Regularity and singularity in solutions of the three-dimensional Navier-Stokes equations

摘要

Higher moments of the vorticity field Ω_m(t) in the form of L~(2m)-norms (1 ≤m <∞) are used to explore the regularity problem for solutions of the three-dimensional incompressible Navier-Stokes equations on the domain [0, L]~3_(per). It is found that the set of quantities D_m(t)=Ω~(αm)_m, α_m = 2m/4m-3, provide a natural scaling in the problem resulting in a bounded set of time averages (D_m)_T on a finite interval of time [0, T]. The behaviour of D_(m+1)/D_m is studied on what are called 'good' and 'bad' intervals of [0, T], which are interspersed with junction points (neutral) τ_i. For large but finite values of m with large initial data (Ω_m(0) < ω?_0O(Gr~4)), it is found that there is an upper bound Ω_m ≤ c~2_(av)ω? _0Gr~4, ω?_0 = νL~(-2), which is punctured by infinitesimal gaps or windows in the vertical walls between the good/bad intervals through which solutions may escape. While this result is consistent with that of Leray (Leray 1934 Acta Math. 63, 193-248 (doi:10.1007/BF02547354)) and Scheffer (Scheffer 1976 Pacific J. Math. 66, 535-552), this estimate for Ω_m corresponds to a length scale well below the validity of the Navier-Stokes equations.