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Projected no-slip Stokes Hilbert space

Definition

Assume the full Axiom of Choice. Let $d \geq 2$ , let $R > 0$ , and let $Ω \subset (- R, R)^{d}$ be a nonempty bounded open set. Define $W^{1, 2} (Ω; R^{d})$ componentwise: $v = (v_{1}, \dots, v_{d})$ belongs to it exactly when each $v_{i}$ belongs to the scalar $W^{1, 2} (Ω; R)$ defined in the earlier integer-order Sobolev-space definition. Its weak derivatives are $\partial_{j} v_{i}$ , its gradient is the matrix $\nabla v = (\partial_{j} v_{i})_{i, j = 1}^{d}$ , and its norm is $∥ v ∥_{W^{1, 2} (Ω; R^{d})}^{2} = \sum_{i = 1}^{d} (∥ v_{i} ∥_{L^{2} (Ω)}^{2} + \sum_{j = 1}^{d} ∥ \partial_{j} v_{i} ∥_{L^{2} (Ω)}^{2}) = \int_{Ω} (∣ v ∣^{2} + ∣ \nabla v ∣_{F}^{2}) d x,$ where $v \cdot w = \sum_{i} v_{i} w_{i}$ , $∣ v ∣^{2} = \sum_{i} ∣ v_{i} ∣^{2}$ , $A : B = \sum_{i, j} A_{i j} B_{i j}$ , and $∣ A ∣_{F}^{2} = \sum_{i, j} ∣ A_{i j} ∣^{2}$ . Define $H_{0}^{1} (Ω; R^{d})$ as the closure of the $d$ -tuples of compactly supported smooth real functions in this norm. Define $div v = 0$ distributionally to mean $\int_{Ω} v \cdot \nabla ϕ = 0$ for every $ϕ \in C_{c}^{\infty} (Ω; R)$ , and set $V = {v \in H_{0}^{1} (Ω; R^{d}) : div v = 0}$ . Equip $V$ with the inherited $H^{1}$ inner product $\int_{Ω} (v \cdot w + \nabla v : \nabla w)$ . The phrase no-slip here means membership in this closure; no trace characterization on arbitrary $Ω$ is part of the definition.

Definition references

Martin Vohralík, A posteriori error estimates for efficiency and error control in numerical simulations, Course NM497 lecture notes (2010–2011).

Theo Bühler and Dietmar Salamon, Functional Analysis.

Facts & Assumptions

Given: The hypotheses stated in the formulation, including the explicitly adopted choice principle where indicated.

[F1]

The Axiom of Choice: every family of nonempty sets has a choice function; full AC is adopted for the Sobolev completeness supplier used here.

[F2]

Integer-order Sobolev spaces and their norms: $W^{1, 2}$ consists of $L^{2}$ classes with first weak derivatives in $L^{2}$ , with the sum-of-squares Sobolev norm.

[F3]

Integer-order Sobolev spaces are Banach: under its stated full Axiom of Choice hypothesis, the stated $W^{1, 2}$ space is complete.

[F4]

Real and complex inner-product spaces and their induced length: a real inner product is a bilinear, symmetric, positive-definite pairing; these axioms must be checked for the displayed Sobolev pairing.

[F5]

Hilbert space: a real inner-product space is Hilbert when its induced norm is complete.

Proof

Given: Full AC, $d \geq 2$ , and nonempty bounded open $Ω \subset (- R, R)^{d}$ with the weak-derivative convention in the Definition.

1.1givenF1F2F3F4algebra

First check the pairing. Every product in its finite sum is integrable because $2 ∣ a b ∣ \leq ∣ a ∣^{2} + ∣ b ∣^{2}$ for $L^{2}$ coordinates or weak derivatives $a, b$ . Null-set changes do not change its integrals, and [F2] supplies unique weak derivative classes. Adding and scaling the weak-derivative test identities proves linearity of weak differentiation, so linearity of integration and commutativity of real multiplication give bilinearity and symmetry. The pairing of $v$ with itself equals the displayed squared Sobolev norm. It is nonnegative and vanishes exactly when every component is the zero $L^{2}$ class, whose weak derivatives are zero. Thus it satisfies [F4], and its induced norm is exactly the vector Sobolev norm. By [F2] and [F3], each scalar component space $W^{1, 2} (Ω; R)$ is complete under the stated AC assumption. A Cauchy sequence in the finite product is Cauchy in each component; completeness gives a limit in each component, and the finite sum in the vector norm gives convergence to the vector of those limits. Thus the vector space is complete. Its closed linear subspace $H_{0}^{1}$ , defined by norm closure, is therefore complete in that same inner-product norm.

2.1givenstep 1.1algebra

For each fixed $ϕ \in C_{c}^{\infty} (Ω)$ the functional $v \mapsto \int_{Ω} v \cdot \nabla ϕ$ is continuous in the $H^{1}$ norm, since its absolute value is at most $∥ v ∥_{L^{2}} ∥ \nabla ϕ ∥_{L^{2}}$ . Its kernel is closed; the intersection of these kernels is exactly $V$ . Thus $V$ is a closed real linear subspace of $H_{0}^{1}$ and is complete.

3.1givenF4F5step 2.1∎

The inherited pairing is positive definite and complete on $V$ , so [F5] makes $V$ a real Hilbert space. The only boundary condition used is the defining closure $H_{0}^{1}$ ; no converse density or arbitrary-domain trace theorem has been invoked.

Projected no-slip Stokes Hilbert space

Definition

Definition references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources