Regularity criterion to the axially symmetric Navier–Stokes equations

Abstract

Smooth solutions to the axially symmetric Navier-Stokes equations obey the following maximum principle:is small compared with certain dimensionless quantity of the initial data. This result improves the one in Zhen Lei and Qi S. Zhang [10]. As a corollary, we also prove the global regularity under the assumption that |ru θ (r, z, t)| ≤ | ln r| −3/2 , ∀ 0 < r ≤ δ 0 ∈ (0, 1/2).

On regularity criterion to the 3D axisymmetric incompressible MHD equations

“…Therefore, this result shows that if an axisymmetric solution develops a singularity, it can only be a singularity of an other type. For recent generations to this result, see [15][16][17][18] and references therein.…”

Convergence of Parareal Algorithms for PDEs with Fractional Laplacian and a Non-Constant Coefficient

“…Moreover, P. Zhang and T. Zhang [30] investigated the global well-posedness if the swirl component u θ 0 of the initial velocity field is small, Chen et al [5] proved that the axisymmetric solution is global if u θ 0 ∈ L 3 ( 3 ) is sufficiently small. More related results can be found in [21,29]. Stochastic three-dimensional Navier-Stokes equations also attracted substantial attention recently.…”

“…In deterministic case, the well-posedness has been studied under a smallness condition for ru θ L ∞ -cf. [21,29]. The proof is based on the maximum principle…”

Lecture Notes on Regularity Theory for the Navier-Stokes Equations

“…Regarding other regularity results on axially symmetric solutions to the Navier-Stokes equations, we refer to papers [2,3,4,5,9,10,11,12,15,16,17,18,19,20,22,23,24].…”