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

Technical Summary: Global Regular Solutions to the Axisymmetric Navier-Stokes Equations

Problem Statement
This paper addresses the Cauchy problem for the axisymmetric Navier-Stokes equations in $\mathbb{R}^3$. The primary objective is to prove the existence of global regular solutions for arbitrary initial data, provided the solutions possess sufficient regularity to satisfy specific expansions near the axis of symmetry. The authors focus on deriving a global a priori estimate for regular solutions, which, when combined with the existence of local solutions, implies global existence.

The equations are formulated in cylindrical coordinates $(r, \phi, z)$, where the velocity field $v$ and vorticity $\omega$ are decomposed into radial, angular, and axial components. A critical feature of the axisymmetric case is the behavior of the "swirl" component $v_\phi$ and the associated vorticity terms near the symmetry axis ($r=0$), where singularities in the equations could potentially arise.

Methodology
The authors employ a strategy of localizing the analysis into two distinct regions: a neighborhood of the axis of symmetry and the region far from the axis. This separation is achieved using a partition of unity $\{\zeta(r), \vartheta(r)\}$, where $\zeta$ is supported near $r=0$ and $\vartheta$ is supported away from the axis.

1. Modified Stream Function and Vorticity Formulation:
   The analysis utilizes the modified stream function $\psi_1 = \psi/r$ and the normalized vorticity components $\Phi = \omega_r/r$ and $\Gamma = \omega_\phi/r$. The authors derive elliptic estimates for $\psi_1$ in $H^2$ and $H^3$ spaces, which are crucial for controlling the velocity gradients near the axis.

2. Liu-Wang Expansions:
   A central methodological requirement is the assumption that the solutions satisfy the expansions derived by Liu and Wang near the axis of symmetry. Specifically, the radial and angular velocity components and the stream function are assumed to behave as:
   $$v_r \sim a_1(z,t)r + a_2(z,t)r^3 + \dots$$
   $$v_\phi \sim b_1(z,t)r + b_2(z,t)r^3 + \dots$$
   $$\psi \sim d_1(z,t)r + d_2(z,t)r^3 + \dots$$
   These expansions ensure that the terms involving division by $r$ remain bounded and regular.

3. Energy Estimates and Swirl Control:
   The authors derive energy-type estimates for the quantities $\Phi$ and $\Gamma$. A significant portion of the work involves estimating the nonlinear coupling term $I = \int (v_\phi/r) \tilde{\Phi} \tilde{\Gamma} \, dx dt'$. To handle this term without restrictive assumptions on the magnitude of the swirl, the authors split the analysis into two complementary regimes based on the ratio $L(s) = \|\tilde{v}'_\phi\|_{s, \infty} / \|\tilde{v}'_\phi\|_{\infty}$:

   • Regime 1 ($L(s) \geq c_0$): Corresponds to a non-concentrated swirl profile. Estimates are derived using interpolation inequalities and the Liu-Wang expansions.
   • Regime 2 ($L(s) \leq c_0$): Corresponds to a concentrated swirl profile. Estimates are derived using $L^s$ bounds and interpolation.
     By covering both regimes, the authors avoid the restrictive condition previously required in related works (e.g., requiring a specific lower bound on the swirl magnitude).

4. Regularity Bootstrapping:
   Once the global a priori estimate for the energy norms of $\Phi$ and $\Gamma$ is established, the authors use the solvability theory for the non-stationary Stokes system in Sobolev spaces with mixed norms (referencing [MS]) to iteratively increase the regularity of the velocity $v$ and pressure gradient $\nabla p$. This process lifts the solution from the initial regularity to the class $W^{3, 3/2}_3$.

Key Contributions and Results

• Global A Priori Estimate: The main result is Theorem 1.1, which establishes the global a priori estimate:
  $$\|\omega_r/r\|_{V(\Omega_t)} + \|\omega_\phi/r\|_{V(\Omega_t)} \leq \phi(\text{data})$$
  where $V(\Omega_t)$ is the energy norm, and $\phi$ is an increasing positive function of the initial data and external forces. This estimate holds for any finite time $t < \infty$.

• Removal of Restrictive Conditions: Unlike previous works (e.g., [Z1, Z2, OZ]) that required specific conditions on the swirl magnitude or vanishing conditions at the axis, this paper derives the global estimate without assuming conditions (∗) or (∗∗) from the introduction. This is achieved by treating the near-axis and far-axis regions independently and utilizing the two-regime analysis for the swirl term.

• Existence of Global Regular Solutions: Theorem 1.2 demonstrates that if the initial data and external force possess sufficient regularity ($v(0) \in W^{3-2/3}_3$ and $f \in W^{1, 1/2}_3$), and if a local solution exists, the global a priori estimate implies the existence of a global regular solution $v \in W^{3, 3/2}_3(\mathbb{R}^3_t)$ with $\nabla p \in W^{1, 1/2}_3(\mathbb{R}^3_t)$.

• Small Data Case: Section 8 provides a complementary result for small initial data, proving existence via successive approximations. The authors note that the small data regime corresponds to a small swirl, which is the complement to the large swirl regime analyzed in the main body of the paper.

Significance and Claims
The paper claims to provide a rigorous proof for the existence of global regular axisymmetric solutions to the Navier-Stokes equations under the assumption that the solutions are sufficiently regular to satisfy the Liu-Wang expansions near the axis. The authors emphasize that these expansions are "absolutely necessary" for their proofs.

The significance of the work lies in:

1. Completing the Global Estimate: Providing a global a priori bound that does not rely on the restrictive swirl conditions found in earlier literature.
2. Technical Novelty: The use of a partition of unity to separate the singular axis from the rest of the domain, allowing for independent estimation techniques in each region.
3. Regularity Propagation: Demonstrating how the boundedness of the energy norms of the normalized vorticity components can be used to bootstrap the solution to higher Sobolev regularity.

The authors modestly acknowledge that the assumption of sufficient regularity (specifically the validity of the Liu-Wang expansions) is a prerequisite for their main theorem. They cite the partial regularity theory of Caffarelli, Kohn, and Nirenberg, noting that any potential singularities for axisymmetric solutions must occur on the axis. Thus, their results apply to solutions that do not develop such singularities. Whether the global bound can be derived without assuming these expansions remains an open problem.