Liouville-type theorems for the 3D stationary Navier-Stokes equations in variable Lebesgue spaces

Abstract: In \cite{CV23}, Chamorro and Vergara-Hermosilla established several Liouville-type theorems to the Navier-Stokes equations in the framework of the variable Lebesgue spaces. These results may allow the variable exponent $p(\cdot)$ beyond the range of $[3,\frac{9}{2}]$ in some non-negligible regions in $\mathbb{R}<sup>3$. In this paper we find two new non-negligible regions, in which the Liouville-type theorems still hold under some assumptions imposed on $p(\cdot)$ in these regions. Our results can be regarded as the marginal cases of the results in \cite{CV23}.

• The paper proves that weak stationary Navier–Stokes solutions vanish under variable-exponent integrability conditions on two new logarithmically defined exceptional regions.
• A logarithmically thickened parabolic region permits exponents above 9/2, provided they remain below the threshold (3γ+3)/γ, while the complement stays below 9/2.
• A logarithmically damped marginal region allows an infinite exponent and resolves an endpoint case through volume-growth estimates, although the general finite-Dirichlet-integral Liouville problem remains open.

Background and problem

The paper studies the Liouville-type problem for the three-dimensional incompressible stationary Navier-Stokes equations

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

where $u$ is the velocity field and $\pi$ the pressure. The classical question asks whether a weak solution $(u,\pi)$ with finite Dirichlet integral $D(u)=\int_{\mathbb{R}^3}|\nabla u|^2\,dx$ and decaying at infinity must vanish identically; this remains open in general [L16]. Known affirmative results impose integrability or decay hypotheses on $u$: Galdi's criterion $u\in L^{9/2}$ [G11], Chae and Wolf's logarithmic refinement of it [CW16], Chae's condition $\Delta u\in L^{6/5}$ [C14], Seregin's $L^6\cap BMO^{-1}$ result [S16], Kozono–Terasawa–Wakasugi's vorticity decay and weak-$L^{9/2}$ smallness conditions [KTW17], and Chamorro–Jarrín–Lemarié-Rieusset's theorem that $u$0 with $u$1 forces $u$2 [CJL21].

The immediate motivation is the work of Chamorro and Vergara-Hermosilla [CV23], who transplanted these Liouville-type results to variable Lebesgue spaces $u$3, defined via the Luxemburg norm associated with the modular $u$4. Their framework permits the exponent $u$5 to exceed the critical range $u$6 on certain "non-negligible" subsets of $u$7 — sets of the form $u$8 and $u$9 — provided $\pi$0 stays within $\pi$1 on the complement. The present paper enlarges this class of admissible regions.

Main results

The paper proves two theorems. Both assume a weak solution $\pi$2, $\pi$3 of the stationary system with $\pi$4 and $\pi$5, and conclude $\pi$6.

Theorem 1 (logarithmically thickened parabolic region). Let

$\pi$7

If $\pi$8 off $\pi$9 and $(u,\pi)$0 on $(u,\pi)$1, then $(u,\pi)$2.

Theorem 2 (logarithmic marginal case). Let

$(u,\pi)$3

If $(u,\pi)$4 off $(u,\pi)$5 and $(u,\pi)$6 on $(u,\pi)$7, then $(u,\pi)$8.

Both results are strict extensions of Theorems 2 and 3 in [CV23]: since $(u,\pi)$9, Theorem 1 covers strictly larger regions than the power-law region $D(u)=\int_{\mathbb{R}^3}|\nabla u|^2\,dx$0; and $D(u)=\int_{\mathbb{R}^3}|\nabla u|^2\,dx$1 corresponds to the endpoint $D(u)=\int_{\mathbb{R}^3}|\nabla u|^2\,dx$2 of the region $D(u)=\int_{\mathbb{R}^3}|\nabla u|^2\,dx$3, which was excluded there because $D(u)=\int_{\mathbb{R}^3}|\nabla u|^2\,dx$4 diverges. The logarithmic factor $D(u)=\int_{\mathbb{R}^3}|\nabla u|^2\,dx$5 restores convergence of $D(u)=\int_{\mathbb{R}^3}|\nabla u|^2\,dx$6, making the marginal case tractable. In both theorems the exponent may be arbitrarily large (indeed infinite, in Theorem 2) on the exceptional region, so no uniform bound $D(u)=\int_{\mathbb{R}^3}|\nabla u|^2\,dx$7 is required globally.

Method of proof

The argument follows the cut-off scheme of [CV23]. One multiplies the momentum equation by $D(u)=\int_{\mathbb{R}^3}|\nabla u|^2\,dx$8, where $D(u)=\int_{\mathbb{R}^3}|\nabla u|^2\,dx$9 is a standard smooth cut-off supported in $u$0, and integrates by parts using $u$1 to obtain the localized energy identity

$u$2

with

$u$3

Since $u$4 on $u$5, showing each $u$6 as $u$7 yields $u$8, hence $u$9 by Sobolev embedding into $u\in L^{9/2}$0.

The estimates split the annulus $u\in L^{9/2}$1 into its intersection with the exceptional region ($u\in L^{9/2}$2 or $u\in L^{9/2}$3) and its complement ($u\in L^{9/2}$4 or $u\in L^{9/2}$5), then apply Hölder's inequality in variable Lebesgue spaces. For $u\in L^{9/2}$6 one uses the conjugate exponent $u\in L^{9/2}$7 against $u\in L^{9/2}$8; for $u\in L^{9/2}$9 and $\Delta u\in L^{6/5}$0 one uses $\Delta u\in L^{6/5}$1 against $\Delta u\in L^{6/5}$2 and $\Delta u\in L^{6/5}$3, respectively. On the complement, where $\Delta u\in L^{6/5}$4 are bounded away from their endpoints, the standard bounds $\Delta u\in L^{6/5}$5 and $\Delta u\in L^{6/5}$6 combined with $\Delta u\in L^{6/5}$7 give vanishing limits because $\Delta u\in L^{6/5}$8 and $\Delta u\in L^{6/5}$9.

The essential new input is the volume growth of the exceptional regions. For Theorem 1,

$L^6\cap BMO^{-1}$0

so that $L^6\cap BMO^{-1}$1. The hypothesis $L^6\cap BMO^{-1}$2 is exactly what makes $L^6\cap BMO^{-1}$3, and since $L^6\cap BMO^{-1}$4 for any $L^6\cap BMO^{-1}$5, the limit vanishes. Analogously, $L^6\cap BMO^{-1}$6 gives $L^6\cap BMO^{-1}$7. For Theorem 2, the exponent $L^6\cap BMO^{-1}$8 (the conjugate of $L^6\cap BMO^{-1}$9) reduces the estimate to the raw volume bound $L^{9/2}$0, giving $L^{9/2}$1 and similarly for $L^{9/2}$2. This is why the logarithmic correction is needed: without it, the volume of the $L^{9/2}$3 region would grow like $L^{9/2}$4 but the corresponding integral test in [CV23] fails at the endpoint.

Significance and limitations

The contribution is quantitative rather than structural: the proofs reuse the localization technique of [CV23] verbatim, and the new content lies in identifying two additional geometries — a logarithmically widened parabolic region and a logarithmically damped marginal region — whose volume growth rates still drive all boundary-layer terms to zero. The upper bounds $L^{9/2}$5 on $L^{9/2}$6 and the requirement $L^{9/2}$7 are sharp within this method: they are precisely the thresholds at which the powers $L^{9/2}$8 and $L^{9/2}$9 change sign, so any further enlargement of $u$00 or relaxation of the exponent range on it would require a different mechanism.

Several assumptions deserve note. The results require the pressure to satisfy $u$01, an a priori integrability condition not derived from the equation itself. The solution class $u$02 with $u$03 is assumed rather than constructed, and the paper does not address whether nontrivial solutions exist when these hypotheses fail — i.e., whether the identified regions are optimal. The underlying Liouville problem for solutions with merely finite Dirichlet integral remains open, and this paper does not close it; it only enlarges the family of weighted-integrability conditions under which triviality can be certified.

Conclusion

The paper extends the variable-exponent Liouville theory for the 3D stationary Navier-Stokes equations to two new non-negligible regions: a logarithmically thickened parabolic region $u$04 with exponent bounded by $u$05 there, and a logarithmic marginal region $u$06 carrying an infinite exponent. Both extend the earlier results of Chamorro and Vergara-Hermosilla, the latter resolving the previously divergent endpoint case $u$07 through a logarithmic correction. The proofs are cut-off energy estimates whose convergence hinges on explicit volume-growth computations, and the resulting exponent thresholds appear to be method-sharp. The central open question — whether every suitable weak solution with finite Dirichlet integral is trivial — remains untouched.