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Periodic Leray and pressure maps

Definition

Assume Countable Choice and fix $d \in {1, 2, 3}$ . For a vector Fourier coefficient $a (k) \in C^{d}$ , define $(P a) (k) = P_{k} a (k)$ for every $k \in Z^{d}$ , where $P_{0} = I$ and $P_{k} = I - k k^{T} / ∣ k ∣^{2}$ for $k \neq 0$ . For a tensor field $F = (F_{i j})_{i, j = 1}^{d}$ with coefficients in any $H^{s} (Q_{d})$ , define the scalar pressure map $Q$ by $\hat{Q F} (0) = 0$ and $\hat{Q F} (k) = - \sum_{i, j = 1}^{d} k_{i} k_{j} ∣ k ∣^{- 2} {\hat{F}}_{i j} (k)$ for $k \neq 0$ . For $q \in H^{s} (Q_{d})$ , define $\partial_{j} q \in H^{s - 1} (Q_{d})$ coefficientwise by $\hat{\partial_{j} q} (k) = i k_{j} \hat{q} (k)$ ; for vector fields set $\hat{\nabla q} (k) = i k \hat{q} (k)$ , and for tensor fields set ${\hat{div F}}_{i} (k) = i \sum_{j} k_{j} {\hat{F}}_{i j} (k)$ . These derivative maps are bounded from $H^{s}$ to $H^{s - 1}$ . The multipliers $P$ and $Q$ are real-preserving and bounded on every $H^{s}$ , $s \in R$ ; $P$ is an orthogonal projection and commutes with these derivatives. For every $F$ and every $k \in Z^{d}$ , their exact identity is the coefficientwise equality $\hat{\nabla Q F} (k) = - (I - P_{k}) \hat{div F} (k) .$ In particular, both sides have zero coefficient at $k = 0$ .

Definition references

Erick Schulz, Introduction to the Navier–Stokes Equations, McGill MATH 595 notes (2014).

Terence Tao, 254A Notes 1: Local well-posedness of the Navier–Stokes equations (2018).

Facts & Assumptions

Given: The hypotheses and conventions stated in the formulation.

[F1]

Periodic fluid Fourier Sobolev spaces: Fourier coefficients represent the real periodic Sobolev coefficient space, with $\hat{f} (- k) = \overline{\hat{f} (k)}$ .

[F2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is an explicit assumption of the periodic Fourier coefficient framework.

Proof

Given: $d \in {1, 2, 3}$ , Countable Choice, and the coefficient spaces in [F1].

1.1givenF1algebra

For $k \neq 0$ , $P_{k}$ is real symmetric, $P_{k}^{2} = P_{k}$ , and its range is $k^{⊥}$ ; $P_{0} = I$ . Thus each multiplier is an orthogonal projection of norm at most one. The scalar multiplier defining $Q$ has absolute value at most $\sum_{i, j} ∣ k_{i} k_{j} ∣ / ∣ k ∣^{2} \leq d$ , so both maps preserve every weighted $ℓ^{2}$ norm up to a dimension-dependent constant.

2.1givenF1F2step 1.1algebra∎

Since $P_{- k} = P_{k}$ and the coefficient multiplier for $Q$ is real and even in $k$ , both maps preserve conjugate symmetry and hence real fields. Multipliers commute with $i k_{j}$ , so $P$ commutes with the coefficientwise derivative maps. For $k \neq 0$ , the $a$ th component of the left side has coefficient $- i \sum_{i, j = 1}^{d} k_{a} k_{i} k_{j} ∣ k ∣^{- 2} {\hat{F}}_{i j} (k),$ which is the $a$ th component of $- (I - P_{k}) \hat{div F} (k)$ because $I - P_{k} = k k^{T} / ∣ k ∣^{2}$ . At $k = 0$ both sides vanish by the zero-frequency derivative convention.

Periodic Leray and pressure maps

Definition

Definition references

Facts & Assumptions

Proof

Depends on

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Sources