Besov Space Regularity Conditions for Weak Solutions of the Navier–Stokes Equations

Consider a bounded domain Ω ⊆ R 3 with smooth boundary, some initial value u 0 ∈ L σ 2 ( Ω ) , and a weak solution u of the Navier–Stokes system in [ 0 , T ) × Ω , 0 < T ≤ ∞ . Our aim is to develop regularity and uniqueness conditions for u which are based on the Besov space B q , s ( Ω ) : = v ∈ L σ 2 ( Ω ) ; ‖ v ‖ B q , s ( Ω ) : = ∫ 0 ∞ e - τ A v q s d τ 1 / s < ∞ with 2 < s < ∞ , 3 < q < ∞ , 2 s + 3 q = 1 ; here A denotes the Stokes operator. This space, introduced by Farwig et al. (Ann. Univ. Ferrara 55:89–110, 2009 and J. Math. Fluid Mech. 14: 529–540, 2012 ), is a subspace of the well known Besov space B q , s - 2 / s ( Ω ) , see Amann (Nonhomogeneous Navier–Stokes Equations with Integrable Low-Regularity Data. Int. Math. Ser. pp. 1–28. Kluwer/Plenum, New York, 2002 ). Our main results on the regularity of u exploits a variant of the space B q , s ( Ω ) in which the integral in time has to be considered only on finite intervals (0, δ ) with δ → 0 . Further we discuss several criteria for uniqueness and local right-hand regularity, in particular, if u satisfies Serrin’s limit condition u ∈ L loc ∞ ( [ 0 , T ) ; L σ 3 ( Ω ) ) . Finally, we obtain a large class of regular weak solutions u defined by a smallness condition ‖ u 0 ‖ B q , s ( Ω ) ≤ K with some constant K = K ( Ω , q ) > 0 .