ZAMM · Z. Angew. Math. Mech. 84, No. 2, 124 – 127 (2004) / DOI 10.1002/zamm.200310096 Short Communication A note on coupling of velocity components in the Navier-Stokes equations Zdenˇ ek Skal ´ ak ∗ and Petr Kuˇ cera Department of Mathematics, Faculty of Civil Engineering, Czech Technical University, Th´ akurova 7, 166 29 Prague 6, Czech Republic Received 2 June 2000, accepted 13 January 2003 Published online 9 January 2004 Key words Navier-Stokes equations, suitable weak solution, regularity, Hausdorff measure MSC (2000) 35Q30 We show that if the third component v3 of the velocity v in the Navier-Stokes equations belongs to L 4 (0,T,W 1,3 (D)), where D is a subdomain of Ω, then v has no singular points in D × (0,T ). c  2004 WILEY-VCH Verlag GmbH & Co. KGaA,Weinheim Introduction Let Ω be a bounded domain in R 3 with smooth boundary ∂Ω, let T> 0 and Q T = Ω × (0,T ). Consider the Navier-Stokes initial-boundary value problem describing the evolution of the velocity v(x,t) and the pressure p(x,t) in Q T : ∂ v ∂t - ν Δv + v ·∇v + ∇p = f , (1) div v =0, (2) v = 0 on ∂Ω × (0,T ), (3) v| t=0 = v 0 , (4) where ν> 0 is the viscosity coefficient. The initial data v 0 satisfy the compatibility conditions v 0 | ∂Ω = 0 and div v 0 =0. Assume, for simplicity, that f = 0. It has been known for a long time that the problem (1)–(4) has a weak solution (for the definition of the weak solution and the proof of its existence see Temam [9]). However, the uniqueness and the regularity of the weak solution is still an open problem. We say that a weak solution v to (1)–(4) is regular in a time interval (t 1 ,t 2 ) ⊆ (0,T ), t 1 <t 2 , if v ∈ L ∞ (t 1 ,t 2 ,W 1,2 (Ω) 3 ). L. Caffarelli, R. Kohn, and L. Nirenberg introduced and thoroughly discussed the concept of a suitable weak solution (see [1]). They proved that the 1-dimensional Hausdorff measure of the set S(v) ⊆ Q T of all interior singular points of a suitable weak solution is equal to zero. A point (x 0 ,t 0 ) ∈ Ω × (0,T ) is called a regular point of the weak solution v if v is essentially bounded in a space-time neighbourhood U of (x 0 ,t 0 ), that is if v ∈ L ∞ (U ). A point (x 0 ,t 0 ) ∈ Ω × (0,T ) is called singular if it is not regular. Neustupa and Penel studied the question whether the components of velocity are coupled in such a way that information about a higher regularity of one of them already implies the higher regularity of all of them. They showed in [6] that in the case of a suitable weak solution the interior essential boundedness of one of the velocity components implies the interior regularity of all the components. They improved this result later in [5] together with Novotn´ y and proved that if v is a suitable weak solution and its component v 3 belongs to L r,s (D) 1 with 2/r +3/s ≤ 1/2, r ∈ [4, ∞],s ∈ (6, ∞], and D being a subdomain of Q T then all the components of v are regular in D (that is regular in all the points of D). Similar problems were also studied in other papers (see [2] or [10]) where sufficient conditions for regularity of v were obtained using the vorticity rather than the velocity. The concept of the suitable weak solution was used in a nice paper by Neustupa (see [4]). He proved that a suitable weak solution from the space L ∞ (0,T,L 3 (Ω) 3 ) can have at most a finite number of interior singular points at every instant of time. ∗ Corresponding author, e-mail: skalak@mat.fsv.cvut.cz 1 If D ⊂ R 3 × R then f ∈ L r,s (D) means that f is a measurable function on D and  ∞ -∞ ( R 3 |f (x, t)| s dx ) r/s dt < ∞, where f is defined as zero outside D. c  2004 WILEY-VCH Verlag GmbH & Co. KGaA,Weinheim