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

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 $(Ω=\mathbb{R}^3)$ $$ |ω_{r/r} |{V(Ω^t)} + |ω{φ/r}|{V(Ω^t)}\leϕ({\rm data}),\ \ t<\infty, \qquad() $$ where $ω_r$ is the radiar component of vorticity, $ω_φ$ 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.

💡 Research Summary

This paper addresses the fundamental question of global regularity for the Cauchy problem of the axially-symmetric Navier-Stokes equations in three-dimensional whole space. The central result is the derivation of a global a priori estimate for sufficiently regular solutions, which, combined with the existence of local-in-time solutions, implies the existence of global smooth solutions.

The analysis is strategically divided by exploiting the geometry of the problem. The authors separately examine the region near the axis of symmetry (the “inner region”) and the region far from it (the “outer region”). This is achieved using a partition of unity with cutoff functions ζ(r) (localizing near the axis) and ϑ(r) (localizing away from it). The core difficulty lies in controlling the behavior near the axis (r=0), where potential singularities for axisymmetric flows are known to concentrate according to the partial regularity theory of Caffarelli, Kohn, and Nirenberg.

The proof relies on a sophisticated combination of estimates for key quantities defined in cylindrical coordinates. The primary variables are the normalized vorticity components Φ = ω_r / r and Γ = ω_φ / r. Their combined energy norm, denoted X(t), is shown to control the solution’s regularity. To estimate X(t), several crucial ingredients are developed for the inner region:

1. Energy estimates for the localized variables ˜Φ and ˜Γ.
2. Maximum principle and energy estimates for the swirl u = r v_φ, yielding bounds for u, u,_z, and u,_r.
3. Elliptic H² and H³ estimates for the modified stream function ψ₁ = ψ / r, which solves the equation -Δψ₁ - (2/r) ∂_r ψ₁ = Γ. These estimates bound ψ₁ in terms of Γ and its derivative.
4. The Liu-Wang expansions: The analysis fundamentally assumes that near the axis, the velocity components and the stream function admit power series expansions in r (e.g., v_φ = b₁(z,t) r + b₂(z,t) r³ + …). These expansions are essential for justifying the boundary conditions and the validity of the elliptic estimates at r=0.

The most delicate part of the proof is handling the critical nonlinear term I = ∫ (v_φ / r) ˜Φ ˜Γ dxdt that appears in the energy inequality for ˜Φ and ˜Γ. The authors introduce an auxiliary quantity L(s) = ||v_φ||{L∞(L^s)} / ||v_φ||{L∞(L∞)} and perform a case-by-case analysis based on its size relative to a threshold c₀(s).

• Regime 1: L(s) ≥ c₀(s): Here, a bound for ||v_φ||{L∞(L^s)} in terms of X(t) is established. Using this and a crucial “local reduction estimate” for ||˜Φ||{L²}, the term I is controlled, leading to a bound for X(t).
• Regime 2: L(s) ≤ c₀(s): In this complementary case, it is assumed that ||v_φ||{L∞(L^s)} is bounded by some constant A. Under this assumption and the condition of “non-small” swirl (||v_φ||{L∞} ≥ 1/c₁), the term I is estimated differently, again yielding a bound for X(t).

Combining the results from both regimes, the main a priori estimate is proven: X(t) ≤ φ(data) for any finite time t, where φ is an increasing function of the initial data and forcing. This estimate, Theorem 1.1, is derived under the assumption that the solution is regular enough for the Liu-Wang expansions to hold.

Finally, in Theorem 1.2, the authors bootstrap this initial regularity. Using the bound on X(t) as a starting point, they iteratively apply the classical solvability theory for the Stokes system in Sobolev spaces with mixed norms. This process successively increases the regularity of the velocity and pressure, ultimately proving that v belongs to W₃^{3, 3/2} and ∇p belongs to W₃^{1, 1/2}, with corresponding norm estimates. This high level of regularity justifies the initial assumption that the Liu-Wang expansions are valid, thereby completing the logical circle and establishing the existence of global regular solutions.

The paper makes a significant contribution by providing a global a priori estimate under a clear, albeit strong, regularity framework. However, it also explicitly notes a key open problem: whether the fundamental estimate for X(t) can be proven independently, without first assuming the high regularity (and hence the Liu-Wang expansions) of the solution.