PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 137, Number 10, October 2009, Pages 3343–3353 S 0002-9939(09)09989-4 Article electronically published on May 29, 2009 SOME ELEMENTARY ESTIMATES FOR THE NAVIER-STOKES SYSTEM JEAN CORTISSOZ (Communicated by Matthew J. Gursky) Abstract. In this paper we study the incompressible Navier-Stokes equations in 3 = [0, 1] 3 with periodic boundary conditions. We show that a weak solution of the Navier-Stokes equations that is small in ∞ (0, ;Φ(2)) is also smooth. We also give elementary proofs of some classical regularity results for the Navier-Stokes equations involving the Sobolev space 1 2 ( 3 ). 1. Introduction Oneoftheoutstandingproblemsinmathematicsistheexistenceofglobalregular solutions to the Navier-Stokes system (1) { − Δ + ⋅∇ + ∇ =0 in 3 × (0, ∞) , (, 0)= , div =0, where 3 =[0, 1] 3 , with periodic boundary conditions. Severalshorttimeexistenceresultsandsmall-initialdata-globalexistenceresults have been shown for different Banach spaces, and the literature on the subject is extensive. Given a function (,) ∈ 2 ( 3 ) we write its Fourier expansion as ∑ k k ()exp(2 ⟨, k⟩) , k =( 1 , 2 , 3 ) ∈ ℤ 3 , and we define the spaces Φ() ⊂ ′ ( ′ is the dual of the set of ∞ periodic functions on 3 ) as follows: Φ()= { : ∣ k ∣≤ ∣k∣ , k ∕=0, 0 =0 } endowed with the norm ∥ ∥ = sup k∈ℤ 3 ∖{0} ∣k∣ ∣ k ∣ , which makes them Banach spaces. Notice that if > 3 2 ,Φ() ⊂ 2 ( 3 ) . These spaces are considered in [1], where the following theorem with =2+ , > 0, is proved. Received by the editors October 14, 2008. 2000 Mathematics Subject Classification. Primary 35Q30. Key words and phrases. Navier-Stokes equations, regularity. c ⃝2009 American Mathematical Society Reverts to public domain 28 years from publication 3343 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use