D002621:Chemistry / Q000379:methods*; D003198:Computer Simulation; D005583:Fourier Analysis; D057446:Hydrodynamics; D007700:Kinetics; D012212:Rheology / Q000379:methods*; D013997:Time Factors