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

Technical Summary: Regularity Estimates for Axially Symmetric Navier–Stokes Equations in a Cylinder

Problem Statement
This paper addresses the global well-posedness problem for the three-dimensional incompressible Navier–Stokes equations under the constraint of axial symmetry, specifically within a finite cylindrical domain $\Omega \subset \mathbb{R}^3$. The authors focus on the regularity of solutions where the velocity field $v$ and vorticity $\omega = \text{curl } v$ satisfy specific boundary conditions: on the lateral boundary $S_1$, the radial velocity $v_r$, azimuthal velocity $v_\phi$, and azimuthal vorticity $\omega_\phi$ vanish; on the top and bottom boundaries $S_2$, the axial velocity $v_z$, $\omega_\phi$, and the axial derivative $\partial_z v_\phi$ vanish.

The central difficulty in establishing regularity for axially symmetric solutions lies in controlling the nonlinear interaction term involving the swirl component $v_\phi$. Specifically, the authors analyze the quantities $\Phi = \omega_r/r$ and $\Gamma = \omega_\phi/r$, which are crucial for bounding the solution. The primary obstruction to closing the energy estimates arises from the integral term $\int_{\Omega_t} \frac{v_\phi}{r} \Phi \Gamma \, dx \, dt'$, whose control depends on the distribution of the angular velocity component $v_\phi$.

Methodology
The authors employ a conditional a priori estimation strategy that partitions the time interval based on the behavior of the $L^s$ and $L^\infty$ norms of the swirl component $v_\phi$.

1. Critical Wedge Identification: The authors define a "critical wedge" $W_{A,c_0}$ in the phase space of time $t$. This regime is characterized by times where the $L^s$ norm of $v_\phi$ is large ($> A$) but the ratio of the $L^s$ norm to the $L^\infty$ norm is small ($< c_0$). Outside this wedge, standard closure mechanisms apply. Inside the wedge, the standard estimates fail to close.
2. Residual Measurement: Instead of asserting unconditional global regularity, the paper introduces a "critical-wedge residual" $E_{W,s}(t)$. This term measures the non-closable portion of the nonlinear interaction specifically when the trajectory enters the critical wedge.
3. Analytical Tools:
   • Modified Stream Function: The analysis utilizes a modified stream function $\psi_1 = \psi/r$, satisfying an elliptic equation with source $\Gamma$. The authors derive $H^2$ and $H^3$ elliptic estimates for $\psi_1$.
   • Liu-Wang Expansions: To handle singularities near the axis of symmetry ($r=0$), the authors rely on the regularity expansions of $v_r, v_\phi, \psi$, and $\psi_1$ near the axis, as established by Liu and Wang. These expansions are essential for estimating boundary terms and integration by parts involving $r$.
   • Order Reduction Estimates: A key technical step involves deriving "order reduction" inequalities that bound higher-order norms (like $\|\Phi\|_{L^2}$) by lower-order norms and the data, effectively reducing the complexity of the nonlinear terms.
   • Sobolev and Besov Spaces: The proof of higher regularity (Theorem 1.4) relies on solvability results for the Stokes system in Sobolev spaces with mixed norms and trace theorems for Besov spaces.

Key Contributions and Results

• Conditional A Priori Estimate (Theorem 1.2): The main result is a conditional estimate for the quantity $X_s(t) = \|\Phi\|_{V(\Omega_t)} + \|\Gamma\|_{V(\Omega_t)}$. The authors prove that:
  $$X_s(t) \leq \Psi_{s,A,c_0}\left(\text{data}, \int_0^t E_{W,s}(\tau) \, d\tau\right)$$
  where $\Psi$ is an increasing positive function. This estimate recovers the original data-dependent bound if the critical-wedge residual vanishes (i.e., if the trajectory does not enter the critical wedge).
• Removal of Previous Restrictions: The paper demonstrates that the restriction $\psi_1|_{r=0} = 0$ (required in previous works by the second author) is not necessary for these estimates. However, the condition relating the $L^s$ and $L^\infty$ norms (previously assumed globally) is relaxed to a conditional requirement: the estimate holds provided the "wedge residual" is finite.
• Higher Regularity (Theorem 1.4): Under additional assumptions on the forcing term and initial data, the authors establish higher Sobolev regularity for the velocity $v$ and pressure gradient $\nabla p$, with the same conditional dependence on the wedge residual.
• Small Data Global Existence (Section 8): The paper includes a separate result proving the existence of global regular solutions for small data. This result is independent of the critical-wedge obstruction and relies on direct control of the angular component.

Significance and Claims
The paper does not claim to solve the global regularity problem for the Navier–Stokes equations unconditionally. Instead, it identifies the specific "concentration regime" (the critical wedge) where the standard energy methods fail. The significance of the work lies in:

1. Precise Characterization of Obstruction: It isolates the exact mechanism (the ratio of $L^s$ to $L^\infty$ norms of $v_\phi$) that prevents the closure of estimates.
2. Conditional Control: It proves that the usual a priori control is recovered whenever the associated wedge residual is finite. This implies that if a solution were to become singular, it must do so by entering this specific critical wedge regime with a non-vanishing residual.
3. Methodological Refinement: By utilizing the Liu-Wang expansions and refined elliptic estimates for the modified stream function, the authors improve upon previous results by removing the artificial boundary condition $\psi_1|_{r=0}=0$ while maintaining rigorous control over the solution's behavior near the axis of symmetry.

In summary, the paper provides a rigorous framework for understanding the regularity of axially symmetric Navier–Stokes solutions by quantifying the "gap" in the estimates as a function of the solution's trajectory relative to a defined critical wedge in phase space.