A Liouville theorem for the two-dimensional stationary hypodissipative Navier--Stokes system

Abstract: We study the two-dimensional stationary incompressible Navier--Stokes equations on $\mathbb R<sup>2$ with fractional dissipation $(-Δ)<sup>s$. In the full range $s \in (0,1)$, we prove that every smooth solution satisfying the natural energy condition $$u\in\dot{\mathrm H}^s(\mathbb R^2;\mathbb R<sup>2)$$ has $u\equiv0$ and constant pressure. This is a fractional counterpart of the planar finite-Dirichlet theorem of Gilbarg and Weinberger at $s=1$. The proof uses different arguments in three ranges. For $0&lt;s&lt;\frac13$, we combine an $\mathrm L<sup>2$-estimate derived from the equation with a stream-function truncation argument. For $\frac13\leq s\leq\frac23$, we use a localized energy estimate whose boundary terms are supported on expanding annuli. For $\frac23&lt;s&lt;1$, we establish regularity and decay via a Lorentz-space bootstrap and then apply the maximum principle to the vorticity. We also treat the stationary damped Euler system at $s=0$ by combining the Bernoulli identity with a cut-off argument under an annular growth condition that includes $u\in \mathrm L^r(\mathbb R<sup>2)$ for every $1\le r\le2$.

• The paper proves that every smooth stationary solution on ℝ² with finite homogeneous energy u∈Ḣˢ is identically zero, with constant pressure, for the full fractional range 0<s<1.
• The authors divide the proof by dissipation strength, using stream-function truncation for low s, annular Bernoulli-flux estimates for 1/3≤s≤2/3, and Lorentz bootstrapping with a vorticity maximum principle for high s.
• The result removes the additional global Lebesgue assumptions required by earlier fractional Liouville theorems and establishes the complete two-dimensional energy-class analogue of the classical Gilbarg–Weinberger rigidity theorem.

Overview and main result

This paper by De Nitti, Niebel, and Yang establishes a Liouville-type rigidity theorem for the two-dimensional stationary incompressible Navier–Stokes system with fractional dissipation on $\mathbb{R}^2$:

$$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$

where $0 < s < 1$. The main theorem states that every smooth solution satisfying only the natural energy condition $u \in \dot{\mathrm H}^s(\mathbb{R}^2;\mathbb{R}^2)$ must satisfy $u \equiv 0$ with constant pressure. This is a fractional counterpart of the classical planar finite-Dirichlet-integral theorem of Gilbarg and Weinberger at $s=1$, and it covers the full range $s\in(0,1)$ — a notable strengthening over prior fractional results, which typically required additional global Lebesgue integrability assumptions such as $u \in L^2$ or $L^{9/2}$.

The natural energy is the homogeneous Gagliardo seminorm $$[u]_{\dot{\mathrm H}^s} = \|(-\Delta)^{s/2}u\|_{L^2}$$, and the homogeneous Sobolev embedding gives $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$0 with $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$1. The central difficulty is that this embedding alone provides neither $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$2-control nor pointwise decay, so $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$3 is not initially admissible as a test function; the formal energy identity $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$4 requires justification.

Context in the literature

For the classical case $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$5, Gilbarg and Weinberger proved triviality under finite Dirichlet integral in two dimensions, while the three-dimensional analogue with $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$6 vanishing at infinity remains open — the paper notes that $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$7 alone appears insufficient.

Prior fractional results were concentrated in three dimensions and imposed extra integrability: Wang and Xiao required $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$8 (with an additional $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$9 condition for $0 < s < 1$0 in the planar compressible setting); Chamorro and Poggi, Jarrín and Vergara-Hermosilla, Zeng, Tan, and others obtained results under various combinations of $0 < s < 1$1 and Lebesgue or Besov assumptions. Lee and Lee's recent work covers dimensions $0 < s < 1$2 but excludes $0 < s < 1$3; its formal planar specialization would give $0 < s < 1$4. The present paper closes the two-dimensional problem across all of $0 < s < 1$5 using only the energy class, and additionally treats the damped Euler endpoint $0 < s < 1$6.

Preliminary reductions

Two structural lemmas underpin all three arguments. First, the pressure normalization lemma: since $0 < s < 1$7 with $0 < s < 1$8, Calderón–Zygmund theory defines $0 < s < 1$9, and a scaling argument against dilated test functions rules out non-constant harmonic polynomial remainders, showing $u \in \dot{\mathrm H}^s(\mathbb{R}^2;\mathbb{R}^2)$0 is constant. This permits working with the projected equation

$u \in \dot{\mathrm H}^s(\mathbb{R}^2;\mathbb{R}^2)$1

and, for $u \in \dot{\mathrm H}^s(\mathbb{R}^2;\mathbb{R}^2)$2, with the representation $u \in \dot{\mathrm H}^s(\mathbb{R}^2;\mathbb{R}^2)$3 modulo polynomials, where $u \in \dot{\mathrm H}^s(\mathbb{R}^2;\mathbb{R}^2)$4 has order $u \in \dot{\mathrm H}^s(\mathbb{R}^2;\mathbb{R}^2)$5. Second, a truncation lemma shows that far-field cutoffs $u \in \dot{\mathrm H}^s(\mathbb{R}^2;\mathbb{R}^2)$6 vanish in $u \in \dot{\mathrm H}^s(\mathbb{R}^2;\mathbb{R}^2)$7 as $u \in \dot{\mathrm H}^s(\mathbb{R}^2;\mathbb{R}^2)$8, which is what makes the cutoff-based energy identities converge to the full energy.

Supporting tools include Lorentz-space Hölder estimates, mapping properties of $u \in \dot{\mathrm H}^s(\mathbb{R}^2;\mathbb{R}^2)$9 (Hardy–Littlewood–Sobolev, endpoint $u \equiv 0$0, Morrey–Sobolev, and Besov–Hölder bootstrapping), a lemma excluding polynomial remainders via cone-growth versus tail-integrability arguments, and a truncation-below-$u \equiv 0$1 result for stream functions based on the boundedness of the modulus map.

The range $u \equiv 0$2: stream-function truncation

The argument first extracts an $u \equiv 0$3 gain from the equation itself: taking Fourier transforms, the multiplier $u \equiv 0$4 is bounded near the origin when $u \equiv 0$5, and Hausdorff–Young gives $u \equiv 0$6 with $u \equiv 0$7, whence $u \equiv 0$8; high frequencies are controlled directly by the energy. Plancherel then yields $u \equiv 0$9.

A second application of the equation controls the stream function $s=1$0 (defined by $s=1$1) at zero frequency: $s=1$2 near the origin is square-integrable precisely because $s=1$3. One obtains $s=1$4 with $s=1$5 at infinity.

The key device is level-set truncation: for $s=1$6, where $s=1$7 clamps values outside $s=1$8, the Sobolev chain rule gives $s=1$9 almost everywhere. Since $s\in(0,1)$0 decays, $s\in(0,1)$1 is compactly supported and divergence-free, so testing against it eliminates both pressure and convection terms exactly. Passing $s\in(0,1)$2 via weak convergence in $s\in(0,1)$3 and strong convergence in $s\in(0,1)$4 yields $s\in(0,1)$5, hence $s\in(0,1)$6. Although used here only below $s\in(0,1)$7, the method works throughout $s\in(0,1)$8.

The intermediate range $s\in(0,1)$9: annular cutoff

Here the equation is tested directly against $u \in L^2$0. The left-hand side converges to the full energy $u \in L^2$1 by the truncation lemma. The right-hand side combines into a single Bernoulli flux $u \in L^2$2 with $u \in L^2$3, bounded by

$u \in L^2$4

which vanishes whenever $u \in L^2$5 because both annular norms tend to zero by absolute continuity. Under the pure energy assumption, Sobolev embedding supplies $u \in L^2$6, and the constraint $u \in L^2$7 translates exactly into $u \in L^2$8: below $u \in L^2$9 the cubic boundary terms are uncontrollable, and above $L^{9/2}$0 the factor $L^{9/2}$1 diverges. More generally, the proposition holds for any $L^{9/2}$2 if $L^{9/2}$3 for some $L^{9/2}$4.

The high range $L^{9/2}$5: Lorentz bootstrap and vorticity maximum principle

The annular estimate fails above $L^{9/2}$6, so the authors instead bootstrap regularity from the representation $L^{9/2}$7. Writing $L^{9/2}$8, the subcritical recurrence $L^{9/2}$9 (with $$[u]_{\dot{\mathrm H}^s} = \|(-\Delta)^{s/2}u\|_{L^2}$$0) strictly decreases $$[u]_{\dot{\mathrm H}^s} = \|(-\Delta)^{s/2}u\|_{L^2}$$1 from $$[u]_{\dot{\mathrm H}^s} = \|(-\Delta)^{s/2}u\|_{L^2}$$2, so finitely many steps reach either the endpoint $$[u]_{\dot{\mathrm H}^s} = \|(-\Delta)^{s/2}u\|_{L^2}$$3 — giving $$[u]_{\dot{\mathrm H}^s} = \|(-\Delta)^{s/2}u\|_{L^2}$$4 via the Lorentz estimate — or the supercritical regime, giving a homogeneous Hölder bound that combined with $$[u]_{\dot{\mathrm H}^s} = \|(-\Delta)^{s/2}u\|_{L^2}$$5 forces uniform continuity and hence decay and boundedness. In both cases $$[u]_{\dot{\mathrm H}^s} = \|(-\Delta)^{s/2}u\|_{L^2}$$6.

A second bootstrap yields $$[u]_{\dot{\mathrm H}^s} = \|(-\Delta)^{s/2}u\|_{L^2}$$7 for some $$[u]_{\dot{\mathrm H}^s} = \|(-\Delta)^{s/2}u\|_{L^2}$$8: choosing $$[u]_{\dot{\mathrm H}^s} = \|(-\Delta)^{s/2}u\|_{L^2}$$9, one or two applications of the Besov–Hölder mapping property of $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$00 lift $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$01 to $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$02, using $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$03. Decay of $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$04, $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$05, and the vorticity $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$06 follows from the uniform-continuity-plus-integrability lemma.

The vorticity satisfies $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$07. At a positive maximum point $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$08, smoothness gives $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$09, so $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$10; but the principal value reduces to an ordinary integral with non-negative integrand, forcing $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$11, contradicting decay. Hence $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$12, each component of $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$13 is harmonic, and the classical Liouville theorem plus $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$14 gives $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$15.

The paper also records an alternative route in this range: once $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$16, a fractional Leibniz argument gives $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$17, interpolation yields $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$18, and a Friedrichs-commutator energy estimate on the mollified vorticity equation gives $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$19 directly.

The damped Euler endpoint $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$20

At $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$21 the system becomes stationary damped Euler, $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$22. The Bernoulli function $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$23 satisfies the dissipative identity $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$24, and consequently

$$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$25

Testing against expanding cutoffs bounds the boundary flux by $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$26. Under the annular growth condition $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$27, Fatou's lemma along a good sequence $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$28 forces $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$29, hence $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$30. The condition is satisfied in particular when $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$31 for any $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$32. This improves on Chae's criterion requiring $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$33, $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$34, and on the $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$35 assumption implicit in Chepyzhov–Ilyin–Zelik.

Limitations and open questions

The main theorem assumes smoothness of $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$36; no claim is made for weak solutions in the energy class, and extending the result beyond $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$37 solutions is not addressed. The three complementary arguments have genuinely disjoint mechanisms — the stream-function method fails at $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$38 (the low-frequency integral $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$39 diverges), the annular estimate closes only on $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$40, and the vorticity maximum principle requires the bootstrap available only for $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$41 — leaving open whether a unified proof exists. The paper does not treat the corresponding three-dimensional problem, where even the classical $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$42 statement with $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$43 vanishing at infinity remains unresolved, nor the question of whether the annular growth condition in the damped Euler theorem can be weakened further. Whether nontrivial solutions exist under weaker decay than $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$44 (e.g., in the complementing ranges without any $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$45 bound for $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$46) is not investigated.

Conclusion

The paper proves that the sole natural energy condition $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$47 forces triviality of smooth stationary solutions of the two-dimensional hypodissipative Navier–Stokes system for every $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$48, completing the fractional analogue of the Gilbarg–Weinberger theorem and removing the auxiliary Lebesgue assumptions required in earlier work. The proof combines a stream-function truncation technique exploiting exact cancellation of convection and pressure, an annular Bernoulli-flux estimate valid precisely on $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$49, and a Lorentz-space bootstrap culminating in a nonlocal vorticity maximum principle for $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$50, together with a Bernoulli-identity argument for the damped Euler endpoint. The result settles the two-dimensional Liouville question in the energy class and delineates sharply the mechanism boundaries at $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$51, $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$52, and $$u\cdot\nabla u + (-\Delta)^s u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$53.