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Expected weak Reynolds decomposition and conditional mean equation

Statement

Assume the Axiom of Countable Choice. Let $(D, D, λ)$ , $d$ , and $H = L^{2} (D, λ; R^{d})$ be as in the Definition. Let $(Ω, F, P)$ be a probability space and let $U : Ω \to H$ be strongly measurable with $E ∥ U ∥_{H}^{2} < \infty$ . Assume a representative $\tilde{U} : Ω \times D \to R^{d}$ of its sections is measurable with respect to the product sigma-algebra $F \otimes D$ . The Bochner mean $\overset{ˉ}{u} : = \int_{Ω} U (ω) d P (ω) \in H$ is well-defined. For $λ$ -almost every $x$ , define $R (x) : = E ⁣ [(\tilde{U} (ω, x) - \overset{ˉ}{u} (x)) \otimes (\tilde{U} (ω, x) - \overset{ˉ}{u} (x))],$ where any measurable representative of $\overset{ˉ}{u}$ may be used. Then $R$ determines an element of $L^{1} (D, λ; R^{d \times d})$ , and for every measurable tensor field $Φ : D \to R^{d \times d}$ with ${ess sup}_{x \in D} ∣ Φ (x) ∣_{F} < \infty$ , $E ⁣ [\int_{D} (\tilde{U} \otimes \tilde{U}) : Φ d λ] = \int_{D} (\overset{ˉ}{u} \otimes \overset{ˉ}{u} + R) : Φ d λ .$ For almost every $x \in D$ , on one common full- $λ$ -measure set for all $z \in R^{d}$ , $R (x)$ is symmetric and $z^{T} R (x) z = E ⁣ [(z \cdot (\tilde{U} (ω, x) - \overset{ˉ}{u} (x)))^{2}] \geq 0.$ For vectors $a, b \in R^{d}$ , $(a \otimes b)_{i j} : = a_{i} b_{j}$ . Here $A : B : = \sum_{i, j = 1}^{d} A_{i j} B_{i j}$ and $∣ \cdot ∣_{F}$ is the Frobenius norm. If $U$ is assigned SI velocity units $m s^{- 1}$ , then $\overset{ˉ}{u}$ has units $m s^{- 1}$ and $R$ has units $m^{2} s^{- 2}$ ; the test field's units are chosen so both pairings have the same units. Here “weak” means tested against an essentially bounded tensor field. $R$ is the covariance tensor; no density factor, stress sign convention, incompressibility, Navier–Stokes equation, time-stationarity, or turbulence closure is assumed.

Conditional time-dependent weak equation. In addition, let $D \subset R^{d}$ , $d \in {2, 3}$ , be bounded and open, let $I = (a, b)$ be a bounded open interval, and let $ν \geq 0$ be fixed. Let $u, p, f$ be real fields jointly measurable for $F \otimes L (I) \otimes L (D)$ on $Ω \times I \times D$ , where $L$ denotes the respective restricted Lebesgue sigma-algebras, with vector $u, f$ and scalar $p$ , such that $E \int_{I \times D} (∣ u ∣^{2} + ∣ p ∣ + ∣ f ∣) d t d x < \infty .$ Assume that for almost every $ω$ , every $φ \in C_{c}^{\infty} (I \times D; R^{d})$ satisfies $\int_{I \times D} (u \cdot \partial_{t} φ + (u \otimes u) : \nabla φ + p div φ + ν u \cdot Δ φ + f \cdot φ) d t d x = 0,$ and every $χ \in C_{c}^{\infty} (I \times D)$ satisfies $\int_{I \times D} u \cdot \nabla χ = 0$ . These identities define the assumed distributional incompressible equation; derivatives act on smooth compactly supported tests, $\nabla φ_{i j} = \partial_{j} φ_{i}$ , and $Δ$ is componentwise. Set $m = E u$ , $\overset{ˉ}{p} = E p$ , $\overset{ˉ}{f} = E f$ , and $R = E [(u - m) \otimes (u - m)]$ , all as almost-everywhere space-time classes. Then $m \in L^{2} (I \times D)^{d}$ , $\overset{ˉ}{p}, \overset{ˉ}{f}, R \in L^{1} (I \times D)$ , $R$ is positive semidefinite almost everywhere, $div m = 0$ distributionally, and $\int_{I \times D} (m \cdot \partial_{t} φ + (m \otimes m + R) : \nabla φ + \overset{ˉ}{p} div φ + ν m \cdot Δ φ + \overset{ˉ}{f} \cdot φ) d t d x = 0.$ For almost every $(t, x) \in I \times D$ , on one common full-measure set for all $z \in R^{d}$ , $R (t, x)$ is symmetric and $z^{T} R (t, x) z = E ⁣ [(z \cdot (u (ω, t, x) - m (t, x)))^{2}] \geq 0.$ This is the conditional expected weak Reynolds equation. If assigned SI units with $u$ a velocity, $p$ is pressure per unit constant density, $f$ is acceleration and $ν$ has units $m^{2} s^{- 1}$ . It closes no equation for $R$ , proves no pathwise solution existence, and asserts no boundary trace, initial trace, physical validity, or commutation with an undefined classical derivative. The same conclusion holds locally, with the space-time function spaces interpreted locally, when the displayed moment condition is imposed on every compact space-time set containing the chosen test support.

Statement references

Finite-second-moment random velocity sensor law

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$

The L2 velocity sensor is well-defined

Quadratic velocity tensors are Bochner integrable

Probability measures and probability spaces

The product sigma-algebra and its finite iterates

Strongly measurable Banach-valued function

Holder's inequality for integrals, including the endpoint cases

Bounded linear maps commute with Bochner integration

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

Fubini's theorem for L^1 functions on a sigma-finite product

$Q$ is countably infinite

Both $Q$ and $R ∖ Q$ are dense in $R$ , and every nonempty open subset of $R$ is uncountable

Facts & Assumptions

Given: Countable Choice; a finite measure space $(D, D, λ)$ , finite-dimensional vector fields in $H = L^{2} (D, λ; R^{d})$ , a probability space $(Ω, F, P)$ , a strongly measurable square-integrable $H$ -valued random variable $U$ , and its jointly measurable representative $\tilde{U}$ from the Statement.

[A1]

The Axiom of Countable Choice, ${A C}_{ω}$ , is assumed (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[A2]

The stated joint representative agrees sectionwise with $U$ for almost every $ω$ , and $E ∥ U ∥_{H}^{2} < \infty$ .

[F1]

The Definition fixes $H$ as the space of almost-everywhere equivalence classes of square-integrable vector fields (Finite-second-moment random velocity sensor law).

[F2]

The sensor pairing $S_{ψ}$ is a bounded linear functional on $H$ for every $ψ \in H$ (The L2 velocity sensor is well-defined).

[F3]

Under ${A C}_{ω}$ , finite-component $L^{2}$ and $L^{1}$ spaces are Banach, and the quadratic tensor map is Bochner integrable with its norm expectation controlled exactly by the second moment (Quadratic velocity tensors are Bochner integrable, Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ ).

[F4]

Strong measurability is almost-everywhere norm convergence of measurable simple Banach-valued maps (Strongly measurable Banach-valued function).

[F5]

Holder's inequality bounds products of $L^{2}$ functions by the product of their $L^{2}$ norms (Holder's inequality for integrals, including the endpoint cases).

[F6]

Tonelli's theorem exchanges integrals for nonnegative product-measurable functions on sigma-finite spaces (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F7]

Fubini's theorem exchanges integrals for absolutely integrable product-measurable functions on sigma-finite spaces (Fubini's theorem for L^1 functions on a sigma-finite product).

[F8]

A bounded linear map commutes with a Bochner integral (Bounded linear maps commute with Bochner integration).

[F9]

The product sigma-algebra is generated by measurable rectangles (The product sigma-algebra and its finite iterates).

[F10]

The triples $(Ω, F, P)$ and $(D, D, λ)$ are measure spaces, and total mass one defines a probability space (Probability measures and probability spaces).

[F11]

The rationals are countable and dense, so rational open boxes form a countable base in each finite-dimensional Euclidean space. The preimage of a basis box under a finite tuple of measurable real functions is a finite intersection of measurable preimages; therefore that tuple is measurable as a Euclidean-valued map, and composing with a continuous finite-dimensional operation preserves measurability ( $Q$ is countably infinite, Both $Q$ and $R ∖ Q$ are dense in $R$ , and every nonempty open subset of $R$ is uncountable).

Proof

Given: The hypotheses and conventions stated above.

1.1A1A2F1F3F4givenalgebra

By [F3], $U$ is Bochner integrable, so $\overset{ˉ}{u} = \int_{Ω} U d P$ exists in the quotient space $H$ of [F1]. Subtracting the constant simple map $\overset{ˉ}{u}$ from the simple approximants in [F4] shows $Z : = U - \overset{ˉ}{u}$ is strongly measurable. Its second moment satisfies $E ∥ Z ∥_{H}^{2} \leq 2 E ∥ U ∥_{H}^{2} + 2 ∥ \overset{ˉ}{u} ∥_{H}^{2} < \infty$ .

2.1A1A2F2F5F6F7F8F10step 1.1algebra

On the product measure space of [F9], [F11] makes $∣ \tilde{U} ∣^{2}$ product-measurable. Tonelli's theorem [F6] gives $\int_{D} E ∣ \tilde{U} (ω, x) ∣^{2} d λ = E ∥ U ∥_{H}^{2} < \infty$ . Since $(P \times λ) (Ω \times D) = λ (D) < \infty$ , Holder [F5] gives $\int_{Ω \times D} ∣ \tilde{U} ∣ d (P \times λ) \leq λ (D)^{1 / 2} (E ∥ U ∥_{H}^{2})^{1 / 2} < \infty$ . Fubini [F7] therefore makes the section expectation $m (x) : = E [\tilde{U} (ω, x)]$ measurable and defined for almost every $x$ ; set $m (x) = 0$ on the exceptional null set. Tonelli [F6] and Holder's inequality on the probability space give $∣ m (x) ∣^{2} \leq E ∣ \tilde{U} (ω, x) ∣^{2}$ almost everywhere, hence $m \in H$ . For each $ψ \in H$ , [F11] makes $\tilde{U} (ω, x) \cdot ψ (x)$ product-measurable, and its absolute integral is finite: $E \int_{D} ∣ \tilde{U} \cdot ψ ∣ d λ \leq E ∥ U ∥_{H} ∥ ψ ∥_{H} \leq (E ∥ U ∥_{H}^{2})^{1 / 2} ∥ ψ ∥_{H} < \infty .$ Fubini [F7] gives $S_{ψ} (m) = E [S_{ψ} (U)]$ . The map $S_{ψ}$ is bounded linear by [F2], so [F8] gives $E [S_{ψ} (U)] = S_{ψ} (\overset{ˉ}{u})$ . Thus $S_{ψ} (m - \overset{ˉ}{u}) = 0$ for every $ψ \in H$ ; taking $ψ = m - \overset{ˉ}{u}$ gives $∥ m - \overset{ˉ}{u} ∥_{H}^{2} = 0$ , hence $m = \overset{ˉ}{u}$ in $H$ and $E [\tilde{U} (ω, x) - \overset{ˉ}{u} (x)] = 0$ for almost every $x$ .

3.1A2F5F6F7F9F11step 1.1step 2.1algebra

On the product measure space of [F9], [F11] makes $\tilde{Z} (ω, x) : = \tilde{U} (ω, x) - \overset{ˉ}{u} (x)$ and its component products product-measurable. Tonelli [F6] yields $\int_{D} E ∣ \tilde{Z} ∣^{2} d λ = E ∥ Z ∥_{H}^{2} < \infty$ . For each $i, j$ , $2 ∣ {\tilde{Z}}_{i} {\tilde{Z}}_{j} ∣ \leq {\tilde{Z}}_{i}^{2} + {\tilde{Z}}_{j}^{2}$ , so ${\tilde{Z}}_{i} {\tilde{Z}}_{j}$ is absolutely integrable on $Ω \times D$ . Fubini [F7] therefore makes each entry $R_{i j} (x) = E [{\tilde{Z}}_{i} {\tilde{Z}}_{j}]$ measurable and finite for almost every $x$ . For each pair $i, j$ , scalar Holder on the probability space gives $∣ R_{i j} (x) ∣^{2} \leq E [{\tilde{Z}}_{i}^{2}] E [{\tilde{Z}}_{j}^{2}]$ ; summing over $i, j$ yields $∣ R (x) ∣_{F} \leq E ∣ \tilde{Z} (ω, x) ∣^{2}$ . Hence $\int_{D} ∣ R ∣_{F} d λ < \infty$ . Changing the representative of $\overset{ˉ}{u}$ on a spatial null set does not change the resulting $L^{1}$ class. On the common full-measure set where $E ∣ \tilde{Z} ∣^{2} < \infty$ , every real vector $z$ satisfies $z^{T} R z = E [(z \cdot \tilde{Z})^{2}] \geq 0$ ; this single set works for all $z$ because every component and the second moment are finite there.

4.1A2F3F5F7F9F10F11step 2.1step 3.1algebragiven

Fix an admissible $Φ$ . By [F11], each component product ${\tilde{U}}_{i} {\tilde{U}}_{j} Φ_{i j}$ is measurable, and the finite sum $(\tilde{U} \otimes \tilde{U}) : Φ$ is product-measurable on [F9]. Finite-dimensional Cauchy--Schwarz and the tensor norm identity in [F3] give $E \int_{D} ∣ (\tilde{U} \otimes \tilde{U}) : Φ ∣ d λ \leq ∥ Φ ∥_{\infty} E ∥ U \otimes U ∥_{K} = ∥ Φ ∥_{\infty} E ∥ U ∥_{H}^{2} < \infty .$ Also, Holder [F5] on $Ω \times D$ gives $\int_{Ω \times D} ∣ \overset{ˉ}{u} (x) ∣ ∣ \tilde{Z} (ω, x) ∣ d (P \times λ) \leq ∥ \overset{ˉ}{u} ∥_{H} (E ∥ Z ∥_{H}^{2})^{1 / 2} < \infty,$ so the expanded cross terms remain absolutely integrable after multiplication by the bounded $Φ$ . Fubini [F7] exchanges expectation and spatial integration. Expanding $\tilde{U} = \overset{ˉ}{u} + \tilde{Z}$ pointwise, the two cross terms have expectation zero for almost every $x$ by step 2.1; the constant term averages to itself since $P (Ω) = 1$ by [F10], and the remaining random term is $R$ . The deterministic tensor is integrable by Holder [F5], and $R$ is integrable by step 3.1, so the claimed identity follows.

4.2F5F6F7step 3.1givenalgebra

For the additional space-time data, Tonelli [F6] and the assumed moment bound give a common full-measure space-time set where $E ∣ u ∣^{2}$ , $E ∣ p ∣$ and $E ∣ f ∣$ are finite. Scalar Holder on the probability space yields $∣ m ∣^{2} \leq E ∣ u ∣^{2}$ , so $m \in L^{2}$ . Fubini [F7] gives the measurable $L^{1}$ means $\overset{ˉ}{p}, \overset{ˉ}{f}$ . The component argument of step 3.1 on $I \times D$ makes $R$ an $L^{1}$ positive-semidefinite tensor. At every point of that common set, expanding $u = m + (u - m)$ and using $E (u - m) = 0$ yields $E [u \otimes u] = m \otimes m + R$ .

5.1F5F6F7step 4.2givenalgebra∎

Fix a test $φ$ in the additional clause. Its derivatives are bounded on a compact support. The second-moment hypothesis controls the quadratic term, Holder controls the linear velocity terms on that finite-volume support, and the assumed $L^{1}$ moments control pressure and forcing. Each tested summand is therefore absolutely integrable on probability times space-time. Fubini [F7] takes expectation of the assumed pathwise weak identity and substitutes the means and covariance from step 4.2, proving the displayed expected equation. The same argument for $u \cdot \nabla χ$ proves $div m = 0$ . It works on each compact support under the local version of the hypotheses, without a global moment bound. No common exceptional event for uncountably many tests is needed: for each fixed test its assumed almost-sure identity suffices.

Expected weak Reynolds decomposition and conditional mean equation

Statement

Statement references

Facts & Assumptions

Proof

Depends on

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Sources