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Reconstruction of the periodic pressure from the velocity: $\nabla p = - (I - P) ((u \cdot \nabla) u)$ and $Δ p = - \sum_{i, j} \partial_{i} \partial_{j} (u_{i} u_{j})$

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $T > 0$ , let $u : [0, T] \times T^{3} \to R^{3}$ be a field such that $u \in C ([0, T], H^{m})$ for every real $m \geq 0$ , $t \mapsto u (t)$ is differentiable into $H^{m}$ with $\partial_{t} u \in C ([0, T], H^{m})$ for every $m$ , $u (t)$ is divergence-free for every $t$ , and $u$ solves the projected equation $\partial_{t} u = ν Δ u - P ((u \cdot \nabla) u)$ for some real $ν \geq 0$ (all in the sense of the spaces of Periodic vector Sobolev spaces on the three-torus, the weight family, and the Leray projection).

Define the real scalar field $p$ on $[0, T] \times T^{3}$ by its Fourier coefficients $\hat{p} (t, 0) : = 0, \hat{p} (t, k) : = - \frac{1}{∣ k ∣_{2}^{2}} \sum_{i, j = 1}^{3} k_{i} k_{j} \hat{u_{i} u_{j}} (t, k) (k \neq 0),$ where $\hat{u_{i} u_{j}} (t, k)$ is the $k$ -th Fourier coefficient of the product field $u_{i} (t, \cdot) u_{j} (t, \cdot)$ . Then:

$p \in C ([0, T], H^{m})$ for every $m \geq 0$ , $t \mapsto p (t)$ is differentiable into every $H^{m}$ , $p \in C^{\infty} ([0, T] \times T^{3})$ , and $\int_{T^{3}} p (t, x) d x = 0$ for every $t$ .
The following identities hold on the strip: $\nabla p = - (I - P) ((u \cdot \nabla) u), \partial_{t} u + (u \cdot \nabla) u = ν Δ u - \nabla p, Δ p = - \sum_{i, j = 1}^{3} \partial_{i} \partial_{j} (u_{i} u_{j}) .$
Uniqueness. If $q \in C^{\infty} ([0, T] \times T^{3})$ has $\int_{T^{3}} q (t, \cdot) d x = 0$ and $\nabla q = \nabla p$ for every $t$ , then $q = p$ ; equivalently, $p$ is the unique smooth mean-zero pressure making $u$ solve the primitive-variable system.

Facts & Assumptions

Given: The field $u$ with the stated continuity, differentiability and divergence-free hypotheses, and the operators of Periodic vector Sobolev spaces on the three-torus, the weight family, and the Leray projection.

[F1]

The scalar product rule and $\sum_{j} \partial_{j} u_{j} = 0$ give, componentwise, $((u \cdot \nabla) u)_{i} = \sum_{j} \partial_{j} (u_{i} u_{j})$ , so $\hat{((u \cdot \nabla) u)_{i}} (k) = 2 π i \sum_{j} k_{j} \hat{u_{i} u_{j}} (k)$ (Periodic vector Sobolev spaces on the three-torus, the weight family, and the Leray projection).

[F2]

For $k \neq 0$ , the coefficient of $(I - P) v$ at $k$ is $(k \cdot \hat{v} (k) / ∣ k ∣_{2}^{2}) k$ , and the coefficient of $\nabla p$ is $2 π i k \hat{p} (t, k)$ ; $P$ kills gradient fields: $P \nabla q = 0$ (Periodic vector Sobolev spaces on the three-torus, the weight family, and the Leray projection).

[F3]

For $m > 3 / 2$ , product bounds $∥ f g ∥_{H^{m}} \leq C_{m} ∥ f ∥_{H^{m}} ∥ g ∥_{H^{m}}$ and $∥ \partial_{j} f ∥_{H^{m}} \leq ∥ f ∥_{H^{m + 1}}$ hold (Discrete Young inequality, the Sobolev product bound, and the transport product bound on the torus, claim 2; Periodic vector Sobolev spaces on the three-torus, the weight family, and the Leray projection).

[F4]

If $h (t, \cdot) \in H^{m}$ for all $m$ with $∣ k ∣^{2} ∣ \hat{h} (t, k) ∣$ bounded by $C_{N} w_{k}^{- N}$ for all $N$ , uniformly on compact time sets, then $h$ has continuous derivatives of all orders in $x$ ; a curve with $\partial_{t}^{r} h \in C ([0, T], H^{m})$ for all $r, m$ is $C^{\infty}$ on the strip (Sobolev embedding on the torus: $H^{σ} \subseteq C^{j}$ for $σ > j + 3 / 2$ , with continuous representatives, claims 1 and 3).

Proof

technique · direct; define the pressure coefficientwise, verify the multiplier identities, and control decay and time differentiation to obtain smoothness and uniqueness

Given: The field $u$ and the coefficient definition of $p$ , with facts [F1]-[F4].

1.1F1F2

The multiplier identity. Put $v = (u \cdot \nabla) u$ and $S_{k} : = \sum_{i, j} k_{i} k_{j} \hat{u_{i} u_{j}} (k)$ . By [F1], $k \cdot \hat{v} (k) = 2 π i S_{k}$ for $k \neq 0$ . The projection formula [F2] gives $\hat{(I - P) v} (k) = 2 π i k S_{k} / ∣ k ∣_{2}^{2}$ , while the stated pressure definition gives $\hat{\nabla p} (k) = 2 π i k \hat{p} (k) = - 2 π i k S_{k} / ∣ k ∣_{2}^{2}$ . At $k = 0$ both coefficients are zero, because gradients have zero constant coefficient and $P$ is the identity on that mode. Thus $\nabla p = - (I - P) v$ coefficientwise; the smoothness and coefficient identification justified by the smoothness argument below turn it into the claimed field identity.

1.2F3F4given

Bounded multiplier and smoothness. Since $∣ k_{i} k_{j} ∣ / ∣ k ∣_{2}^{2} \leq 1$ for $k \neq 0$ , the pressure coefficient map satisfies $∥ p ∥_{H^{r}} \leq \sum_{i, j} ∥ u_{i} u_{j} ∥_{H^{r}}$ by the triangle inequality, for every real $r$ . For $r > 3 / 2$ each product is continuous into $H^{r}$ by [F3]; larger orders give the assertion also for any smaller order by the monotonicity of the Sobolev weights. The hypotheses of the smoothness bootstrap A strong solution of the projected periodic equation with all Sobolev norms continuous in time is $C^{\infty}$ hold for $u$ , so all its time derivatives are continuous into every $H^{r}$ . The bilinear product rule makes the same true for $u_{i} u_{j}$ , and the bounded coefficient multiplier defining $p$ commutes with those derivatives. Applying [F4] at arbitrarily high orders proves joint spatial and time smoothness and identifies the coefficients of all derivatives. Finally $\hat{p} (0) = 0$ gives the zero mean. This proves claim 1.

2.1step 1.1F2given

Primitive equation. Since $P$ is a projection, $P ((u \cdot \nabla) u) = (u \cdot \nabla) u - (I - P) ((u \cdot \nabla) u) = (u \cdot \nabla) u + \nabla p$ by step 1.1. Substituting into $\partial_{t} u = ν Δ u - P ((u \cdot \nabla) u)$ gives $\partial_{t} u = ν Δ u - (u \cdot \nabla) u - \nabla p$ , i.e. $\partial_{t} u + (u \cdot \nabla) u = ν Δ u - \nabla p$ : the second identity of claim 2.

2.2step 1.1F1

Poisson identity. By [F1] and the definitions, the coefficient of $Δ p$ at $k \neq 0$ is $- 4 π^{2} ∣ k ∣_{2}^{2} \hat{p} (t, k) = 4 π^{2} \sum_{i, j} k_{i} k_{j} \hat{u_{i} u_{j}} (k)$ , while the coefficient of $- \sum_{i, j} \partial_{i} \partial_{j} (u_{i} u_{j})$ is $- \sum_{i, j} (2 π i k_{i}) (2 π i k_{j}) \hat{u_{i} u_{j}} (k) = 4 π^{2} \sum_{i, j} k_{i} k_{j} \hat{u_{i} u_{j}} (k)$ ; at $k = 0$ both sides vanish (for $Δ p$ because $\hat{p} (0) = 0$ , and for the double divergence because $\hat{- \sum \partial_{i} \partial_{j} (u_{i} u_{j})} (0) = 0$ ). Hence $Δ p = - \sum_{i, j} \partial_{i} \partial_{j} (u_{i} u_{j})$ : the third identity of claim 2.

3.1step 1.2F4∎

Uniqueness. If $\nabla q = \nabla p$ then for every $k \neq 0$ , $2 π i k (\hat{q} (k) - \hat{p} (k)) = 0$ forces $\hat{q} (t, k) = \hat{p} (t, k)$ , while at $k = 0$ both means vanish, so $\hat{q} (t, 0) = \hat{p} (t, 0) = 0$ ; hence $q - p$ has all Fourier coefficients zero. Injectivity of the scalar $L^{2}$ coefficient map in Periodic vector Sobolev spaces on the three-torus, the weight family, and the Leray projection gives $q - p = 0$ almost everywhere. Both $q, p$ are continuous, so $q - p = 0$ everywhere by the uniqueness of continuous representatives (Sobolev embedding on the torus: $H^{σ} \subseteq C^{j}$ for $σ > j + 3 / 2$ , with continuous representatives, claim 2).

Remarks

The pressure is a mean-zero solution of the periodic Poisson equation $Δ p = - \sum_{i, j} \partial_{i} \partial_{j} (u_{i} u_{j})$ ; it is not an independent evolution variable, and it is unique once the mean is fixed.
The sign convention is the one of the primed equations: with the coefficient formula above, the primitive system reads $\partial_{t} u + (u \cdot \nabla) u = - \nabla p + ν Δ u$ , matching the standard physical normalization used by the consuming mechanics page.

Reconstruction of the periodic pressure from the velocity: $\nabla p = - (I - P) ((u \cdot \nabla) u)$ and $Δ p = - \sum_{i, j} \partial_{i} \partial_{j} (u_{i} u_{j})$

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Reconstruction of the periodic pressure from the velocity: ∇p=−(I−P)((u⋅∇)u)∇p=−(I−P)((u⋅∇)u) and Δp=−∑i,j∂i∂j(uiuj)Δp=−∑i,j​∂i​∂j​(ui​uj​)

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Reconstruction of the periodic pressure from the velocity: $\nabla p = - (I - P) ((u \cdot \nabla) u)$ and $Δ p = - \sum_{i, j} \partial_{i} \partial_{j} (u_{i} u_{j})$