\[H^s_{\mathrm{div}}(\mathbb{T}^3) := \bigl\{ \mathbf{u} \in H^s(\mathbb{T}^3)^3 : \nabla \cdot \mathbf{u} = 0 \bigr\}\]

Remark 2.1 (Temporal Parameterization and Construction Order)

The parameter \(\tau \in [0,\infty)\) serves as the primary evolution variable. Rather than reparameterizing a pre-existing physical solution, we construct the solution directly in \(\tau\)-coordinates and subsequently recover physical time as a derived quantity.

The Coupled System. For each Galerkin truncation \(N \geq 1\), we solve the autonomous system

with initial conditions \(\mathbf{U}_N(0) = P_N \mathbf{u}^\circ\) and \(\varphi_N(0) = 0\), where \(\Phi : \mathbb{R}_{\geq 0} \to [\varphi_{\min}, \varphi_{\max}]\) is continuous with \(0 < \varphi_{\min} \leq \varphi_{\max} < \infty\). This construction generalizes the classical Sundman transformation from celestial mechanics, with vorticity magnitude replacing collision distance as the regularizing variable.

Well-Posedness. The right-hand side is locally Lipschitz on the finite-dimensional Galerkin subspace, since \(\Phi(\|\nabla \times \mathbf{U}_N\|_{L^\infty}) \geq \varphi_{\min} > 0\) is bounded away from zero. The Picard–Lindelöf theorem yields local existence and uniqueness. Taking the \(L^2\) inner product with \(\mathbf{U}_N\) and applying the standard Galerkin cancellation \(\langle (\mathbf{U}_N \cdot \nabla)\mathbf{U}_N, \mathbf{U}_N \rangle = 0\), which holds on \(\mathbb{T}^3\) by incompressibility and periodicity, we obtain

This implies \(\|\mathbf{U}_N(\tau)\|_{L^2} \leq \|\mathbf{u}^\circ\|_{L^2}\) for all \(\tau \geq 0\), precluding finite-time blow-up and establishing global existence of the pair \((\mathbf{U}_N, \varphi_N)\) in \(\tau\).

Recovery of Physical Time. The map \(\varphi_N : [0,\infty) \to [0,\infty)\) satisfies \(\varphi_N'(\tau) \in [\varphi_{\min}, \varphi_{\max}]\) with \(\varphi_{\min} > 0\), hence is a \(C^1\) diffeomorphism. Physical time is defined as the image of this map: \(t := \varphi_N(\tau)\). The global coverage property \(\varphi_N(\tau) \geq \varphi_{\min} \cdot \tau \to \infty\) as \(\tau \to \infty\) ensures that every \(t \in [0,\infty)\) is attained.

Physical Solution. The velocity field \(\mathbf{u}_N(x,t) := \mathbf{U}_N(x, \varphi_N^{-1}(t))\) solves the standard Galerkin system in physical time. This follows from the chain rule: since \(\partial_\tau = \varphi_N'(\tau) \partial_t\) when acting on functions of \(\varphi_N(\tau)\), the lifted equation transforms to the physical Navier–Stokes equation upon multiplication by \(\varphi_N'(\tau)\).

Proposition 2.2 (Global Existence of Galerkin Approximations)

For each \(N \geq 1\), the coupled Galerkin system admits a unique global solution \((\mathbf{U}_N, \varphi_N) \in C^\infty([0,\infty))\).

Proof. The right-hand side of the Galerkin ODE is locally Lipschitz on the finite-dimensional subspace, with \(\Phi(\|\nabla \times \mathbf{U}_N\|_{L^\infty}) \geq \varphi_{\min} > 0\) bounded away from zero. Picard–Lindelöf yields local existence. The energy estimate \(\|\mathbf{U}_N(\tau)\|_{L^2} \leq \|\mathbf{u}^\circ\|_{L^2}\) precludes finite-time blow-up, establishing global existence. ∎

\[\frac{\partial u_i}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} = \nu \Delta u_i - \frac{\partial p}{\partial x_i} + f_i(x,t)\]

\[\nabla \cdot \mathbf{u} = 0\]

\[\mathbf{u}(x,0) = \mathbf{u}^\circ(x), \qquad \nabla \cdot \mathbf{u}^\circ = 0, \qquad \int_{\mathbb{T}^3} \mathbf{u}^\circ(x)\,dx = 0.\]

\[\partial_t u_i + u_j\,\partial_j u_i = \nu \Delta u_i - \partial_i p + f_i\]

\[\frac{1}{\varphi'(\tau)}\,\partial_\tau U_i + U_j\,\partial_j U_i = \nu \Delta U_i - \partial_i P + F_i\]

Remark 2.3 (BKM Invariance Under Reparameterization)

Let \(t=\varphi(\tau)\) with \(\varphi'(\tau)\in[\phi_{\min},\phi_{\max}]\). Then for any \(T>0\),

Thus finiteness or divergence of the Beale–Kato–Majda integral is preserved under this bounded, strictly increasing change of variables. No further property of \(\varphi\) is required.

\[\int_{0}^{T^{*}} \|\omega(t)\|_{L^\infty}\, dt = \infty.\]

Definition 2.4 (Bounded Vorticity–Response Functional)

A vorticity–response functional is a continuous, monotonically increasing function

with \(0 < \varphi_{\min} \le \varphi_{\max} < \infty\). If the temporal lifting \(t=\varphi(\tau)\) is defined by \(\varphi'(\tau) = \Phi\!\left( \|\Omega(\tau)\|_{L^\infty} \right)\) with \(\varphi(0)=0\), then the bounds on \(\Phi\) imply \(\varphi_{\min} \le \varphi'(\tau) \le \varphi_{\max}\) for all \(\tau\ge0\), and therefore \(\varphi\) is a \(C^{1}\) diffeomorphism of \([0,\infty)\) onto itself.

Definition 2.5 (Temporal Lifting as a Coupled System)

Let \(\Phi : \mathbb{R}_{\ge 0} \to [\varphi_{\min},\varphi_{\max}]\) be continuous and monotonically increasing, with \(0 < \varphi_{\min} \le \varphi_{\max} < \infty\).

where \(\mathcal{N}(U) = -(U \cdot \nabla)U - \nabla P + \nu \Delta U\) denotes the Navier–Stokes operator with the incompressibility constraint \(\nabla \cdot U = 0\) imposed in the weak sense.

\[\varphi'(\tau) = \Phi\!\left(\|\Omega(\tau)\|_{L^\infty}\right), \qquad \Omega(\tau) := \nabla \times U(\tau).\]

\[\partial_t u_i + u_j \partial_j u_i + \partial_i p - \nu \Delta u_i = 0, \qquad \nabla\cdot\mathbf{u} = 0\]

\[\frac{1}{\varphi'(\tau)}\,\partial_\tau U_i + U_j \partial_j U_i + \partial_i P - \nu \Delta U_i = 0, \qquad \nabla\cdot\mathbf{U} = 0\]

Proposition 2.6 (Equivalence of the Two Forms)

The physical and lifted equations are equivalent, related by the chain rule \(\partial_\tau U_i(x,\tau) = (\partial_t u_i)(x,\varphi(\tau))\,\varphi'(\tau)\).

Proof. Since \(U(x,\tau)=u(x,\varphi(\tau))\), the chain rule gives \(\partial_\tau U_i = (\partial_t u_i)(x,\varphi(\tau))\,\varphi'(\tau)\). Thus \(\partial_t u_i = \varphi'(\tau)^{-1}\partial_\tau U_i\). Substituting directly into the physical equation yields the lifted equation. All spatial terms are unchanged because they act only on \(x\). ∎

Lemma 2.7 (Uniform Vorticity Bounds for Galerkin Approximations)

Let \(\mathbf{u}^\circ \in H^s_{\mathrm{div}}(\mathbb{T}^3)\) with \(s > 5/2\). For the Galerkin approximations \(\mathbf{u}_N\), there exists a constant \(C = C(s, T, \mathbf{u}^\circ)\), independent of \(N\), such that

for all \(N \geq 1\) and any \(T > 0\).

Proposition 2.8 (BKM Coordinate Invariance)

Let \(\varphi : [0,\infty) \to [0,\infty)\) be a strictly increasing \(C^1\) map with \(\varphi'(\tau) > 0\) for all \(\tau \ge 0\) and \(\varphi(0)=0\). Define the lifted vorticity \(\Omega(\tau) := \omega(\varphi(\tau))\). Then for every \(T>0\),

Consequently, the Beale–Kato–Majda integral is finite in the coordinate index \(t\) if and only if it is finite in the parameter index \(\tau\).

Remark 2.9 (Decoupling of Spatial and Temporal Resolution)

For each \(N\ge1\), the Galerkin system is a globally defined smooth ODE (Proposition 2.2), and the vorticity satisfies uniform bounds on finite intervals independent of \(N\) (Lemma 2.7). The parameter \(N\) determines the spatial scales represented in the approximation, while the parameter \(\tau\) supplies a temporal indexing that depends only on the solution trajectory, not on the truncation level. Since the bounds hold uniformly in \(N\), passage to the limit \(N\to\infty\) on any finite interval \([0,T]\) is compatible with the lifted formulation, yielding a solution of the full Navier–Stokes equations on \(\mathbb{T}^3\).

\[(\mathbf{U}, \varphi) \in \bigl(L^\infty(0,\infty; H^s(\mathbb{T}^3)) \cap C^\infty(\mathbb{T}^3 \times (0,\infty))\bigr) \times C^1([0,\infty))\]

Corollary 3.2 (Existence and Smoothness on \(\mathbb{R}^3 / \mathbb{Z}^3\))

For any smooth, divergence-free initial data \(\mathbf{u}^\circ\) on \(\mathbb{R}^3 / \mathbb{Z}^3\), there exist smooth functions \(\mathbf{u}(x,t)\) and \(p(x,t)\) on \((\mathbb{R}^3 / \mathbb{Z}^3) \times [0,\infty)\) that solve the incompressible Navier–Stokes equations.

Theorem 3.3 (Global Regularity via BKM Syllogism)

Smooth global solutions to the incompressible Navier–Stokes equations exist on \(\mathbb{T}^3\) for smooth divergence-free initial data.