Interior Regularity Criteria for Suitable Weak Solutions of the Navier-Stokes Equations

Abstract

We present new interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations: a suitable weak solution is regular near an interior point z if either the scaled L p,q x,t -norm of the velocity with 3/p+2/q ≤ 2, 1 ≤ q ≤ ∞, or the L p,q x,t -norm of the vorticity with 3/p+2/q ≤ 3, 1 ≤ q < ∞, or the L p,q x,t -norm of the gradient of the vorticity with 3/p+2/q ≤ 4, 1 ≤ q, 1 ≤ p, is sufficiently small near z.

Interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations

“…The proofs are the same as those in [7] (see [7] from Lemma 3.6 to the end). Since arguments are just tedious repetitions, we omit the details.…”

On the boundary regularity of suitable weak solutions to the Navier-Stokes equations

“…also [18] [14]. For further results concerning the partial regularity of suitable weak solutions of the Navier-Stokes equations see [10,25,27,28]. Concerning Serrin's condition for the local regularity we refer to [22,26].…”

Some criteria concerning the vorticity and the problem of global regularity for the 3D Navier–Stokes equations

“…Conditions concerning the (scaled) growth of this term in order to have local regularity, have been treated by Gustafson et al [35], in the spirit of partial regularity results à la Caffarelli, Kohn, and Nirenberg. We observe now that the scale-invariance argument implies that regularity should be obtained if …”