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

Abstract

Higher moments of the vorticity field Ω m ( t ) in the form of L 2 m -norms ( ) are used to explore the regularity problem for solutions of the three-dimensional incompressible Navier–Stokes equations on the domain . It is found that the set of quantities 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)≤ ϖ 0 O ( Gr 4 )), it is found that there is an upper bound 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.