Global exponential stability for the three-dimensional
Navier-Stokes equations on hyperbolic space
May 21, 2026
We prove that the three-dimensional incompressible Navier-Stokes equations with the deformation Laplacian on hyperbolic 3-space \(\mathbb{H}^3\) admit a unique global mild solution for sufficiently small initial data in \(L^3(\mathbb{H}^3)\), and that this solution decays exponentially to zero. The exponential decay rate is \(\mu\lambda_\mathrm{Def}^{(3)}\), where \(\mu\) is the kinematic viscosity and \(\lambda_\mathrm{Def}^{(3)} = 26/9\) is the effective spectral gap of the deformation Laplacian in \(L^3\). On flat \(\mathbb{R}^3\), the corresponding Kato-type result gives only algebraic decay. The exponential stability is a macroscopic consequence of the spectral gap provided by negative curvature. We also show that the \(L^2\) norm is supercritical on \(\mathbb{H}^3\) (as on \(\mathbb{R}^3\)), with the obstruction arising from the local ultraviolet scaling of the heat kernel, which is insensitive to global geometry. The boundary between what curvature can and cannot improve is located exactly: the Fujita-Kato integral has a scaling exponent \(1/2 - 3/(2p)\) that depends only on the integrability of the initial data, not on the geometry of the manifold. For \(p \geq 3\) (the Kato critical space), the integral is bounded and the spectral gap contributes exponential time decay. For \(p < 3\), the integral diverges at \(t = 0\) (and strictly diverges for all \(t>0\) when \(p \le 3/2\)) regardless of the curvature.
The incompressible Navier-Stokes equations on a Riemannian manifold \((M,g)\), \[\label{eq:NS} \partial_t u + \nabla_u u + \nabla p = \mu\,\Delta_L u,\qquad \mathrm{div}\,u = 0,\qquad u(0) = u_0,\tag{1}\] require a choice of Laplacian \(\Delta_L\) acting on vector fields. On flat \(\mathbb{R}^3\), all natural candidates (Hodge, Bochner, deformation) agree. On a curved manifold, they differ by multiples of the Ricci curvature, and the choice affects both the analytical properties and the physical content of the equations (see, e.g., the recent survey by Czubak [1]).
In a companion paper [2], we established that the deformation Laplacian \(\Delta_\mathrm{Def}= \Delta_B + \mathrm{Ric}\) (where \(\Delta_B\) is the Bochner Laplacian) is the unique choice consistent with the Lagrangian kinematics of the fluid, and proved global well-posedness in two dimensions on manifolds with strictly negative curvature. This intrinsic mandate is further reinforced from an extrinsic perspective, as the deformation Laplacian universally emerges in the thin-shell limit of ambient flows under stress-free boundary conditions [3]. Supported by these dual physical foundations, we are justified in focusing exclusively on the deformation operator to leverage its unique spectral advantages. The present paper addresses the three-dimensional case on hyperbolic 3-space \(\mathbb{H}^3\) (constant sectional curvature \(-1\)).
In three dimensions, the global regularity of the Navier-Stokes equations is open even on flat \(\mathbb{R}^3\). The classical result of Kato [4] gives global existence for small data in \(L^3(\mathbb{R}^3)\) (the critical space determined by scaling), with algebraic decay \(\|u(t)\|_{L^3} \to 0\) as \(t\to\infty\). The \(L^2\) norm is supercritical: the energy estimate alone does not give uniqueness or regularity.
On the hyperbolic plane \(\mathbb{H}^2\), Chan and Czubak [5], [6] showed that Leray-Hopf weak solutions are non-unique even with the deformation Laplacian derived from Ebin and Marsden. Khesin and Misiołek [7] showed that this non-uniqueness is a consequence of the Hodge decomposition specific to dimension two and does not occur on \(\mathbb{H}^n\) for \(n \geq 3\). In three dimensions, Lichtenfelz [8] proved non-uniqueness on a negatively curved 3-manifold, but only for the Hodge Laplacian: harmonic 1-forms are stationary solutions of the Hodge-based equation but not of the deformation equation (since \(Lh = 2\mathrm{Ric}(h) \neq 0\)). These results show that in three dimensions the deformation Laplacian is better behaved than the Hodge Laplacian with respect to uniqueness. The present paper works exclusively with the deformation Laplacian, whose spectral gap (established below) provides analytical advantages absent for the Hodge Laplacian.
On \(\mathbb{H}^3\), the scaling symmetry of the Navier-Stokes equations is broken by the curvature. The deformation Laplacian has a spectral gap: the Stokes operator \(A = -\mathbb{P}\Delta_\mathrm{Def}\) (where \(\mathbb{P}\) is the Leray projector onto divergence-free fields) satisfies \(\sigma(A) \subseteq [4,\infty)\) on \(L^2\), and the effective gap in \(L^3\) is \(\lambda_\mathrm{Def}^{(3)} = 26/9\). This spectral gap provides exponential time decay in the heat semigroup, which has no flat-space analogue.
The Fujita-Kato approach for the Navier-Stokes equations on non-compact manifolds with negative curvature was pioneered by Pierfelice [9], who established crucial dispersive and smoothing estimates. Building on this framework, Balentine [10] studied the equations on \(\mathbb{H}^n\) using the deformation (Ebin-Marsden) Laplacian, proving global existence for small \(L^n\) initial data and demonstrating qualitative exponential time decay. While our geometric setting overlaps with these works, the present paper advances the theory on \(\mathbb{H}^3\) by introducing several explicit novelties. First, whereas Balentine demonstrates generic exponential decay without explicitly evaluating the rates, we leverage the exact \(L^2\) spectral gap of the deformation Laplacian on \(\mathbb{H}^3\) to derive the explicit optimal exponential decay rate \(\mu\lambda_{\mathrm{Def}}^{(3)} = \mu \cdot 26/9\). Furthermore, our explicit computation captures the full effective spectral gap rather than a half-gap. This avoids the suboptimal decay rates that occur when mathematically trading spectral bounds to absorb polynomial time singularities. In addition, while prior works primarily treat the deformation Laplacian as a mathematically convenient analytical choice, we establish a physical connection by grounding the operator in physical principles, utilizing the kinematic selection derivation established in our companion paper [2]. Finally, we explicitly formulate and prove a short-time UV obstruction (Theorem 2), which neither Pierfelice nor Balentine addressed. This theorem rigorously locates the boundary between what global curvature can and cannot improve, proving that the local Euclidean structure dominates the short-time scaling, keeping \(L^2\) initial data supercritical on \(\mathbb{H}^3\).
The main result of this paper is:
Theorem 1 (Main theorem). Let \(\mathbb{H}^3\) be hyperbolic 3-space with sectional curvature \(-1\), and let \(A = -\mathbb{P}\Delta_\mathrm{Def}\) be the Stokes operator with the deformation Laplacian. There exists \(\epsilon_0 > 0\) such that for any divergence-free \(u_0 \in L^3(\mathbb{H}^3)\) with \(\|u_0\|_{L^3} < \epsilon_0\), the integral equation \[\label{eq:mild} u(t) = e^{-t\mu A}u_0 - \int_0^t e^{-(t-s)\mu A}\mathbb{P}\nabla\cdot (u(s)\otimes u(s))\,ds\tag{2}\] has a unique global mild solution \(u\in C([0,\infty); L^3_\sigma)\) satisfying \[\label{eq:decay} \|u(t)\|_{L^6} \leq C\,\|u_0\|_{L^3}\,t^{-1/4}\, {e^{-\mu\lambda_\mathrm{Def}^{(3)}t}},\qquad t > 0,\tag{3}\] where \(\lambda_\mathrm{Def}^{(3)} = 26/9\) and \(L^3_\sigma\) denotes the closure of smooth compactly supported divergence-free vector fields in \(L^3\).
The exponential factor \(e^{-\mu\lambda_\mathrm{Def}^{(3)}t}\) is the qualitative novelty: on \(\mathbb{R}^3\), the corresponding decay is algebraic (\(t^{-1/4}\) alone). The curvature of \(\mathbb{H}^3\) converts the large-time behaviour from power-law to exponential.
We also establish the following negative result, which delineates the boundary of what geometry can achieve:
Theorem 2 (UV obstruction). The Fujita-Kato contraction argument for the deformation Laplacian on \(\mathbb{H}^3\) with \(L^p\) initial data has a scaling integral \(I(t)\) satisfying for short times \(t \ll 1\): \[\label{eq:scaling-intro} {I(t) \sim C_0\,t^{1/2-3/(2p)},}\tag{4}\] where the scaling exponent is independent of the spectral gap. For \(p < 3\), the integral diverges as \(t\to 0\), and the contraction argument does not close, regardless of the spectral gap. For \(p \leq 3/2\), the integral strictly diverges for all \(t > 0\). In particular, \(L^2\) data remains supercritical on \(\mathbb{H}^3\).
The exponent \(1/2 - 3/(2p)\) is determined by the local (short-time) scaling of the heat kernel, which is identical on \(\mathbb{H}^3\) and \(\mathbb{R}^3\). The spectral gap contributes only the exponential factor, which controls the large-time behaviour but cannot cure the short-time singularity. The boundary between what curvature helps and what it cannot is located precisely at Kato’s critical space \(L^3\).
The paper is organised as follows. Section 2 establishes the spectral theory of the Stokes operator on \(\mathbb{H}^3\), including the exact \(L^2\) spectral gap and the \(L^p\) extension via the diamagnetic inequality. Section 3 proves the commutation of the Leray projector with the deformation Laplacian on \(\mathbb{H}^3\). Section 4 establishes the bilinear (Oseen-Stokes) estimate via a duality argument. Section 5 carries out the Fujita-Kato contraction, proving both the exponential stability (Theorem 1) and the UV obstruction (Theorem 2). Section 6 discusses the results in context.
On \(\mathbb{H}^3\) with constant sectional curvature \(-1\), the Ricci tensor acts as \(\mathrm{Ric}= -2g\) on vector fields. The deformation Laplacian is therefore \(\Delta_\mathrm{Def}= \Delta_B - 2\), where \(\Delta_B\) is the Bochner (rough) Laplacian. Since \(\mathrm{Ric}= -2\,\mathrm{Id}\) commutes with \(\Delta_B\) (both are \(SO(3)\)-equivariant operators on the isometry group of \(\mathbb{H}^3\)), the deformation semigroup factorises: \[\label{eq:factorisation} e^{t\mu\Delta_\mathrm{Def}} = e^{t\mu(\Delta_B - 2)} = e^{-2\mu t}\, e^{t\mu\Delta_B}.\tag{5}\]
Proposition 3. The Stokes operator \(A = -\mathbb{P}\Delta_\mathrm{Def}\) on \(L^2\) divergence-free vector fields on \(\mathbb{H}^3\) has spectrum \(\sigma(A) \subseteq [4,\infty)\).
Proof. The Weitzenböck formula on \(\mathbb{H}^3\) reads \(\Delta_H = -\Delta_B + \mathrm{Ric}= -\Delta_B - 2\), where \(\Delta_H = d\delta + \delta d \geq 0\) is the (positive-semidefinite) Hodge Laplacian. Rearranging: \(-\Delta_B = \Delta_H + 2\). Hence \[\label{eq:A-hodge} A = -\Delta_\mathrm{Def}= -\Delta_B + 2 = \Delta_H + 4.\tag{6}\]
For divergence-free vector fields \(u\) (equivalently, co-closed 1-forms: \(\delta u^\flat = 0\)), the Hodge Laplacian reduces to \(\Delta_H u^\flat = d\delta u^\flat + \delta du^\flat = \delta du^\flat\). By Hodge duality on \(\mathbb{H}^3\), the exterior derivative \(d\) maps co-exact 1-forms isometrically to exact 2-forms, so the spectrum of \(\Delta_H\) on co-exact 1-forms equals the spectrum on exact 2-forms.
By Donnelly’s theorem [11] on the spectral geometry of \(\mathbb{H}^n\), the \(L^2\) spectrum of \(\Delta_H\) on exact \(k\)-forms is bounded below by \((n-2k+1)^2/4\). For \(n = 3\) and \(k = 2\): \((3-4+1)^2/4 = 0\). Since \(\Delta_H \geq 0\), we have \(\sigma(\Delta_H|_\mathrm{co\text{-}exact\;1\text{-}forms}) \subseteq [0,\infty)\), and by 6 , \(\sigma(A) \subseteq [4,\infty)\). ◻
Remark 4. The Donnelly bound is sharp: the continuous spectrum of \(\Delta_H\) on exact 2-forms on \(\mathbb{H}^3\) starts at exactly \(0\). So \(\lambda_\mathrm{Def}^{(2)} = 4\) is the exact infimum, not merely a lower bound. This improves the crude estimate \(\lambda_\mathrm{Def}\geq \min(1,\kappa^2) = 1\) from the general coercivity bound [2].
The scalar heat kernel on \(\mathbb{H}^3\) is known explicitly [12]: \[\label{eq:scalar-heat} p_t(r) = \frac{1}{(4\pi t)^{3/2}}\,\frac{r}{\sinh r}\, e^{-t - r^2/(4t)},\tag{7}\] where \(r\) is the geodesic distance. The factor \(e^{-t}\) reflects the bottom of the \(L^2\) scalar spectrum at \(\lambda_0^{(2)} = 1\). The \(L^p\) spectral bottom is \(\lambda_0^{(p)} = 4(p-1)/p^2\) (see [12]), giving the scalar semigroup bounds: \[\label{eq:scalar-Lp-Lq} \|e^{t\Delta_\mathrm{scalar}}f\|_{L^q} \leq C\, t^{-\frac{3}{2}(\frac{1}{p}-\frac{1}{q})}\,e^{-\lambda_0^{(p)}t}\, \|f\|_{L^p}.\tag{8}\]
The Hess-Schrader-Uhlenbrock diamagnetic inequality [13] bounds the Bochner heat semigroup on 1-forms pointwise by the scalar semigroup: \(|e^{t\Delta_B}u|(x) \leq e^{t\Delta_\mathrm{scalar}}|u|(x)\). Combined with the factorisation 5 and the commutation of \(\mathbb{P}\) with \(\Delta_\mathrm{Def}\) (proved in the next section), we obtain:
Proposition 5. The Stokes semigroup on \(\mathbb{H}^3\) satisfies, for \(1 \leq p \leq q \leq \infty\): \[\label{eq:stokes-Lp-Lq} \|e^{-t\mu A}f\|_{L^q} \leq C\,t^{-\frac{3}{2}(\frac{1}{p} -\frac{1}{q})}\,e^{-\mu\lambda_\mathrm{Def}^{(p)}t}\,\|f\|_{L^p},\qquad{(1)}\] where \(\lambda_\mathrm{Def}^{(p)}\) is the effective \(L^p\) spectral gap, bounded below via the diamagnetic inequality by \(\lambda_0^{(p)} + 2 = 4(p-1)/p^2 + 2\). In particular, for \(p=3\) this yields \(\lambda_\mathrm{Def}^{(3)} \geq 26/9\). For \(p=2\), the formula gives a lower bound of \(3\), but Proposition 3 establishes the exact gap on divergence-free fields is sharper: \(\lambda_\mathrm{Def}^{(2)} = 4\).
Proposition 6. On \(\mathbb{H}^3\), the Leray-Helmholtz projector \(\mathbb{P}\) satisfies:
\(\mathbb{P}\) is bounded on \(L^p(\mathbb{H}^3)\) for all \(1 < p < \infty\).
\(\mathbb{P}\) commutes with \(\Delta_\mathrm{Def}\): \([\mathbb{P}, \Delta_\mathrm{Def}] = 0\).
Proof. (a) The projector is \(\mathbb{P}= I - d(-\Delta_\mathrm{scalar})^{-1}\delta\), which factors through the Riesz transforms \(\mathcal{R} = d(-\Delta_\mathrm{scalar})^{-1/2}\). On complete Riemannian manifolds with bounded geometry and strictly negative curvature, the Riesz transforms are bounded singular integral operators on \(L^p\) for \(1 < p < \infty\), by the Calderón-Zygmund theory of Strichartz [14] and Lohoué [15].
(b) On any Riemannian manifold, the Hodge Laplacian \(\Delta_H = d\delta + \delta d\) algebraically commutes with the exterior derivative \(d\) and codifferential \(\delta\) (since \(\Delta_H d = (d\delta + \delta d)d = d\delta d = d(d\delta+\delta d) = d \Delta_H\)), and therefore universally commutes with \(\mathbb{P}\). Because \(\mathbb{H}^3\) is an Einstein manifold with \(\mathrm{Ric}= -2\,\mathrm{Id}\), the Weitzenböck identity gives \(\Delta_\mathrm{Def}= -\Delta_H - 4\) globally on all vector fields. Since \(\Delta_\mathrm{Def}\) differs from \(-\Delta_H\) only by a constant multiple of the identity, it inherits this commutation: \([\mathbb{P}, \Delta_\mathrm{Def}] = [\mathbb{P}, -\Delta_H - 4] = 0\). ◻
Remark 7. The commutation \([\mathbb{P}, \Delta_\mathrm{Def}] = 0\) holds on any Einstein manifold (where \(\mathrm{Ric}= c\,g\) for a constant \(c\)), not only on space forms. On a manifold with non-constant Ricci curvature, \([\mathbb{P}, \Delta_\mathrm{Def}] \neq 0\) in general, and the analysis requires commutator estimates (see [16]).
As a consequence, the Stokes semigroup \(e^{-t\mu A}\) restricted to divergence-free fields equals the deformation semigroup \(e^{t\mu\Delta_\mathrm{Def}}\) applied to divergence-free fields, with no commutator correction. The bounds of Proposition 5 hold for the Stokes semigroup without modification.
The nonlinear term in 2 requires bounding the composite operator \(T_\tau = e^{-\tau\mu A}\mathbb{P}\nabla\cdot : L^r \to L^q\) for \(\tau > 0\). We establish this via duality.
Proposition 8. For \(1 < r \leq q < \infty\) and \(\tau > 0\): \[\label{eq:oseen} \|e^{-\tau\mu A}\mathbb{P}\nabla\cdot F\|_{L^q} \leq C\, (\tau^{-\delta} + \tau^{-\delta_0})\, e^{-\mu\gamma\tau}\,\|F\|_{L^r},\qquad{(2)}\] where \(F\) is a symmetric tensor field, \(\delta = \frac{1}{2}+\frac{3}{2}(\frac{1}{r}-\frac{1}{q})\), \(\delta_0 = \frac{3}{2}(\frac{1}{r}-\frac{1}{q})\), and \(\gamma = 2 + \lambda_0^{(r')}/2 + \lambda_0^{(r)}/2\) is the effective bilinear spectral gap.
Proof. By duality, \(\|T_\tau F\|_{L^q} = \sup_{\|\Phi\|_{L^{q'}}=1} |\langle T_\tau F, \Phi\rangle|\), where \(q' = q/(q-1)\). The adjoint is \(T_\tau^* = -\nabla e^{-\tau\mu A}\mathbb{P}\) (using self-adjointness of \(A\) and \(\mathbb{P}\), and that \(\nabla\cdot\) and \(-\nabla\) are formal adjoints). We bound \(\|T_\tau^*\Phi\|_{L^{r'}}\) with \(r' = r/(r-1)\).
Splitting the gradient via the fractional Bochner Laplacian: \[\label{eq:split} \nabla e^{-\tau\mu A}\mathbb{P}= e^{-2\mu\tau}\bigl[\nabla(-\Delta_B)^{-1/2} \bigr]\circ\bigl[(-\Delta_B)^{1/2}e^{\tau\mu\Delta_B}\mathbb{P}\bigr].\tag{9}\] The factorisation 5 and the commutation of \(\mathbb{P}\) (Proposition 6) have been used.
The bundle Riesz transform \(\nabla(-\Delta_B)^{-1/2}\) is bounded on \(L^{r'}(\mathbb{H}^3)\) for \(1 < r' < \infty\), by the result of Bakry [17] on manifolds with Ricci curvature bounded below.
For the fractional smoothing, we halve the time: \((-\Delta_B)^{1/2}e^{\tau\mu\Delta_B} = [(-\Delta_B)^{1/2} e^{(\tau/2)\mu\Delta_B}]\circ e^{(\tau/2)\mu\Delta_B}\). Since the spectrum of \(-\Delta_B\) on \(L^{r'}\) is bounded strictly below by \(\lambda_0^{(r')} > 0\), the diamagnetic inequality and the explicit scalar heat kernel imply that the operator norm of \((-\Delta_B)^{1/2}e^{(\tau/2)\mu\Delta_B}\) is bounded by \(\sup_{x \geq \lambda_0^{(r')}} x^{1/2} e^{-\mu\tau x/2}\). For small \(\tau\), this supremum is \(O(\tau^{-1/2})\), but for large \(\tau\), the maximum occurs at the spectral boundary \(x = \lambda_0^{(r')}\). This yields the uniform bound \(\|(-\Delta_B)^{1/2}e^{(\tau/2)\mu\Delta_B}\|_{L^{r'}\to L^{r'}} \leq C(\tau^{-1/2} + 1)\,e^{-\mu\lambda_0^{(r')}\tau/2}\). The second half provides the \(L^{q'}\to L^{r'}\) transition. On divergence-free fields, \(e^{(\tau/2)\mu\Delta_B}\mathbb{P} = e^{\mu\tau}\,e^{-(\tau/2)\mu A}\mathbb{P}\) (absorbing the Weitzenböck shift). By duality, the norm of the heat semigroup from \(L^{q'} \to L^{r'}\) is identically equal to its norm from \(L^r \to L^q\). Applying Proposition 5 on \(L^r \to L^q\) gives: \(\|e^{(\tau/2)\mu\Delta_B}\mathbb{P}\Phi\|_{L^{r'}} \leq C\, \tau^{-\frac{3}{2}(\frac{1}{r}-\frac{1}{q})}\, e^{-\mu\lambda_0^{(r)}\tau/2}\,\|\Phi\|_{L^{q'}}\).
Assembling (and noting \(1/q' - 1/r' = 1/r - 1/q\)): \[\|T_\tau^*\Phi\|_{L^{r'}} \leq C\,e^{-2\mu\tau}\,(\tau^{-1/2} + 1)\, e^{-\mu\lambda_0^{(r')}\tau/2}\,\tau^{-\frac{3}{2}(\frac{1}{r}-\frac{1}{q})}\, e^{-\mu\lambda_0^{(r)}\tau/2}\,\|\Phi\|_{L^{q'}}.\] By duality, \(\|T_\tau\|_{L^r\to L^q}\) has the same bound. Collecting exponentials gives the effective spectral gap \(\gamma = 2 + \lambda_0^{(r')}/2 + \lambda_0^{(r)}/2\). For our application to the Navier-Stokes equations, \(q = 6\) and \(1/r = 2/q\), which gives \(r = 3\) and \(r' = 3/2\). Thus \(\lambda_0^{(3/2)} = \frac{4(3/2-1)}{(3/2)^2} = 8/9\) and \(\lambda_0^{(3)} = \frac{4(3-1)}{3^2} = 8/9\). This yields exactly \(\gamma = 2 + 4/9 + 4/9 = 26/9 = \lambda_\mathrm{Def}^{(3)}\). ◻
For the Navier-Stokes nonlinearity, the tensor \(F = u\otimes v\) satisfies \(\|u\otimes v\|_{L^r} \leq \|u\|_{L^q}\|v\|_{L^q}\) with \(1/r = 2/q\). Substituting into ?? : \[\label{eq:bilinear-applied} \|e^{-\tau\mu A}\mathbb{P}\nabla\cdot(u\otimes v)\|_{L^q} \leq C\, (\tau^{-\delta} + \tau^{-\delta_0})\,e^{-\mu\gamma\tau}\, \|u\|_{L^q}\|v\|_{L^q},\tag{10}\] where \(\gamma = 26/9\) collects the exponential factors. The exponent \(\delta = 1/2 + 3/(2q)\) and the large-time correction \(\delta_0 = 3/(2q)\) capture the temporal singularities of the bilinear term.
We seek a mild solution of 2 in the space \[X = \bigl\{u \in C((0,\infty); L^q_\sigma) : \|u\|_X < \infty\bigr\},\] with norm \[\label{eq:X-norm} \|u\|_X = \sup_{t>0}\,e^{\alpha t}\,t^\beta\,\|u(t)\|_{L^q},\tag{11}\] where \(q > 3\), \(\beta = 3/(2p) - 3/(2q)\) (matching the linear semigroup decay for \(L^p\) data), and \(\alpha = \mu\gamma = \mu\lambda_\mathrm{Def}^{(3)}\) (the full effective spectral gap). We take \(p = 3\) and \(q = 6\), giving \(\beta = 1/4\).
The linear term satisfies, by Proposition 5: \[e^{\alpha t}t^\beta\|e^{-t\mu A}u_0\|_{L^6} \leq C\, e^{(\alpha - \mu\lambda_\mathrm{Def}^{(3)})t}\,\|u_0\|_{L^3} {= C\,\|u_0\|_{L^3}},\] since \(\alpha = \mu\lambda_\mathrm{Def}^{(3)}\) by construction. So \(\|e^{-t\mu A}u_0\|_X \leq C_1\|u_0\|_{L^3}\).
For the bilinear term \(B(u,v)(t) = -\int_0^t T_{t-s}(u(s)\otimes v(s))\,ds\), using 10 : \[\begin{align} e^{\alpha t}t^\beta\|B(u,v)(t)\|_{L^6} &\leq C\|u\|_X\|v\|_X\, e^{\alpha t}t^\beta\int_0^t \bigl((t-s)^{-\delta} + (t-s)^{-\delta_0}\bigr)\,e^{-\mu\gamma(t-s)}\, e^{-2\alpha s}\,s^{-2\beta}\,ds\nonumber\\ &\equiv C\|u\|_X\|v\|_X\cdot (I_1(t) + I_2(t)).\label{eq:bilinear-X} \end{align}\tag{12}\]
Lemma 1. With \(\alpha = \mu\gamma\), the integral splits into two pieces \(I_1(t)\) and \(I_2(t)\). The primary integral satisfies \[\label{eq:I1-exact} I_1(t) \leq \mathcal{B}(1-2\beta, 1-\delta)\,t^{1/2-3/(2p)},\tag{13}\] where \(\mathcal{B}\) is the Beta function. The secondary integral \(I_2(t)\) has the large-time asymptotic scaling \(I_2(t) \sim O(t^{\beta-\delta_0})\). In particular:
For \(p = 3\): choosing \(q = 6\) yields \(\beta = 1/4\), \(\delta = 3/4\), and \(\delta_0 = 1/4\). Thus \(I_1(t) \leq \mathcal{B}(1/2, 1/4) < \infty\). For \(I_2(t)\), the large-time behavior scales as \(t^{\beta-\delta_0} = t^0 = O(1)\). Both integrals are uniformly bounded, preserving the exact exponential decay rate.
For \(p \leq 3/2\): convergence of the Beta integral requires \(2\beta < 1 \implies 3/p - 3/q < 1\) and \(\delta < 1 \implies q > 3\), which combined require \(p > 3/2\). Thus, for \(p \le 3/2\), no valid choice of \(q\) exists, the integral has a non-integrable singularity at \(\tau=0\), and it strictly diverges to \(+\infty\) for all \(t > 0\).
For \(3/2 < p < 3\) (e.g., \(L^2\) data with \(p=2\)): Kato’s framework allows freely choosing \(q > \max(3,p)\). Choosing \(q = 4\) yields \(\beta = 3/8\) and \(\delta = 7/8\), so the Beta integral \(\mathcal{B}(1/4, 1/8)\) converges, giving a finite integral for \(t>0\). However, it scales as \(t^{-1/4}\) and blows up as \(t \to 0\), keeping the data supercritical.
Proof. With \(\alpha = \mu\gamma\), the exponential factors combine as \(e^{\alpha t}e^{-\mu\gamma(t-s)}e^{-2\alpha s} = e^{-\alpha s} \leq 1\). For the principal part \(I_1(t)\), the substitution \(s = \tau t\) gives \[\begin{align} I_1(t) &= t^\beta\int_0^1 (t(1-\tau))^{-\delta}\,e^{-\alpha \tau t}\, (t\tau)^{-2\beta}\,t\,d\tau\nonumber\\ &\leq t^{1-\delta-\beta}\int_0^1 (1-\tau)^{-\delta}\,\tau^{-2\beta}\,d\tau = \mathcal{B}(1-2\beta, 1-\delta) t^{1/2-3/(2p)}. \end{align}\] This converges when \(\beta < 1/2\) and \(\delta < 1\). For the secondary part \(I_2(t)\), the same substitution yields \[I_2(t) = t^{1-\delta_0-\beta}\int_0^1 (1-\tau)^{-\delta_0}\, \tau^{-2\beta}\,e^{-\alpha \tau t}\,d\tau.\] For large times \(t \gg 1\), the exponential localizes the integral to \(\tau \sim 1/(\alpha t)\), contributing an internal asymptotic scaling of \(t^{2\beta-1}\). Thus, the overall large-time scaling of \(I_2(t)\) is \(t^{1-\delta_0-\beta} \times t^{2\beta-1} = t^{\beta-\delta_0}\). By choosing \(p=3\) and \(q=6\), we obtain \(\beta-\delta_0 = 1/4 - 1/4 = 0\), explicitly demonstrating the exact mathematical cancellation that bounds \(I_2(t)\) uniformly by a constant. ◻
The \(q\)-dependence cancels exactly: the scaling exponent depends only on the integrability of the initial data. This is a geometric invariant of the problem, reflecting the fact that the short-time behaviour of the heat kernel is locally Euclidean.
With \(p = 3\), \(q = 6\), \(\beta = 1/4\), and \(\delta = 3/4\), the bilinear bound 12 gives \[\|B(u,v)\|_X \leq C_2\,\|u\|_X\|v\|_X,\] where \(C_2 = C\sup_{t>0}(I_1(t) + I_2(t)) < \infty\). The map \(u \mapsto e^{-t\mu A}u_0 + B(u,u)\) is a contraction on the ball \(\{u\in X : \|u\|_X \leq 2C_1\|u_0\|_{L^3}\}\) provided \(4C_1C_2\|u_0\|_{L^3} < 1\), i.e., \(\|u_0\|_{L^3} < \epsilon_0 \equiv (4C_1C_2)^{-1}\).
The unique fixed point satisfies \(\|u\|_X \leq 2C_1\|u_0\|_{L^3}\), which unpacks to \[\|u(t)\|_{L^6} \leq 2C_1\|u_0\|_{L^3}\,t^{-1/4}\, {e^{-\mu\gamma t}},\] giving 3 with the full exponential decay rate \(\gamma = \lambda_\mathrm{Def}^{(3)} = 26/9\). 0◻
For general \(L^p\) data with \(p < 3\), the bounds fail. Specifically, for \(p \leq 3/2\), Lemma 1 shows that \(I_1(t)\) possesses a non-integrable temporal singularity and diverges strictly to \(+\infty\) for all \(t>0\). For \(3/2 < p < 3\), by freely choosing an appropriate Kato exponent \(q > \max(3,p)\), the Beta function converges for any \(t>0\), but the evaluated integral behaves as \(I_1(t) \sim t^{1/2-3/(2p)}\) as \(t\to 0\). The exponent \(1/2 - 3/(2p) < 0\) gives \(\sup_{t>0}(I_1(t) + I_2(t)) = \infty\), and the contraction argument does not close. No choice of \(q\), \(\beta\), or \(\alpha\) can remedy this, because the scaling exponent is independent of \(q\) and the spectral gap contributes only a bounded exponential factor, which is \(O(1)\) as \(t\to 0\). 0◻
On \(\mathbb{R}^3\), the Kato theorem [4] gives global existence for small \(L^3\) data with algebraic decay \(\|u(t)\|_{L^q} \leq C\,t^{-\frac{3}{2}(\frac{1}{3}-\frac{1}{q})}\). On \(\mathbb{H}^3\), the decay acquires an exponential factor: \(\|u(t)\|_{L^q} \leq C\,t^{-\frac{3}{2}(\frac{1}{3}-\frac{1}{q})}\, e^{-\mu\lambda_\mathrm{Def}^{(3)}t}\). The short-time behaviour (\(t \ll 1\)) is identical; the improvement is entirely at large times (\(t \gg 1\)).
The smallness threshold \(\epsilon_0 = (4C_1C_2)^{-1}\) is the same as on \(\mathbb{R}^3\) (since the Beta-function constant \(\mathcal{B}(1/2,1/4)\) is geometry-independent). Curvature does not enlarge the basin of attraction of the zero solution; it only accelerates the decay within that basin.
While the present work establishes global exponential stability explicitly on the constant-curvature space \(\mathbb{H}^3\), the physical mechanism driving this dissipation is not an artifact of constant curvature. In a companion paper [16], we establish that this exponential decay, along with the exact ultraviolet scaling obstruction, extends universally to the broader class of complete, simply connected 3-manifolds with pinched negative sectional curvature. On such manifolds, the deformation Laplacian’s spectral gap continues to dictate the macroscopic stabilization of the fluid, further cementing its role as the analytically optimal viscous operator.
The results locate a sharp boundary between what global geometry (curvature, spectral gap) can and cannot do for the 3D Navier-Stokes equations:
What curvature provides (IR improvement): exponential time decay for \(L^p\) norms with \(p \geq 3\) (by interpolation with the semigroup bounds of Proposition 5), with rate determined by the spectral gap \(\lambda_\mathrm{Def}^{(p)}\). This is a large-scale, low-frequency effect.
What curvature cannot provide (UV obstruction): improved regularity for data rougher than \(L^3\). The obstruction is a short-time, high-frequency phenomenon determined by the local Euclidean structure of the manifold. Because every Riemannian manifold is locally flat, no amount of global curvature can change the ultraviolet scaling.
The results are specific to the deformation Laplacian. With the Hodge Laplacian, the Stokes operator on \(\mathbb{H}^3\) would be \(A_H = \Delta_H\), with spectral gap \(0\) on divergence-free fields (by Proposition 3, the gap is \(\Delta_H \geq 0\) with infimum exactly \(0\)). The exponential decay would be lost, and the large-time behaviour would revert to the flat-space algebraic rate. The Bochner Laplacian would give \(A_B = -\Delta_B = \Delta_H + 2\), with gap \(2\). Only the deformation Laplacian gives the full gap of \(4\) and the corresponding decay rate \(\lambda_\mathrm{Def}^{(3)} = 26/9\).
This illustrates a point made in [2]: the choice of viscous operator has analytical consequences beyond the formal structure of the equations. The deformation Laplacian is not only the kinematically correct choice; it is the analytically optimal one on negatively curved manifolds.