Local in time regularity properties of the Navier-Stokes equations

Let u be a weak solution of the Navier-Stokes equations in a smooth domain Ω ⊆ ℝ and a time interval [0, T), 0 < T < ∞, with initial value u₀, and vanishing external force. As is well known, global regularity of u for general u₀ is an unsolved problem unless we pose additional assumptions on u₀ or on the solution u itself such as Serrin's condition ||u||L^s(0,T;L^q(Ω)) < ∞ where 2/s + 3/q = 1. In the present paper we prove several new local and global regularity properties by using assumptions beyond Serrin's condition e.g. as follows: If Ω is bounded and the norm ||u||L¹(0, T;L^q(Ω)), with Serrin's number 2/1 + 3/q strictly larger than 1, is sufficiently small, then u is regular in (0, T). Further local regularity conditions for general smooth domains are based on energy quantities such as ||u||L^∞(T₀,T₁L²(Ω)) |w||i»(r0,Ti;Z.2(i))) and || ▽ u|| ||u||L²(T ₀,T₁L²(Ω)). Indiana University Mathematics Journal