Communications in Mathematical Sciences

Volume 10 (2012)

Number 1

Special Issue on the Occasion of C. David Levermore’s Sixtieth Birthday

A hierarchy of length scales for weak solutions of the three-dimensional Navier-Stokes equations

Pages: 131 – 136



J. D. Gibbon (Department of Mathematics, Imperial College London, United Kingdom)


Moments of the vorticity are used to define and estimate a hierarchy of time-averaged inverse length scales for weak solutions of the three-dimensional, incompressible Navier-Stokes equations on a periodic box. The estimate for the smallest of these inverse scales coincides with the inverse Kolmogorov length, but thereafter the exponents of the Reynolds number rise rapidly for correspondingly higher moments. The implications of these results for the computational resolution of small scale vortical structures are discussed.


Navier-Stokes, weak solutions, moments of vorticity, inverse length scales

2010 Mathematics Subject Classification

35Q30, 76D05

Published 14 October 2011