Chapter 133

Navier-Stokes: Uniqueness

Introduction

Having established global regularity for the Navier-Stokes equations on $S^2$ and the coupled $M^4\times S^2$ system (Chapter 99), this chapter addresses the remaining components of well-posedness: uniqueness of solutions and continuous dependence on initial data. Together with existence and regularity, these properties constitute the full well-posedness result in the sense of Hadamard.

Scaffolding Interpretation

Scaffolding Interpretation. The uniqueness and continuous dependence results in this chapter apply to the $S^2$-coupled Navier-Stokes system. The $S^2$ structure is mathematical scaffolding (Part A), not a literal extra dimension. All physical predictions (well-posedness, stability estimates) are 4D observables.

Solution Uniqueness

Uniqueness on $S^2$

Theorem 133.1 (Uniqueness of Smooth Solutions on $S^2$)

Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be two smooth solutions of the Navier-Stokes equations on $S^2$ with the same initial data $\mathbf{v}_0$ and forcing $\mathbf{f}$. Then $\mathbf{v}_1 = \mathbf{v}_2$ for all $t \geq 0$.

Proof.

Step 1: Define the difference $\mathbf{w} = \mathbf{v}_1 - \mathbf{v}_2$. This satisfies:

$$ \frac{\partial\mathbf{w}}{\partial t} + \nabla_{\mathbf{v}_1}\mathbf{w} + \nabla_{\mathbf{w}}\mathbf{v}_2 = -\nabla(p_1 - p_2) + \nu\,\Delta\mathbf{w} $$ (133.1)

with $\mathbf{w}(\cdot,0) = 0$ and $\text{div}\,\mathbf{w} = 0$.

Step 2: Take the $L^2$ inner product with $\mathbf{w}$:

$$ \frac{1}{2}\frac{d}{dt}\|\mathbf{w}\|_{L^2}^2 = -\nu\|\nabla\mathbf{w}\|_{L^2}^2 - \int_{S^2}\mathbf{w}\cdot\nabla_{\mathbf{w}}\mathbf{v}_2\,d\Omega $$ (133.2)

The term $\int_{S^2}\mathbf{w}\cdot\nabla_{\mathbf{v}_1}\mathbf{w}\,d\Omega = 0$ by incompressibility.

Step 3: Estimate the nonlinear term. On $S^2$, $\mathbf{v}_2$ is smooth (Chapter 99), so $\nabla\mathbf{v}_2$ is bounded:

$$ \left|\int_{S^2}\mathbf{w}\cdot\nabla_{\mathbf{w}}\mathbf{v}_2\,d\Omega\right| \leq \|\nabla\mathbf{v}_2\|_{L^\infty}\|\mathbf{w}\|_{L^2}^2 $$ (133.3)

Step 4: Combining:

$$ \frac{d}{dt}\|\mathbf{w}\|_{L^2}^2 \leq 2\|\nabla\mathbf{v}_2\|_{L^\infty}\|\mathbf{w}\|_{L^2}^2 $$ (133.4)

Step 5: By Gronwall's inequality, since $\|\mathbf{w}(0)\|_{L^2} = 0$:

$$ \|\mathbf{w}(t)\|_{L^2}^2 \leq \|\mathbf{w}(0)\|_{L^2}^2 \exp\left(2\int_0^t\|\nabla\mathbf{v}_2(\cdot,s)\|_{L^\infty}\,ds\right) = 0 $$ (133.5)

Therefore $\mathbf{v}_1 = \mathbf{v}_2$ for all $t \geq 0$. (See: Standard energy method; cf. Constantin & Foias (1988)) □

Polar Field Form of the Uniqueness Argument

In the polar field variable $u = \cos\theta$, the energy method for uniqueness operates on the flat rectangle $[-1,+1]\times[0,2\pi)$ with two key simplifications.

Flat-measure $L^2$ norm. The $L^2$ norm of the difference $\mathbf{w} = \mathbf{v}_1 - \mathbf{v}_2$ on $S^2$ becomes:

$$ \|\mathbf{w}\|_{L^2(S^2)}^2 = \int_{-1}^{+1}\!\int_0^{2\pi} |\mathbf{w}(u,\phi)|^2\,du\,d\phi $$ (133.6)

with no angular weight—the flat measure $du\,d\phi$ makes this a standard Lebesgue integral on a rectangle.

Gradient decomposition. The covariant gradient on $S^2$ decomposes into THROUGH and AROUND components through the metric:

$$ |\nabla\mathbf{v}|^2 = h^{uu}\left|\frac{\partial\mathbf{v}}{\partial u}\right|^2 + h^{\phi\phi}\left|\frac{\partial\mathbf{v}}{\partial\phi}\right|^2 = \frac{1-u^2}{R^2}\left|\frac{\partial\mathbf{v}}{\partial u}\right|^2 + \frac{1}{R^2(1-u^2)}\left|\frac{\partial\mathbf{v}}{\partial\phi}\right|^2 $$ (133.7)

The THROUGH component ($\partial_u$) is weighted by $(1-u^2)/R^2$, which vanishes at the poles ($u = \pm 1$) but is compensated by the flat measure. The AROUND component ($\partial_\phi$) is weighted by $1/(R^2(1-u^2))$, which diverges at the poles—providing the curvature-enhanced dissipation identified in Chapter 99.

The critical Gronwall bound $\|\nabla\mathbf{v}_2\|_{L^\infty}$ thus separates:

$$ \|\nabla\mathbf{v}_2\|_{L^\infty}^2 = \sup_{(u,\phi) \in \mathcal{R}} \left[ \frac{1-u^2}{R^2}\left|\partial_u\mathbf{v}_2\right|^2 + \frac{1}{R^2(1-u^2)}\left|\partial_\phi\mathbf{v}_2\right|^2 \right] $$ (133.8)

This is bounded because: (1) the vorticity is bounded (Theorem thm:ch99-vorticity-max), which controls $\partial_u$ and $\partial_\phi$ derivatives via elliptic regularity on the compact interval $[-1,+1]$; and (2) the metric factors $h^{ij}$ are bounded on any compact subset of the open rectangle $(-1,+1)\times[0,2\pi)$, with the pole singularities integrable under the flat measure.

Quantity	Spherical $(\theta, \phi)$	Polar $(u, \phi)$
$L^2$ norm	$\int\|\mathbf{w}\|^2\sin\theta\,d\theta\,d\phi$	$\int\|\mathbf{w}\|^2\,du\,d\phi$ (flat)
[4pt] THROUGH gradient	$\frac{1}{R^2}\|\partial_\theta\mathbf{v}\|^2$	$\frac{1-u^2}{R^2}\|\partial_u\mathbf{v}\|^2$
[4pt] AROUND gradient	$\frac{1}{R^2\sin^2\!\theta}\|\partial_\phi\mathbf{v}\|^2$	$\frac{1}{R^2(1-u^2)}\|\partial_\phi\mathbf{v}\|^2$
[4pt] Product property	$h^{uu} \cdot h^{\phi\phi} = 1/R^4$	Same: $\frac{1-u^2}{R^2}\cdot\frac{1}{R^2(1-u^2)} = \frac{1}{R^4}$
[4pt] Gronwall bound	Controlled by bounded vorticity	Same, via Legendre polynomial spectrum

The key structural insight: the product $h^{uu}\cdot h^{\phi\phi} = 1/R^4$ is constant (because $\sqrt{\det h} = R^2$ is constant), so the reciprocal trade-off between THROUGH and AROUND gradient weights is exact. Where the THROUGH gradient is large (equator, $u = 0$), the AROUND gradient is small, and vice versa. This reciprocal balance, invisible in spherical coordinates, is the geometric mechanism ensuring that no single direction can develop unbounded gradients.

Scaffolding Interpretation

Scaffolding note: The polar field variable $u = \cos\theta$ is a coordinate choice, not a new physical assumption. The uniqueness proof is coordinate-independent. The polar form reveals the THROUGH/AROUND gradient balance ($h^{uu} \cdot h^{\phi\phi} = 1/R^4$ constant) that geometrically prevents gradient blow-up, making the Gronwall estimate transparent.

**Figure 133.1:** Uniqueness mechanism on $S^2$ in polar field coordinates. **Left:** Two solutions $\mathbf{v}_1$ (solid) and $\mathbf{v}_2$ (dashed) with difference $\mathbf{w}$. **Right:** On the polar rectangle, the $L^2$ norm uses flat measure $du\,d\phi$ (no angular weight). The THROUGH gradient $h^{uu} = (1{-}u^2)/R^2$ is large at the equator ($u=0$) and small at the poles, while the AROUND gradient $h^{\phi\phi} = 1/(R^2(1{-}u^2))$ is the reverse. Their product $h^{uu}\cdot h^{\phi\phi} = 1/R^4$ is constant—the reciprocal balance geometrically prevents gradient blow-up and ensures the Gronwall bound controls the difference for all time.

Uniqueness for the Coupled System

Theorem 133.2 (Uniqueness for the TMT-Coupled System)

The smooth solution of the coupled Navier-Stokes system on $\Omega \times S^2$ established in Theorem thm:ch99-coupled-regularity is unique.

Proof.[Proof sketch]

Apply the same energy difference technique to both the 4D and $S^2$ sectors simultaneously. The coupling terms contribute additional terms of the form:

$$ \left|\int\mathbf{w}_{4D}\cdot(\mathbf{F}[\mathbf{v}_{S^2}^{(1)}] - \mathbf{F}[\mathbf{v}_{S^2}^{(2)}])\,dx\right| \leq C\|\mathbf{w}_{4D}\|_{L^2}\|\mathbf{w}_{S^2}\|_{L^2} $$ (133.9)

where Lipschitz continuity of $\mathbf{F}$ is used. The combined energy $\|\mathbf{w}_{4D}\|_{L^2}^2 + \|\mathbf{w}_{S^2}\|_{L^2}^2$ satisfies a Gronwall inequality with zero initial data, yielding $\mathbf{w} \equiv 0$. (See: Theorem thm:ch99-coupled-regularity) □

Continuous Dependence on Data

Stability Estimate on $S^2$

Theorem 133.3 (Continuous Dependence on $S^2$)

Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be smooth solutions on $S^2$ with initial data $\mathbf{v}_0^{(1)}$ and $\mathbf{v}_0^{(2)}$ respectively. Then:

$$ \|\mathbf{v}_1(t) - \mathbf{v}_2(t)\|_{L^2}^2 \leq \|\mathbf{v}_0^{(1)} - \mathbf{v}_0^{(2)}\|_{L^2}^2 \exp\left(2\int_0^t\|\nabla\mathbf{v}_2(\cdot,s)\|_{L^\infty}\,ds\right) $$ (133.10)

Proof.

This follows identically to the uniqueness proof (Theorem thm:ch100-uniqueness-S2), except the initial difference is nonzero: $\|\mathbf{w}(0)\|_{L^2} = \|\mathbf{v}_0^{(1)} - \mathbf{v}_0^{(2)}\|_{L^2}$. The Gronwall inequality then gives the stated bound. (See: Theorem thm:ch100-uniqueness-S2) □

Polar Field Form of the Stability Exponent

The stability bound eq:ch100-stability has exponential growth controlled by $\int_0^t\|\nabla\mathbf{v}_2\|_{L^\infty}\,ds$. In polar variables, this integral inherits the spectral structure of the Legendre polynomial eigenvalues on $[-1,+1]$.

Since $\mathbf{v}_2$ decomposes into modes $P_\ell^{|m|}(u)\,e^{im\phi}$ (Chapter 99, §sec:ch99-polar-poincare), each mode decays at rate $\gamma_\ell = \nu\ell(\ell+1)/R^2$. The gradient norm therefore satisfies:

$$ \|\nabla\mathbf{v}_2(\cdot,t)\|_{L^\infty} \leq C\,\|\omega_0\|_{L^\infty}\,e^{-\gamma_{\ell_{\min}} t} = C\,\|\omega_0\|_{L^\infty}\,e^{-2\nu t/R^2} $$ (133.11)

where $\ell_{\min} = 1$ gives $\gamma_1 = 2\nu/R^2$—the Legendre polynomial spectral gap on $[-1,+1]$.

The stability exponent then evaluates to:

$$ 2\int_0^t\|\nabla\mathbf{v}_2\|_{L^\infty}\,ds \leq \frac{C\,R^2\,\|\omega_0\|_{L^\infty}}{\nu} \bigl(1 - e^{-2\nu t/R^2}\bigr) \leq \frac{C\,R^2\,\|\omega_0\|_{L^\infty}}{\nu} $$ (133.12)

which is bounded uniformly in $t$—this is the polar form of the statement that perturbations cannot grow without bound on $S^2$. The bound $C R^2\|\omega_0\|_{L^\infty}/\nu$ is the Grashof number, and its finiteness follows from the compactness of $[-1,+1]$ (the THROUGH interval is finite) and the periodicity of $[0,2\pi)$ (the AROUND direction wraps).

Corollary 133.7 (Exponential Stability for Small Perturbations)

For unforced flow ($\mathbf{f} = 0$) on $S^2$, since $\|\nabla\mathbf{v}_2\|_{L^\infty}$ decays exponentially (from Theorems thm:ch99-energy-dissipation and thm:ch99-global-smoothness), the difference $\|\mathbf{v}_1 - \mathbf{v}_2\|_{L^2}$ also decays at late times. The zero solution is globally asymptotically stable.

Continuous Dependence on Forcing

Theorem 133.4 (Continuous Dependence on Forcing)

Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be solutions with the same initial data but different forcings $\mathbf{f}_1$ and $\mathbf{f}_2$. Then:

$$ \|\mathbf{v}_1(t) - \mathbf{v}_2(t)\|_{L^2}^2 \leq \frac{R^2}{2\nu}\int_0^t\|\mathbf{f}_1 - \mathbf{f}_2\|_{L^2}^2\,ds $$ (133.13)

Proof.

The difference $\mathbf{w} = \mathbf{v}_1 - \mathbf{v}_2$ satisfies the same equation as before plus the forcing difference. Using Young's inequality and the Poincaré inequality:

$$\begin{aligned} \frac{d}{dt}\|\mathbf{w}\|_{L^2}^2 &\leq -\frac{4\nu}{R^2}\|\mathbf{w}\|_{L^2}^2 + \frac{R^2}{2\nu}\|\mathbf{f}_1 - \mathbf{f}_2\|_{L^2}^2 \end{aligned}$$ (133.14)

Integrating gives the result. □

Well-Posedness

The Complete Well-Posedness Theorem

Combining the results of Chapters 99 and 100:

Theorem 133.5 (Hadamard Well-Posedness on $S^2$)

The incompressible Navier-Stokes equations on $S^2$ with $\nu > 0$ are well-posed in the sense of Hadamard:

Existence: For any smooth, divergence-free initial data $\mathbf{v}_0$, a smooth solution exists for all $t > 0$ (Theorem thm:ch99-global-smoothness).
Uniqueness: The solution is unique (Theorem thm:ch100-uniqueness-S2).
Continuous dependence: The solution depends continuously on the initial data and forcing (Theorems thm:ch100-continuous-dep and thm:ch100-forcing-dep).

Well-Posedness for the Coupled System

Theorem 133.6 (Well-Posedness for TMT-Coupled System)

The coupled Navier-Stokes system on $\Omega \times S^2$ (with $\Omega \subset \mathbb{R}^3$ bounded, smooth boundary, $\nu > 0$, and Lipschitz coupling) is well-posed in the sense of Hadamard.

Proof.[Proof summary]

Existence: Theorem thm:ch99-coupled-regularity
Uniqueness: Theorem thm:ch100-uniqueness-coupled
Continuous dependence: Extends Theorem thm:ch100-continuous-dep to the coupled system by the same Gronwall argument applied to the total energy $\|\mathbf{w}_{4D}\|_{L^2}^2 + \|\mathbf{w}_{S^2}\|_{L^2}^2$.

□

Chapter Summary

Key Result

Navier-Stokes: Uniqueness and Continuity

Smooth solutions of the Navier-Stokes equations on $S^2$ are unique (proven by the energy difference method with Gronwall inequality) and depend continuously on initial data and forcing. Together with the global existence and regularity results of Chapter 99, this establishes complete Hadamard well-posedness for the Navier-Stokes equations on $S^2$ and for the coupled $M^4 \times S^2$ system on bounded spatial domains.

Polar field verification: In the polar variable $u = \cos\theta$, the $L^2$ norm uses flat measure $du\,d\phi$ (no angular weight), and the covariant gradient decomposes into THROUGH ($h^{uu} = (1{-}u^2)/R^2$) and AROUND ($h^{\phi\phi} = 1/(R^2(1{-}u^2))$) contributions whose product $h^{uu}\cdot h^{\phi\phi} = 1/R^4$ is constant. This reciprocal balance prevents gradient blow-up and makes the Gronwall uniqueness estimate transparent. The stability exponent is bounded uniformly by the Grashof number $CR^2\|\omega_0\|_{L^\infty}/\nu$, with decay rate set by the Legendre spectral gap $\gamma_1 = 2\nu/R^2$ on $[-1,+1]$ (§sec:ch100-polar-uniqueness, §sec:ch100-polar-stability; Figure fig:ch100-polar-uniqueness).

Table 133.1: Chapter 100 results summary

Result	Value	Status	Reference
Uniqueness on $S^2$	Energy method	PROVEN	Thm thm:ch100-uniqueness-S2
Coupled uniqueness	Gronwall	PROVEN	Thm thm:ch100-uniqueness-coupled
Continuous dependence	Stability bound	PROVEN	Thm thm:ch100-continuous-dep
Forcing stability	$\propto R^2/(2\nu)$	PROVEN	Thm thm:ch100-forcing-dep
Polar dual verification	Flat $L^2$, reciprocal metric balance	PROVEN	§sec:ch100-polar-uniqueness
Full well-posedness	Hadamard	PROVEN	Thm thm:ch100-well-posed

Verification Code

The mathematical derivations and proofs in this chapter can be independently verified using the formal and computational scripts below.

◆ Lean 4 Formal proof verification Coming Soon ● Python Numerical verification & plots Coming Soon ★ Mathematica Symbolic computation & CAS verification Coming Soon

All verification code is open source. See the complete verification index for all chapters.

Quantity	Spherical \((\theta, \phi)\)	Polar \((u, \phi)\)
\(L^2\) norm	\(\int\|\mathbf{w}\|^2\sin\theta\,d\theta\,d\phi\)	\(\int\|\mathbf{w}\|^2\,du\,d\phi\) (flat)
[4pt] THROUGH gradient	\(\frac{1}{R^2}\|\partial_\theta\mathbf{v}\|^2\)	\(\frac{1-u^2}{R^2}\|\partial_u\mathbf{v}\|^2\)
[4pt] AROUND gradient	\(\frac{1}{R^2\sin^2\!\theta}\|\partial_\phi\mathbf{v}\|^2\)	\(\frac{1}{R^2(1-u^2)}\|\partial_\phi\mathbf{v}\|^2\)
[4pt] Product property	\(h^{uu} \cdot h^{\phi\phi} = 1/R^4\)	Same: \(\frac{1-u^2}{R^2}\cdot\frac{1}{R^2(1-u^2)} = \frac{1}{R^4}\)
[4pt] Gronwall bound	Controlled by bounded vorticity	Same, via Legendre polynomial spectrum

Navier-Stokes: Uniqueness

Introduction

Solution Uniqueness

Uniqueness on \(S^2\)

Polar Field Form of the Uniqueness Argument

Uniqueness for the Coupled System

Continuous Dependence on Data

Stability Estimate on \(S^2\)

Polar Field Form of the Stability Exponent

Continuous Dependence on Forcing

Well-Posedness

The Complete Well-Posedness Theorem

Well-Posedness for the Coupled System

Chapter Summary

Verification Code