Liouville Theorems Above the Critical $9/2$ Threshold for Stationary Navier-Stokes Equations

Abstract: We establish new Liouville-type theorems for the stationary Navier--Stokes equations in $\mathbb{R}<sup>3$. A central open problem in this context is whether the classical $L<sup>{9/2}(\mathbb{R}<sup>3)$ condition of G.~Galdi can be relaxed. In this note we show that this global integrability requirement can indeed be weakened. More precisely, we prove that triviality already follows under assumptions of the form $$u \in L^{9/2 + \varepsilon(\cdot)}(\mathbb{R}<sup>3)$$, where $\varepsilon(\cdot)&gt;0$. As a consequence, we obtain a localized Liouville theorem: it is sufficient to impose this integrability condition only at infinity, with no additional assumptions on the behavior of $u$ inside a compact set. This highlights that the mechanism enforcing triviality is purely asymptotic. Our approach relies on a general uniqueness result in the framework of Lebesgue spaces with variable exponents, which naturally captures the coexistence of different integrability regimes across the domain.

• The paper proves a new Liouville theorem showing that stationary Navier–Stokes solutions in \(\dot H^1(\mathbb R^3)\) are trivial under a variable-exponent condition \(u\in L^{9/2+\varepsilon(\cdot)}\), where \(\varepsilon(x)\to0\) at infinity.
• The proof uses localized energy estimates, variable-exponent Hölder inequalities, annular decay, and Riesz-transform bounds to make both diffusion and nonlinear pressure remainders vanish without frequency decompositions or smallness assumptions.
• The result demonstrates that integrability is needed only outside an arbitrary compact set, while the critical \(L^{9/2}\) case and the broader \(\dot H^1\) uniqueness problem remain open.

This paper establishes new Liouville-type theorems for the stationary Navier–Stokes equations

$$-\Delta u + u\cdot abla u + \nabla P = 0, \qquad \nabla\cdot u = 0$$

in $\mathbb{R}^3$, weakening the classical global integrability hypothesis $u \in L^{9/2}(\mathbb{R}^3)$ due to Galdi to a variable-exponent condition $$u \in L^{\frac{9}{2}+\varepsilon(\cdot)}(\mathbb{R}^3)$$, where $\varepsilon(x) \to 0$ as $|x|\to\infty$. The main conceptual contribution is a demonstration that the rigidity mechanism enforcing triviality is purely asymptotic: integrability need only be controlled at infinity, with no assumptions on $u$ inside an arbitrarily large compact set.

Background and the open problem

The motivating question, raised by Galdi and Seregin, asks whether every solution $u \in \dot{H}^1(\mathbb{R}^3)$ vanishing at infinity must be identically zero. Existence of nontrivial solutions in $(\dot{H}^1, \dot{H}^{1/2})$ is known (via Lemarié-Rieusset's construction), but uniqueness in this class remains open, so additional decay or integrability hypotheses are needed to force triviality.

Prior results form a hierarchy of relaxations of Galdi's $L^{9/2}$ condition:

• Chae–Wolf: the logarithmic refinement $\mathbb{R}^3$0 suffices.
• Lerner: it suffices that only the low-frequency projection $\mathbb{R}^3$1 belongs to $\mathbb{R}^3$2.
• Kozono–Terasawa–Wakasugi, extended by Seregin–Wang: a smallness condition on the weak-$\mathbb{R}^3$3 norm relative to $\mathbb{R}^3$4 suffices.
• Chae: $\mathbb{R}^3$5 implies triviality; Seregin: $\mathbb{R}^3$6 with $\mathbb{R}^3$7 implies triviality.

The present work departs from all of these by working entirely in physical space with variable exponents, requiring no frequency decomposition and no smallness assumption.

Main results

The central uniqueness theorem is formulated in the variable-exponent Lebesgue space $\mathbb{R}^3$8, defined via the Luxemburg norm associated with the modular $\mathbb{R}^3$9.

Theorem (general criterion). Let $u \in L^{9/2}(\mathbb{R}^3)$0 and let $u \in L^{9/2}(\mathbb{R}^3)$1 solve the stationary system. Suppose $u \in L^{9/2}(\mathbb{R}^3)$2 satisfies: (i) $u \in L^{9/2}(\mathbb{R}^3)$3 on $u \in L^{9/2}(\mathbb{R}^3)$4; (ii) $u \in L^{9/2}(\mathbb{R}^3)$5 is continuous, radially decreasing, and $u \in L^{9/2}(\mathbb{R}^3)$6 everywhere; (iii) $u \in L^{9/2}(\mathbb{R}^3)$7 for $u \in L^{9/2}(\mathbb{R}^3)$8, with $u \in L^{9/2}(\mathbb{R}^3)$9. If $$u \in L^{\frac{9}{2}+\varepsilon(\cdot)}(\mathbb{R}^3)$$0, then $$u \in L^{\frac{9}{2}+\varepsilon(\cdot)}(\mathbb{R}^3)$$1.

Two concrete corollaries follow immediately:

• Explicit exponent theorem ($$u \in L^{\frac{9}{2}+\varepsilon(\cdot)}(\mathbb{R}^3)$$2): with $$u \in L^{\frac{9}{2}+\varepsilon(\cdot)}(\mathbb{R}^3)$$3 for $$u \in L^{\frac{9}{2}+\varepsilon(\cdot)}(\mathbb{R}^3)$$4 and $$u \in L^{\frac{9}{2}+\varepsilon(\cdot)}(\mathbb{R}^3)$$5 for $$u \in L^{\frac{9}{2}+\varepsilon(\cdot)}(\mathbb{R}^3)$$6, membership $$u \in L^{\frac{9}{2}+\varepsilon(\cdot)}(\mathbb{R}^3)$$7 forces $$u \in L^{\frac{9}{2}+\varepsilon(\cdot)}(\mathbb{R}^3)$$8. Note that $$u \in L^{\frac{9}{2}+\varepsilon(\cdot)}(\mathbb{R}^3)$$9 near the origin, so the hypothesis coincides exactly with what Sobolev embedding already provides on bounded sets, while decaying toward the critical value $\varepsilon(x) \to 0$0 at infinity at a quantified rate $\varepsilon(x) \to 0$1.
• Localized Liouville theorem: the same conclusion holds if the variable-exponent condition is imposed only on $\varepsilon(x) \to 0$2, since the interior contribution is covered automatically by $\varepsilon(x) \to 0$3. Since $\varepsilon(x) \to 0$4 is arbitrary, no assumption whatsoever is made on $\varepsilon(x) \to 0$5 inside any fixed compact set — the paper's strongest qualitative claim, namely that triviality is enforced purely by asymptotic behavior.

Method of proof

The argument follows the classical energy/localization scheme: testing the equation against $\varepsilon(x) \to 0$6 with a standard cut-off yields

$\varepsilon(x) \to 0$7

where $\varepsilon(x) \to 0$8 collects the cut-off Laplacian term and $\varepsilon(x) \to 0$9 the convective and pressure terms. Triviality follows once both remainder terms vanish as $|x|\to\infty$0.

The technical core lies in estimating these remainders using variable-exponent Hölder inequalities on annuli $|x|\to\infty$1. Three lemmas drive the asymptotics:

1. Since $|x|\to\infty$2 is continuous, radially decreasing, and approaches $|x|\to\infty$3 at rate $|x|\to\infty$4, the essential supremum satisfies $|x|\to\infty$5.
2. Consequently $|x|\to\infty$6, and hence $|x|\to\infty$7.
3. For any $|x|\to\infty$8, one has $|x|\to\infty$9.

Combining these with Lemma showing $u$0 (a dominated-convergence argument on the modular), both $u$1 and $u$2 vanish. The pressure term is handled via the Riesz-transform representation $u$3 together with boundedness of Riesz transforms on $u$4 for globally log-Hölder exponents $u$5 — a point verified directly for the exponents considered here. This last point is significant: earlier variable-exponent Liouville results (Chamorro–Vergara-Hermosilla) used discontinuous exponents on infinite-measure regions, where Riesz transform boundedness fails, forcing simultaneous hypotheses on $u$6 and $u$7. The log-Hölder framework here removes that restriction, allowing conditions on the velocity alone.

The proof concludes with $u$8, hence $u$9 by Sobolev embedding.

Limitations and open questions

Several caveats should be noted plainly. First, the result does not resolve the original Problem of Galdi–Seregin: the exponent $u \in \dot{H}^1(\mathbb{R}^3)$0 still exceeds the critical threshold everywhere, and the rate of approach $u \in \dot{H}^1(\mathbb{R}^3)$1 is quantified; whether pure $u \in \dot{H}^1(\mathbb{R}^3)$2, or even weaker asymptotic conditions, suffice remains open. Second, the theorem requires $u \in \dot{H}^1(\mathbb{R}^3)$3 in the decay condition (iii), a constraint tied to the specific scaling of the estimates rather than an obviously sharp requirement. Third, the argument relies on regularity of $u \in \dot{H}^1(\mathbb{R}^3)$4 obtained from $u \in \dot{H}^1(\mathbb{R}^3)$5 (via Galdi's Theorem X.1.1), which holds here because $u \in \dot{H}^1(\mathbb{R}^3)$6; extensions to rougher solution classes are not addressed. Finally, the paper leaves open whether the Sobolev embedding information $u \in \dot{H}^1(\mathbb{R}^3)$7 can be upgraded to a global condition implying triviality in $u \in \dot{H}^1(\mathbb{R}^3)$8 — the remark in the introduction notes this would immediately settle the problem but "remains an open problem."

Conclusion

The paper proves that the classical $u \in \dot{H}^1(\mathbb{R}^3)$9 Liouville threshold for stationary Navier–Stokes in $(\dot{H}^1, \dot{H}^{1/2})$0 can be relaxed to a spatially varying integrability condition approaching $(\dot{H}^1, \dot{H}^{1/2})$1 from above at infinity, and — more strikingly — that this condition need only hold outside an arbitrary compact set. The mechanism is shown to be purely asymptotic, and the variable-exponent framework with log-Hölder continuity provides the functional analytic tool that makes a velocity-only formulation possible. The precise boundary between these hypotheses and the full $(\dot{H}^1, \dot{H}^{1/2})$2 conjecture remains unresolved.