Liouville Theorems for Stationary Navier-Stokes Equations via the Radial Velocity Component

Technical Summary: Liouville Theorems for Stationary Navier-Stokes Equations via the Radial Velocity Component

Problem Statement
The paper addresses the Liouville problem for the stationary Navier-Stokes equations in $\mathbb{R}^3$:
$$\begin{cases} -\Delta u + (u \cdot \nabla)u + \nabla \pi = 0, \\ \nabla \cdot u = 0, \end{cases}$$
where $u$ is the velocity field and $\pi$ is the pressure. The central question, known as Conjecture 1.1 (proposed by Galdi and Seregin), asks whether any solution $u \in \dot{H}^1(\mathbb{R}^3)$ that decays to zero at infinity must be identically zero. While the two-dimensional case is resolved (Gilbarg-Weinberger) and the four-dimensional case is settled (Galdi), the three-dimensional case remains a significant open problem.

Existing results typically rely on global isotropic integrability conditions (e.g., $u \in L^{9/2}(\mathbb{R}^3)$) or anisotropic conditions involving mixed norms of Cartesian components. The author notes that while $u \in \dot{H}^1(\mathbb{R}^3)$ implies $u \in L^6(\mathbb{R}^3)$ via Sobolev embedding, this decay is insufficient on its own to prove triviality.

Methodology
The paper employs a cut-off function technique combined with energy estimates and Hölder inequalities in both standard and variable-exponent Lebesgue spaces.

1. Cut-off and Energy Identity: The proofs utilize a radial cut-off function $\psi_R$ to localize the energy identity. By testing the equations with $\psi_R u$ and integrating by parts, the author derives an inequality bounding the local Dirichlet energy $\int_{B(R/2)} |\nabla u|^2$ by boundary terms involving the pressure and velocity on the annulus $C(R) = \{x : R/2 < |x| < R\}$.
2. Radial Decomposition: A key methodological shift is the decomposition of the nonlinear term and the pressure term based on the radial component of the velocity, $u_\rho$. The author exploits the fact that the gradient of the radial cut-off function, $\nabla \psi_R$, is purely radial. This allows the integral terms involving $\nabla \psi_R \cdot u$ to be rewritten as $\partial_\rho \psi_R u_\rho$, isolating the radial velocity component.
3. Variable Exponent Spaces: For the second main result, the author utilizes the framework of Lebesgue spaces with variable exponents, $L^{p(\cdot)}(\mathbb{R}^3)$. This allows for a unified treatment of integrability conditions that vary spatially, specifically transitioning from a stronger condition near the origin to a critical condition at infinity.
4. Pressure Control: The pressure $\pi$ is controlled via the Riesz transform representation $\pi = \sum R_i R_j (u_i u_j)$, leveraging the boundedness of Riesz transforms on $L^r$ spaces to relate $\|\pi\|_{L^3}$ to $\|u\|_{L^6}^2$.

Key Contributions and Results

Theorem 1.1 (Radial Integrability Condition):
The author proves that if $u \in \dot{H}^1(\mathbb{R}^3)$ is a solution and its radial component $u_\rho$ satisfies $u_\rho \in L^p(\mathbb{R}^3)$ for $3/2 < p \le 3$, then $u \equiv 0$.

• Significance: This result improves upon previous isotropic conditions (like $L^{9/2}$) and anisotropic Cartesian conditions. It demonstrates that triviality can be deduced by imposing integrability constraints only on the radial component of the velocity, without any restrictions on the tangential components.
• Mechanism: The proof relies on a specific distribution of the cubic nonlinearity in the energy estimate. By applying Hölder's inequality with exponents $p_1=6$ (for the full velocity $u$, justified by Sobolev embedding) and $p_2=p$ (for the radial component $u_\rho$), the author balances the terms to allow for $p$ as low as $3/2$.

Theorem 1.2 (Variable Exponent Condition):
The author establishes a more refined uniqueness result where the integrability condition on the radial component is allowed to vary. Specifically, if $u_\rho \in L^{3+\eta(\cdot)}(\mathbb{R}^3)$, where the exponent $\eta(x)$ is defined such that it equals 3 for $|x| < R_0$ (implying $u_\rho \in L^6$ locally) and decays as $3R_0/|x|$ for $|x| \ge R_0$ (approaching the critical exponent 3 at infinity), then $u \equiv 0$.

• Significance: This theorem shows that the critical $L^3$ condition for the radial component need only be satisfied asymptotically. Stronger integrability (up to $L^6$) is required only on a fixed compact set, while the condition relaxes to the critical threshold at infinity. This highlights the "rigidity" of the system driven by localized and radial properties rather than uniform global conditions.

Theorem 3.1 (General Variable Exponent Result):
This is the technical engine behind Theorem 1.2. It proves that for any continuous, radially decreasing variable exponent $p(x)$ that equals 6 near the origin and approaches 3 at infinity with a specific decay rate ($|p(x)-3| \le C/|x|$), the condition $u_\rho \in L^{p(\cdot)}(\mathbb{R}^3)$ implies $u \equiv 0$.

Significance and Claims
The paper claims to offer a "different perspective" on the Liouville problem by shifting the focus from isotropic or Cartesian anisotropic conditions to radial anisotropy.

• Weaker Assumptions: The results impose conditions strictly weaker than known isotropic results (e.g., $L^{9/2}$) and distinct from existing anisotropic results that rely on mixed norms of Cartesian components.
• Localization: The analysis reveals that the rigidity of the stationary Navier-Stokes system can be driven by localized integrability properties. Theorem 1.2 specifically demonstrates that the critical integrability threshold can be approached asymptotically, allowing for a transition from strong local behavior to critical behavior at infinity.
• Methodological Novelty: The use of variable-exponent Lebesgue spaces provides a flexible framework to capture these spatially varying integrability behaviors, overcoming the limitations of fixed-exponent spaces in this context.

The author explicitly states that these results do not resolve the full Conjecture 1.1 (which requires no additional integrability assumptions beyond $\dot{H}^1$ and decay) but provide new sufficient conditions that are structurally distinct and less restrictive on the velocity field's components than previously known.