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J. Math. Fluid Mech. (2026) 28:50c ? 2026 The Author(s)1422-6928/26/030001-33https://doi.org/10.1007/s00021-026-01022-9Journal of MathematicalFluid MechanicsA Regularity Criterion for the Angular Component of Velocity in the normL q (0,T;L p (Ω)),3p+2q< 1, q < ∞, in Axisymmetric Navier-Stokes Equations in aCylinderWies? law J. Grygierzec and Wojciech M. Zaj¸ aczkowskiCommunicated by G. P. GaldiAbstract. We consider the axisymmetric Navier-Stokes equations in a f i nite cylinder Ω ⊂ R 3 . We assume that v r , v ? , ω ?vanish on the lateral part of boundary ∂Ω of the cylinder, and that v z , ω ? , ∂ z v ? vanish on the top and bottom parts ofthe boundary ∂Ω, where we used standard cylindrical coordinates, and we denoted by ω = curlv the vorticity f i eld. We useH 3 Sobolev estimates for the modif i ed stream function (stream function divided by radius) and energy type estimates forgradient of swirl to derive two order reduction estimates. Finally, using the estimate?v ? ? L q (0,T;L p (Ω)) ≤ A,where A is a given number and3p+2q< 1, q < ∞ we prove the existence of global regular axially-symmetric solutions.Mathematics Subject Classif i cation. 35A01, 35B01, 35B65, 35Q30, 76D03, 76D05.Keywords. Navier-Stokes equations, Axially-symmetric solutions, Cylindrical domain.1. IntroductionWe are concerned with the 3D incompressible Navier-Stokes equations,∂ t v − νΔv + v · ∇v + ∇p = f,divv = 0 in Ω T ,(1.1)under the axisymmetry constraint, where Ω T := Ω × (0,T), T > 0, v = v(x,t) ∈ R 3 denotes the velocityf i eld, p = p(x,t) ∈ R denotes the pressure function, f = f(x,t) ∈ R 3 denotes the external force f i eld,ν > 0 denotes the viscosity, and x = (x 1 ,x 2 ,x 3 ) denotes the Cartesian coordinates. As for Ω we focus onthe case of a f i nite cylinder,Ω = {x ∈ R 3 : x 21 + x22 < R2 ,|x 3 | < a},where a, R > 0 are constants. We note thatS := ∂Ω = S 1 ∪ S 2 ,whereS 1 = {x ∈ R 3 :?x 21 + x 2 2 = R, x 3 ∈ [−a,a]},S 2 = {x ∈ R 3 :?x 21 + x 2 2 < R, x 3 ∈ {−a,a}}denote the lateral boundary and the top and bottom parts of the boundary, respectively.In order to state the boundary conditions stating our main result we use the cylindrical coordinatesr, ?, z def i ned byx 1 = rcos?, x 2 = rsin?, x 3 = z,