On the Cauchy problem to the axially-symmetric solutions to the Navier-Stokes equations

TL;DR

The paper addresses the global existence of regular axially symmetric solutions to the Navier–Stokes Cauchy problem on ${\\mathbb{R}}^3$. It develops a global a priori framework by splitting the domain into axis-near and axis-far regions, deriving energy-type, swirl, and high-regularity bounds for the modified stream function and vorticity, and employing Liu-Wang axis expansions near the symmetry axis. A partition of unity allows independent control in each region, which are then merged to yield a global regular solution and corresponding bounds $(*)$ and $(**)$ that ensure regularity persists in time. The work also establishes local-to-global extension for regular solutions and outlines a path to global regularity under small data, contributing a rigorous pathway to global axisymmetric Navier–Stokes regularity via detailed local estimates and axis-expansion techniques.

Abstract

We consider the Cauchy problem to the axisymmetric Navier-Stokes equations. To prove an existence of global regular solutions we examine the Navier-Stokes equations near the axis of symmetry and far from it separately. We derive only a global a priori estimate. To show it near the axis of symmetry we need the energy estimate, $L_\infty$-estimate for swirl, $H^2$ and $H^3$ estimates for the modified stream function (stream function divided by radius) and also expansions of velocity and modified stream function found by Liu-Wang. The estimate for solutions far from the axis of symmetry follows easily. Hence, having so regular solutions that Liu-Wang expansions hold we have the global a priori estimate $(Ω=\mathbb{R}^3)$ $$ \|ω_{r/r} \|_{V(Ω^t)} + \|ω_{\varphi/r}\|_{V(Ω^t)}\leφ({\rm data}),\ \ t<\infty, \qquad(*) $$ where $ω_r$ is the radiar component of vorticity, $ω_\varphi$ the angular, $V(Ω^t)$ is the energy norm. Estimate $(*)$ can be treated as an a priori estimate derived on sufficiently regular solutions. Increasing regularity of solutions $(*)$ we derive the estimate $$\eqalign{ &\|v\|_{W_3^{3,3/2}(Ω^t)}+\|\nabla p\|_{W_3^{1,1/2}(Ω^t)}\cr &\leφ(φ({\rm data}),\|f\|_{W_3^{1,1/2}(Ω^t)},\|v(0)\|_{W_3^{3-2/3}(Ω)}),\cr} \qquad(**) $$ where $φ$ is an increasing positive function. The estimate is proved on the local solution. Estimate $(**)$ plus existence of local solutions imply the existence of global regular solutions to the Cauchy problem.

Theorems & Definitions (68)

• Theorem 1.1
• proof : Proof of Theorem 1.1
• Theorem 1.2
• proof : Proof of Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Lemma 2.1
• proof
• Remark 2.2