On regularity estimates for axially symmetric Navier-Stokes equations in a cylinder and the critical-wedge occurrence problem

2025/11/07 by Wiesław J. Grygierzec, Wiesław Grygierzec, Wojciech M. Zajączkowski +2
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations #math.AP #msc:35A01 #msc:35B01 #msc:35B65 #msc:35Q30 #msc:76D03 #msc:76D05

paper · pdf · doi:10.48550/arxiv.2511.05098

52 pages. Substantially revised version. The unconditional global regularity claim has been replaced by a conditional estimate involving the critical-wedge residual. Title, abstract, main results, proofs, and conclusions were revised. arXiv admin note: substantial text overlap with arXiv:2507.14964, arXiv:2405.16670, arXiv:2501.18302

openalex publication_date 2025/11/07 · openalex created_date 2025/11/11 · arxiv created 2026/07/25 · arxiv updated 2026/07/30 · openalex updated_date 2026/07/31

Abstract

We consider the axisymmetric Navier-Stokes equations in a finite cylinder Ω⊂ℝ³ with slip-type boundary conditions. Our aim is to estimate X_s(t):=‖ω_r/r‖_V(Ω^t)+‖ω_φ/r‖_V(Ω^t). The closure mechanism depends on the relation between the L^s and L^∞ norms of the angular component v_φ. For fixed A,c₀>0, we identify the critical wedge W_A,c0:=\t∈(0,T):‖v_φ(t)‖_L^s(Ω)>A, ‖v_φ(t)‖_L^s(Ω)/‖v_φ(t)‖_L^∞(Ω)

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