Forced self-similar solutions to the stationary Navier--Stokes equations in a half-space

Technical Summary: Forced Self-Similar Solutions to the Stationary Navier–Stokes Equations in a Half-Space

Problem Statement
This paper investigates the existence and uniqueness of axisymmetric, self-similar solutions to the stationary incompressible Navier–Stokes equations in the three-dimensional upper half-space $\mathbb{R}^3_+ = \{x \in \mathbb{R}^3 : x_3 > 0\}$ subject to the no-slip boundary condition. The system is driven by an axisymmetric external force $F$ that is homogeneous of degree $-3$. The velocity field $u$, pressure $p$, and force $F$ satisfy the scaling invariance:
$$u(\lambda x) = \lambda^{-1}u(x), \quad p(\lambda x) = \lambda^{-2}p(x), \quad F(\lambda x) = \lambda^{-3}F(x).$$
The study focuses on determining whether such solutions exist for given forces, particularly analyzing the dependence on the magnitude of the force and the presence of "swirl" (azimuthal velocity components).

Methodology
The authors employ a reduction of the partial differential equations to ordinary differential equations (ODEs) by exploiting the self-similar and axisymmetric structure.

1. Coordinate Transformation: The velocity is expressed in spherical coordinates $(\rho, \theta, \phi)$ as $u = \rho^{-1}(f(\phi)e_\rho + g(\phi)e_\phi + h(\phi)e_\theta)$. The divergence-free condition allows the radial component $f$ to be expressed in terms of the meridional component $g$.
2. Swirl-Free Reduction: In the absence of swirl ($h \equiv 0$), the system is integrated and transformed via the change of variables $t = \cos \phi$ and $L(t) = g(\phi)\sin \phi$. This reduces the problem to a second-order scalar equation, which is further normalized into a Riccati initial-value problem:
   $$M'(t) + \frac{1}{2}M(t)^2 = \lambda N(t), \quad M(0) = 0,$$
   where $N(t)$ is a functional determined by the external force, and $\lambda$ is a scaling parameter for the force magnitude.
3. Swirling Case: When swirl is present ($Z \neq 0$), the azimuthal equation is solved for $h$ in terms of $g$ and the force. This defines a nonlinear map from the meridional profile $g$ to the Riccati source term. The authors utilize a contraction mapping argument on a space of bounded angular profiles to establish existence for small forces.
4. Extension to Cones: The analysis is generalized to solid cones $K_{\phi_0}$ by adjusting the boundary conditions and the integration intervals, tracking the dependence of estimates on the opening angle $\phi_0$.

Key Contributions and Results

• Existence for Small Tangential Curl (Theorem 1.1):
  The authors prove that if the tangential component of the curl of the external force, $(\text{curl } F)_{\text{tan}}$, is sufficiently small in the $C^0$ norm on the unit sphere, there exists a unique small axisymmetric self-similar solution.

  • Crucial Distinction: The smallness condition applies only to the tangential curl. The full curl of the force may be arbitrarily large. The paper provides an example where the tangential curl vanishes as a parameter tends to zero, while the radial component of the curl blows up, yet a unique solution still exists.
  • Swirl-Free Uniqueness: If the external force is swirl-free, any axisymmetric self-similar solution is necessarily swirl-free and unique.

• Global Existence Interval for Swirl-Free Forces (Theorem 1.2):
  For a fixed swirl-free force $F$, the authors introduce a scaling parameter $\lambda$ and analyze the system with force $\lambda F$. They prove that solutions exist precisely for $\lambda$ in an open interval $(\underline{\lambda}, \overline{\lambda})$ containing zero.

  • The finiteness of the endpoints $\underline{\lambda}$ and $\overline{\lambda}$ is determined by the sign of an explicit discriminant $N(\phi)$, which is an integral functional of the force components.
  • If $N(\phi)$ does not change sign, one endpoint is finite (corresponding to a blow-up of the solution at the symmetry axis) and the other is infinite. If $N(\phi)$ changes sign, both endpoints are finite. If $N \equiv 0$, the interval is $(-\infty, +\infty)$, corresponding to the trivial solution.

• Extension to Solid Cones (Theorem 5.1 & 5.2):
  The results are extended to solid cones with opening angle $\phi_0$.

  • The existence threshold $\varepsilon(\phi_0)$ for the smallness of the tangential curl depends on $\phi_0$.
  • As the cone becomes narrower ($\phi_0 \to 0$), the allowable magnitude of the external force increases significantly (the threshold $\varepsilon(\phi_0)$ scales roughly as $\phi_0^{-5}$ for swirl-free forces and $\phi_0^{-1}$ for swirling forces). This implies that narrower cones can sustain larger external forces while maintaining a unique self-similar solution.

• Blow-up Characterization:
  The paper characterizes the blow-up of solutions at the boundary of the existence interval. Finite endpoints correspond to solutions that blow up exactly at the north pole ($\phi = 0$), meaning the solution cannot be extended continuously to the symmetry axis.

Significance
The paper addresses a fundamental gap in the theory of stationary Navier–Stokes equations in domains with boundaries. While homogeneous solutions in the whole space (Landau solutions) are well-understood, the presence of a boundary (the half-space) typically forces axisymmetric solutions to vanish unless sustained by a non-potential external force.
This work establishes that:

1. Self-similar solutions exist in the half-space driven by specific homogeneous forces, provided the tangential curl is small.
2. The condition for existence is distinct from the whole-space case; no global compatibility condition on the force is required in the half-space.
3. The relationship between the force magnitude and solution existence is precisely quantified via a discriminant derived from the Riccati reduction, offering a complete classification of the existence interval for swirl-free forces.
4. The geometry of the domain (specifically the opening angle of a cone) plays a critical role in relaxing the smallness constraints on the external force, allowing for solutions under much larger forcing in narrow geometries.

The authors frame these results as providing crucial insights into the regularity theory and asymptotic behavior of fluid flows near singularities or at large distances, extending the classical understanding of Landau-type solutions to forced, bounded domains.