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Normalization, periodization, and derivative commutation for the positive fluid filter

Statement

Assume the Axiom of Countable Choice required by the normalized periodic torus convention. Let $d \in {1, 2, 3}$ , $L > 0$ , $T_{L}^{d} = (R / L Z)^{d}$ , and $d μ_{L} = L^{- d} d x$ . Let $K : R^{d} \to [0, \infty)$ be smooth, supported in a compact subset of $B_{1} (0)$ , with $\int K (s) d s = 1$ , and let $0 < ℓ < L / 2$ . Define $G_{ℓ} (r) = \sum_{n \in Z^{d}} ℓ^{- d} K ((r + L n) / ℓ)$ and ${\overline{q}}_{ℓ} (x) = \int_{T_{L}^{d}} G_{ℓ} (r) q (x - r) d x$ . The locally finite sum defining $G_{ℓ}$ is smooth, nonnegative, and $L$ -periodic, and $\int_{T_{L}^{d}} G_{ℓ} (r) d x = 1$ . For every smooth periodic finite-dimensional field $q$ , the filter also has the Euclidean representation

${\overline{q}}_{ℓ} (x, t) = \int_{R^{d}} K (s) q (x - ℓ s, t) d s,$

where $q$ on the right is its periodic lift. If $I \subseteq R$ is open and each component of $q : I \times T_{L}^{d} \to R^{m}$ and of its time derivative has continuous spatial derivatives of every order, then the filtered field has the same regularity and, for every spatial multi-index $β$ ,

$\partial_{x}^{β} {\overline{q}}_{ℓ} = {\overline{\partial_{x}^{β} q}}_{ℓ}, \partial_{t} \partial_{x}^{β} {\overline{q}}_{ℓ} = {\overline{\partial_{t} \partial_{x}^{β} q}}_{ℓ} .$

For every $γ > 0$ , the kernel moments

$M_{γ} : = \int_{R^{d}} K (s) ∣ s ∣^{γ} d s, N_{γ} : = \max_{1 \leq j \leq d} \int_{R^{d}} ∣ \partial_{j} K (s) ∣ ∣ s ∣^{γ} d s$

are finite.

Statement references

Positive periodic convolution filter and subfilter stress

Countable Choice axiom

A compactly supported Riemann integrand admits the global change-of-variables formula from a diffeomorphism near the relevant compact preimage

Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral

Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections

University of Washington, ME 544, Large-Eddy Simulation lecture notes, §2, printed pp. 5–6, filter definition and commutation conditions

Gregory L. Eyink, The ‘Multifractal Model’ of Turbulence and A Priori Estimates in Large-Eddy Simulation, §II, printed pp. 5–7, increment covariance and filter-moment estimates

Facts & Assumptions

Given: Countable Choice and the parameters and fields in the statement, with $d \in {1, 2, 3}$ , $L > 0$ , $0 < ℓ < L / 2$ , and $K$ smooth, nonnegative, compactly supported in $B_{1} (0)$ , and of mass one.

[A1]

Adopt Countable Choice: every countable family of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )); the Fourier-Sobolev torus convention is stated and proved under this axiom.

[F1]

Positive periodic convolution filter and subfilter stress: $G_{ℓ} (r) = \sum_{n \in Z^{d}} ℓ^{- d} K ((r + L n) / ℓ)$ and ${\overline{q}}_{ℓ} (x) = \int_{T_{L}^{d}} G_{ℓ} (r) q (x - r) d x$ .

[F2]

A compactly supported Riemann integrand admits the global change-of-variables formula from a diffeomorphism near the relevant compact preimage: let $n \geq 1$ , let $V \subseteq R^{n}$ be open, and let $g : V \to R^{n}$ be injective and $C^{1}$ with invertible derivative throughout $V$ . If $f : R^{n} \to R$ is compactly supported Riemann integrable and $supp f \subseteq g (V)$ , then the function equal to $f (g (s)) ∣ \det D g (s) ∣$ on $V$ and zero outside $V$ is compactly supported Riemann integrable, and its integral over $R^{n}$ equals $\int_{R^{n}} f$ .

[F3]

Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral: if the integrand and its parameter derivative extend continuously to a compact rectangle, then the derivative of its one-dimensional Riemann integral is the integral of that derivative.

[F4]

Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections: an integrable function on a product rectangle with integrable sections has equal multiple and iterated Riemann integrals.

[F5]

Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ : the multidimensional Riemann integral is linear and order preserving, and $∣ \int f ∣ \leq \int ∣ f ∣$ for integrable $f$ .

[F6]

Riemann's criterion on a nondegenerate rectangle in $R^{m}$ : integrability is equivalent to arbitrarily small Darboux gaps: a bounded function on a nondegenerate rectangle is Riemann integrable when its Darboux gaps can be made arbitrarily small.

[F7]

Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line: closed boxes in $R^{d}$ are compact.

[F8]

Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous: a continuous function on a compact metric space is uniformly continuous.

[F9]

Lower and upper Darboux sums over a grid partition in $R^{m}$ : the Darboux gap is the finite sum of each cell's oscillation multiplied by its volume; Grid partitions of a rectangle in $R^{m}$ , their cells, refinements and mesh: the cells cover the box and their volumes sum to its volume; Laws of finite sums and finite products: finite sums preserve inequalities.

[F10]

For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ : for each $η > 0$ there is $N \geq 1$ with $1 / N < η$ , so a compact box admits uniform grids of arbitrarily small mesh.

Proof

1.1A1F1givenalgebra

For $r$ in a bounded period cell, a nonzero summand requires $∣ r + L n ∣ < ℓ < L / 2$ . Only finitely many integer vectors $n$ meet this bound, uniformly on the cell, so the sum is locally finite there; all its derivatives are locally finite sums as well. Replacing $r$ by $r + L m$ and reindexing $n$ by $n - m$ proves periodicity, and each summand is nonnegative.

1.2F6F7F8F9F10given

Every continuous integrand used below is supported in a fixed nondegenerate compact box $Q$ (enlarge the support box if necessary). By [F7] and [F8], it is bounded and uniformly continuous on $Q$ . Write $V = vol (Q) > 0$ and let $δ > 0$ be a uniform-continuity radius for tolerance $ε / (2 V)$ . Choose $N \geq 1$ with $1 / N < δ / (2 R \sqrt{d})$ by [F10], where $2 R$ is the side length of a box containing $Q$ , and divide each side into $N$ equal parts. Each cell's Euclidean diameter is at most $2 R \sqrt{d} / N < δ$ , so each cell oscillation is below that tolerance. By [F9], the Darboux gap is at most $(ε / (2 V)) \sum_{i} vol (Q_{i}) = ε / 2 < ε$ ; the criterion [F6] makes the integrand Riemann integrable. This applies componentwise to the compactly supported kernel factors and their products with the periodic field and its derivatives.

2.1F2F5step 1.1step 1.2

Choose the centered closed period cell $Q = [- L / 2, L / 2]^{d}$ . For $r \in Q$ and $n \in Z^{d} ∖ {0}$ , one coordinate of $r + L n$ has absolute value at least $L / 2 > ℓ$ , so its kernel summand is zero. Thus on $Q$ the periodized kernel is $f (r) = ℓ^{- d} K (r / ℓ)$ , whose support $ℓ supp K$ is compactly contained in $B_{ℓ} (0) \subset int Q$ . Its period-cell integral is therefore its integral over $R^{d}$ . Step 1.2 gives compact-support Riemann integrability. Apply [F2] on the open domain $V = R^{d}$ with $g (s) = ℓ s$ . This map is a global $C^{1}$ diffeomorphism with image $R^{d}$ , $∣ \det D g ∣ = ℓ^{d}$ , and hence $supp f \subseteq g (V)$ ; zero extension outside $V$ is vacuous. The transformed integrand is $K (s)$ , giving $\int_{T_{L}^{d}} G_{ℓ} = \int_{R^{d}} K = 1$ .

2.2F5step 1.2

For $γ > 0$ , the functions $K (s) ∣ s ∣^{γ}$ and $∣ \partial_{j} K (s) ∣ ∣ s ∣^{γ}$ are continuous, bounded, and compactly supported. Step 1.2 makes them Riemann integrable, and their integrals are finite by boundedness on a compact box; the finite maximum over $j$ is finite as well.

3.1F1F2F5step 1.1step 1.2step 2.1

With the additional smooth periodic factor $q (x - r, t)$ , the centered-cell argument in step 2.1 identifies the convolution with $\int_{R^{d}} f_{x, t} (y) d y$ , where $f_{x, t} (y) = ℓ^{- d} K (y / ℓ) q (x - y, t)$ , componentwise. Each such integrand has support contained in the same compact set $ℓ supp K$ and is Riemann integrable by step 1.2. Apply [F2] with the same global dilation $g (s) = ℓ s$ on $V = R^{d}$ ; its image contains this support, its derivative is invertible, and zero extension outside $V$ is vacuous. The transformed integrand is $K (s) q (x - ℓ s, t)$ , proving the stated Euclidean representation.

4.1F3F4step 1.2step 3.1

On a compact time interval inside $I$ and a compact parameter interval for any coordinate $x_{j}$ , the Euclidean representation has a continuous integrand and continuous parameter derivative on the fixed compact support box of $K$ . By [F4] the multiple integral is an iterated integral; applying [F3] to each one-dimensional factor and then [F4] again passes $\partial_{t}$ and $\partial_{x_{j}}$ through the integral. Repeating this argument for each spatial derivative gives the stated multi-index identities and continuous derivatives.

5.1step 1.1step 2.1step 3.1step 4.1step 2.2∎

Steps 1.1–4.1 establish the kernel, convolution, derivative, and moment claims.

Remarks

The scale change $x = L y$ sends normalized Haar measure on $T_{L}^{d}$ to normalized Haar measure on the unit torus. In particular, $L = 2 π$ gives the current $x = 2 π y$ Fourier convention. The lemma is a smooth periodic statement; it does not assert a commutator estimate for spatially varying filters or boundary-truncated kernels.

Normalization, periodization, and derivative commutation for the positive fluid filter

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

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