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

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Bounded no-slip solenoidal closure spaces

Definition

Let $Ω \subset R^{3}$ be a nonempty bounded open set. Let $C_{c}^{\infty} (Ω; R^{3})$ denote smooth compactly supported vector fields and set $D_{σ} (Ω) = {a \in C_{c}^{\infty} (Ω; R^{3}) : div a = 0} .$ Let $H$ be the closure of $D_{σ} (Ω)$ in $L^{2} (Ω; R^{3})$ , and let $V$ be its closure in $H_{0}^{1} (Ω; R^{3})$ , where $H_{0}^{1}$ is the $H^{1}$ closure of $C_{c}^{\infty} (Ω; R^{3})$ and weak derivatives define the $H^{1}$ norm. Use $∥ a ∥_{H} = ∥ a ∥_{2}$ and $∥ a ∥_{V}^{2} = ∥ a ∥_{2}^{2} + ∥ \nabla a ∥_{2}^{2}$ . Under Countable Choice, each $V$ element has zero extension in $H^{1} (R^{3})$ and distributional divergence zero. This closure definition does not identify $V$ with every divergence-free member of $H_{0}^{1} (Ω; R^{3})$ on an arbitrary open set, and it makes no pointwise boundary trace assertion.

Definition references

Berselli and Spirito, On the Existence of Leray–Hopf Weak Solutions, §§2.1–2.2, pp. 4–5.

Leonid Ryzhik, Math 256B Lecture Notes, §5.2, pp. 39–43.

Facts & Assumptions

Given: A nonempty bounded open $Ω \subset R^{3}$ , the Sobolev and weak-derivative conventions in the Definition, and Countable Choice for the zero-extension and divergence conclusions.

[F1]

Integer-order Sobolev spaces and their norms and Zero-boundary Sobolev space as a norm closure: $H^{1}$ classes have $L^{2}$ weak derivatives, and $H_{0}^{1}$ is the closure of compactly supported smooth functions in the Sobolev norm.

[F2]

Weak derivative of a locally integrable function and Distributional derivative: weak derivatives agree with distributional derivatives when represented by locally integrable functions.

[F3]

Zero extension of W_0^{1,p} has no boundary derivative: under Countable Choice, zero extension maps $H_{0}^{1} (Ω)$ into $H^{1} (R^{3})$ and preserves weak derivatives.

Proof

1.1givenF1F2algebra

The set $D_{σ} (Ω)$ is a real vector subspace because the divergence operator is linear and sums and scalar multiples of compactly supported smooth fields remain compactly supported and smooth. Its closures in the two stated norms are therefore linear subspaces and define the sets $H$ and $V$ without a boundary trace convention.

2.1givenF1F2F3step 1.1∎

If $a_{n} \in D_{σ}$ converges to $a \in V$ in $H_{0}^{1}$ , then [F3] makes their zero extensions converge in $H^{1} (R^{3})$ and preserves their derivatives. Their distributional divergences converge to $div a$ when paired with any compact smooth scalar test by [F2]. Since each $div a_{n} = 0$ , the limit divergence is zero, and the zero extension represents the $H^{1}$ limit. This verifies the final properties in the Definition without claiming a converse density theorem.

Bounded no-slip solenoidal closure spaces

Definition

Definition references

Facts & Assumptions

Proof

Depends on

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