Physics theoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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Weak continuation agrees with smooth periodic evolution

Physical scope

Smooth divergence-free initial data for the three-dimensional constant- density, constant-viscosity, unforced periodic model. Every Leray–Hopf weak continuation agrees with the smooth solution only on its local smooth interval. No global 3D regularity or uniqueness among arbitrary weak solutions is claimed.

Statement

Let $L > 0$ be the numerical period in the declared SI metre unit, with physical side length $L m$ ; let $ρ_{0} > 0$ in $k g m^{- 3}$ , $μ > 0$ in $P a s$ , and $ν = μ / ρ_{0} > 0$ in $m^{2} s^{- 1}$ . Let $s \geq 4$ be an integer and let $u_{0} \in H_{σ}^{s} (T_{L}^{3}; R^{3}) \cap C^{\infty} (T_{L}^{3}; R^{3})$ be a divergence-free initial velocity in $m s^{- 1}$ . Here $H_{σ}^{s}$ consists of real Fourier-weighted fields with dimensionless weights $(1 + L^{2} ∣ κ_{k} ∣^{2})^{s}$ and $k \cdot {\hat{u}}_{0} (k) = 0$ for $k \neq 0$ , with the constant Fourier mode unrestricted. Assume Countable Choice and Dependent Choice. Then there are $T > 0$ , a smooth periodic velocity $v \in C^{\infty} ([0, T] \times T_{L}^{3}; R^{3})$ , and a smooth mean-zero kinematic pressure $π_{v}$ , with $v (0) = u_{0}$ , such that $\partial_{t} v + div (v \otimes v) + \nabla π_{v} = ν Δ v, div v = 0.$ Every global unforced periodic Leray–Hopf weak continuation $u$ of the same model and data satisfies $u (t) = v (t)$ in $H$ for all $t \in [0, T]$ . The physical pressure is $p_{v} = ρ_{0} π_{v}$ in $P a$ . The conclusion applies only while this smooth branch exists; it does not assert global smoothness or uniqueness among arbitrary weak solutions.

Statement references

Published local smooth periodic Euler/viscous existence theorem on the unit torus

Published smooth mean-zero periodic pressure reconstruction

Conditional weak–smooth uniqueness on the smooth interval

Lenya Ryzhik, Lecture Notes for Math 256B, Version 2024, §6.1, printed pp. 53–54

Berselli and Spirito, On the Existence of Leray–Hopf Weak Solutions to the Navier–Stokes Equations, §3.1 and §4, printed pp. 5–6 and 8–12

Facts & Assumptions

Given: The periodic model parameters, a smooth divergence-free initial velocity on $T_{L}^{3}$ , Countable Choice and Dependent Choice.

[A1]

Weak continuation of the adopted periodic viscous model: under Countable Choice, a global weak periodic Leray–Hopf velocity with the same $ν$ and $u_{0}$ exists.

[F1]

Local smooth solutions of the periodic incompressible Euler and constant-viscosity equations on $T^{3}$ : on the unit torus $T^{3} = (R / Z)^{3}$ defined by Periodic vector Sobolev spaces on the three-torus, the weight family, and the Leray projection, smooth divergence-free data in $H_{σ}^{s} \cap C^{\infty}$ , integer $s \geq 4$ , and $ν \geq 0$ give a smooth projected-equation solution on a positive interval; its proof assumes Countable Choice.

[F2]

Reconstruction of the periodic pressure from the velocity: $\nabla p = - (I - P) ((u \cdot \nabla) u)$ and $Δ p = - \sum_{i, j} \partial_{i} \partial_{j} (u_{i} u_{j})$ : for a smooth divergence-free solution of the projected equation on $T^{3}$ , the Fourier pressure with zero mean is smooth and satisfies $\nabla π = - (I - P) ((v \cdot \nabla) v)$ ; it is the unique smooth mean-zero pressure.

[F3]

Weak–smooth uniqueness on a smooth interval: this draft mathematical supplier compares a global unforced periodic Leray–Hopf velocity $u$ with a smooth divergence-free velocity $v \in C^{\infty} ([0, T] \times T_{L}^{d}; R^{d})$ and smooth pressure $q \in C^{\infty} ([0, T] \times T_{L}^{d})$ solving the same equation with the same initial data and viscosity; its proof gives $u (t) = v (t)$ in $H$ on $[0, T]$ . We use that comparison under the Statement's Countable Choice and Dependent Choice assumptions: the supplier's absolutely-continuous Grönwall argument uses both principles when applying the fundamental theorem for absolutely continuous functions. It is conditional on the given smooth pair and does not assert its existence.

[F4]

Periodic solenoidal Fourier spaces: $T_{L}^{3}$ uses normalized measure $L^{- 3} d x$ , $H^{s}$ has the stated dimensionless Fourier weights, and $H$ has unrestricted mean mode.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every countable family of nonempty sets has a choice function.

[A3]

The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain: for an entire relation on a nonempty set and a specified start, an $N$ -indexed chain exists.

Proof

Given: The data and assumptions in the Statement, together with [A1]–[A3] and [F1]–[F4].

1.1F1F4A2

Define $y = x / L$ , $τ = t / L^{2}$ , and ${\tilde{u}}_{0} (y) = L u_{0} (L y)$ . The period- $L$ Fourier weights $(1 + L^{2} ∣ κ_{k} ∣^{2})^{s}$ , with $κ_{k} = 2 π k / L$ , equal the unit-torus weights $(1 + 4 π^{2} ∣ k ∣^{2})^{s}$ , so [F4] makes ${\tilde{u}}_{0}$ a smooth divergence-free datum in the required $H_{σ}^{s}$ . By [F1] and [A2], there is a smooth projected-equation solution $\tilde{v}$ on some interval $[0, \tilde{T}]$ for the same $ν$ . Set $v (t, x) = L^{- 1} \tilde{v} (t / L^{2}, x / L)$ ; each time, convective, and viscous term acquires the same factor $L^{- 3}$ , so $v$ is a smooth solution on $[0, T]$ , where $T = L^{2} \tilde{T} > 0$ , with $v (0) = u_{0}$ .

2.1F2step 1.1

Apply [F2] to $\tilde{v}$ to obtain a smooth mean-zero pressure $\tilde{π}$ for its primitive equation. The rescaled field $π_{v} (t, x) = L^{- 2} \tilde{π} (t / L^{2}, x / L)$ is smooth, has mean zero, and its gradient scales by $L^{- 3}$ ; therefore $v, π_{v}$ satisfy the full equation in the Statement.

3.1A1F4step 1.1step 2.1

By [A1], choose a global weak Leray–Hopf velocity $u$ with the same initial datum and viscosity as $v$ . The exact Fourier convention and initial trace are those in [F4], and both $u$ and $v$ solve the same periodic model.

4.1F3A2A3step 3.1step 1.1step 2.1∎

The hypotheses of [F3] hold: $u$ is global Leray–Hopf by step 3.1, $v$ and $π_{v}$ are smooth by steps 1.1–2.1, and they share $u_{0}$ and $ν$ . With the exact Countable and Dependent Choice assumptions [A2]–[A3], the weak–smooth comparison gives $u (t) = v (t)$ in $H$ for every $t \in [0, T]$ . This is restricted to the smooth interval.

Weak continuation agrees with smooth periodic evolution

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources