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

Abstract: We study Liouville-type results for the stationary Navier--Stokes equations in $\mathbb{R}<sup>3$. We prove that any $\dot{H}<sup>1(\mathbb{R}<sup>3)$ solution is trivial under an integrability condition imposed only on the radial component of the velocity, namely $u_ρ(x) \in L<sup>p(\mathbb{R}<sup>3)$ with $3/2 &lt; p \leq 3$. We also establish a uniqueness result in a variable-exponent setting, where an $L<sup>6$-type condition is required only on a bounded region, while the exponent approaches the critical value $3$ at infinity. Our analysis reveals that the rigidity of the stationary Navier--Stokes system can be driven by localized and radial integrability properties, rather than uniform global conditions.

• The paper proves that a stationary 3D Navier–Stokes solution in \(\dot H^1(\mathbb R^3)\) is identically zero if its radial velocity component belongs to \(L^p\) for some \(3/2<p\leq3\), without assumptions on tangential components.
• The paper establishes a variable-exponent criterion requiring the radial component to have \(L^6\)-type integrability on an arbitrary ball while its exponent relaxes toward the critical value 3 at infinity.
• The proofs use radial cut-off energy estimates, which isolate the radial velocity in the nonlinear term, together with Sobolev, pressure, Riesz-transform, and variable-exponent bounds to force the total energy to vanish.

Overview and context

This paper by G. Vergara-Hermosilla establishes Liouville-type triviality results for the stationary Navier–Stokes system in $\mathbb{R}^3$,

$$-\Delta u + (u\cdot\nabla)u + \nabla\pi = 0, \qquad \nabla\cdot u = 0,$$

under integrability assumptions imposed only on the radial component $u_\rho$ of the velocity field. The motivating problem is the long-standing conjecture attributed to Galdi and Seregin: any solution $u \in \dot{H}^1(\mathbb{R}^3)$ with $u(x)\to 0$ as $|x|\to\infty$ must vanish identically. The two-dimensional analogue is due to Gilbarg–Weinberger and the four-dimensional case to Galdi; the three-dimensional case remains open. The contribution here is anisotropic: whereas most known sufficient conditions are isotropic (invariant under rotations), the author shows that rigidity can be driven by a single directional component of the velocity.

Relation to prior work

The classical result of Galdi states that $u\in L^{9/2}(\mathbb{R}^3)$ forces $u\equiv 0$, later refined logarithmically by Chae–Wolf. More recent refinements include Lerner's low/high frequency decomposition (only the projection onto Fourier modes containing a neighborhood of the origin needs $L^{9/2}$ integrability) and the author's own variable-exponent condition $u\in L^{9/2+\varepsilon(\cdot)}(\mathbb{R}^3)$ (Vergara-Hermosilla, 7 Apr 2026). Anisotropic results are scarcer: Chae proved uniqueness under mixed-norm conditions on each Cartesian component, extended by Zhang–Zu. The present work differs from Chae's condition — which requires each component $$-\Delta u + (u\cdot\nabla)u + \nabla\pi = 0, \qquad \nabla\cdot u = 0,$$0 to lie in $$-\Delta u + (u\cdot\nabla)u + \nabla\pi = 0, \qquad \nabla\cdot u = 0,$$1 — by imposing a range of exponents on a single component only.

Main results

Theorem 1 (fixed exponent). If $$-\Delta u + (u\cdot\nabla)u + \nabla\pi = 0, \qquad \nabla\cdot u = 0,$$2 solves the stationary system and its radial component satisfies $$-\Delta u + (u\cdot\nabla)u + \nabla\pi = 0, \qquad \nabla\cdot u = 0,$$3 for some $$-\Delta u + (u\cdot\nabla)u + \nabla\pi = 0, \qquad \nabla\cdot u = 0,$$4, then $$-\Delta u + (u\cdot\nabla)u + \nabla\pi = 0, \qquad \nabla\cdot u = 0,$$5. No condition is required on the tangential components. This is a strictly weaker additional hypothesis than the isotropic $$-\Delta u + (u\cdot\nabla)u + \nabla\pi = 0, \qquad \nabla\cdot u = 0,$$6 assumption on the full velocity.

The heuristic behind the admissible range is instructive. Galdi's $$-\Delta u + (u\cdot\nabla)u + \nabla\pi = 0, \qquad \nabla\cdot u = 0,$$7 threshold arises from a Hölder balance $$-\Delta u + (u\cdot\nabla)u + \nabla\pi = 0, \qquad \nabla\cdot u = 0,$$8 applied to a cubic term in $$-\Delta u + (u\cdot\nabla)u + \nabla\pi = 0, \qquad \nabla\cdot u = 0,$$9. Here the analogous integral involves two factors of $u_\rho$0 and one factor of $u_\rho$1, giving the balance $u_\rho$2; choosing $u_\rho$3 (consistent with the Sobolev embedding $u_\rho$4) yields $u_\rho$5 as the natural endpoint.

Theorem 2 (variable exponent). Fix $u_\rho$6 and let $u_\rho$7 for $u_\rho$8 and $u_\rho$9 for $u \in \dot{H}^1(\mathbb{R}^3)$0. If $u \in \dot{H}^1(\mathbb{R}^3)$1, then $u \in \dot{H}^1(\mathbb{R}^3)$2. This improves upon the global $u \in \dot{H}^1(\mathbb{R}^3)$3 endpoint: strong ($u \in \dot{H}^1(\mathbb{R}^3)$4-type) integrability of $u \in \dot{H}^1(\mathbb{R}^3)$5 is required only on the ball $u \in \dot{H}^1(\mathbb{R}^3)$6, while at infinity the exponent decays radially toward the critical value $u \in \dot{H}^1(\mathbb{R}^3)$7 at rate $u \in \dot{H}^1(\mathbb{R}^3)$8. Since $u \in \dot{H}^1(\mathbb{R}^3)$9 is arbitrary, the region of stronger integrability may be placed arbitrarily far from the origin. Notably, whether the global condition $u(x)\to 0$0 alone implies triviality remains open; this theorem replaces that global requirement with localized-plus-asymptotic control on one component.

Both results follow from a more general statement (Theorem 3): if $u(x)\to 0$1 is continuous, radially decreasing, equals $u(x)\to 0$2 on $u(x)\to 0$3, and satisfies $u(x)\to 0$4 for $u(x)\to 0$5 with $u(x)\to 0$6, then $u(x)\to 0$7 implies $u(x)\to 0$8.

Method of proof

The argument follows the standard energy-testing strategy. Regularity of $u(x)\to 0$9 follows from local integrability via Galdi's Theorem X.1.1. Testing against $|x|\to\infty$0, where $|x|\to\infty$1 is a radial cut-off, and integrating by parts using $|x|\to\infty$2 yields

$|x|\to\infty$3

where $|x|\to\infty$4 involves $|x|\to\infty$5 and $|x|\to\infty$6 involves $|x|\to\infty$7. Because $|x|\to\infty$8 is radial, $|x|\to\infty$9, so the convective term reduces to an integral involving $u\in L^{9/2}(\mathbb{R}^3)$0 — this is where the radial-component hypothesis enters, and it is the structural observation underlying both theorems.

For Theorem 1, Hölder with the balance $u\in L^{9/2}(\mathbb{R}^3)$1, $u\in L^{9/2}(\mathbb{R}^3)$2, gives $u\in L^{9/2}(\mathbb{R}^3)$3. For the pressure term, the identity $u\in L^{9/2}(\mathbb{R}^3)$4 via Riesz transforms gives $u\in L^{9/2}(\mathbb{R}^3)$5, so $u\in L^{9/2}(\mathbb{R}^3)$6 by Lemma 2.5 (annular norms of $u\in L^{9/2}(\mathbb{R}^3)$7 functions vanish). The cut-off terms satisfy $u\in L^{9/2}(\mathbb{R}^3)$8 uniformly, and $u\in L^{9/2}(\mathbb{R}^3)$9. Hence $u\equiv 0$0 and $u\equiv 0$1.

For the variable-exponent theorem, the key quantitative input is Lemma 2.7: since $u\equiv 0$2 along annuli, one has $u\equiv 0$3, which kills the polynomial prefactor arising from $u\equiv 0$4. The pressure estimate again uses boundedness of Riesz transforms in the constant-exponent spaces involved. The specific exponent of Theorem 2, $u\equiv 0$5, is verified to satisfy all hypotheses of the general theorem (continuity at $u\equiv 0$6, radial decrease, and decay constant $u\equiv 0$7).

Limitations and open questions

Several caveats should be noted. First, the results remain conditional: they do not resolve the full Galdi–Seregin conjecture, which requires no integrability beyond $u\equiv 0$8 and pointwise decay. Second, the restriction $u\equiv 0$9 in Theorem 1 is essential to the Hölder bookkeeping (it ensures $L^{9/2}$0); whether the range can be widened is not addressed. Third, the asymptotic approach to the critical exponent $L^{9/2}$1 in Theorem 2 is constrained to rate $L^{9/2}$2 with a small constant relative to $L^{9/2}$3; slower decay rates toward $L^{9/2}$4 are outside the scope of the general theorem. Fourth, the proof relies on the radiality of the cut-off function to convert $L^{9/2}$5 into $L^{9/2}$6; extending the method to non-radial weights or other directional components would require new ideas. Finally, the paper leaves open whether an analogous single-component condition suffices without the $L^{9/2}$7 framework, and whether the tangential components admit entirely arbitrary behavior consistent with solvability.

Conclusion

The paper contributes two Liouville theorems for the 3D stationary Navier–Stokes equations in which rigidity is enforced exclusively through the radial velocity component: a fixed-exponent result covering $L^{9/2}$8, $L^{9/2}$9, and a variable-exponent result requiring $u\in L^{9/2+\varepsilon(\cdot)}(\mathbb{R}^3)$0-type integrability of $u\in L^{9/2+\varepsilon(\cdot)}(\mathbb{R}^3)$1 only on an arbitrary compact set, with the exponent relaxing to the critical value $u\in L^{9/2+\varepsilon(\cdot)}(\mathbb{R}^3)$2 at infinity. The proofs combine the classical energy-testing identity with variable-exponent Hölder inequalities and a quantitative lemma controlling the polynomial prefactor near the critical exponent. These results demonstrate that localized and directionally restricted integrability assumptions suffice for uniqueness, sharpening the known landscape of partial answers to the open three-dimensional Liouville problem.