Editing Openai/6935580c-34e4-8001-9136-7bbdf1499790 (section)

=== Assistant: I will answer this precisely, because here “sufficiently regular” and “commutes” must be mathematical statements, not h… ===
I will answer this precisely, because here “sufficiently regular” and “commutes” must be mathematical statements, not hand-waving.

==== Let PPP be the Leray projector and let DDD be a differential operator (either ∂t\partial_t∂t or Δ\DeltaΔ). ====

To say that PPP commutes with DDD on a class U\mathcal{U}U means:

: 

This is not automatic; it must be justified operator-by-operator.

==== “Sufficiently regular” means: ====

: 

Concretely, the minimal regularity depends on the operator:

===== We need: =====
* u∈Wloc1,1((0,T);X)u \in W^{1,1}_{\text{loc}}((0,T); X)u∈Wloc1,1((0,T);X) for a Banach space XXX on which PPP is bounded (e.g. X=L2X = L^2X=L2, HsH^sHs, or LpL^pLp with 1<p<∞1<p<\infty1<p<∞).

This ensures:
* ∂tu∈Lloc1((0,T);X)\partial_t u \in L^1_{\text{loc}}((0,T); X)∂tu∈Lloc1((0,T);X),
* Pu(t)P u(t)Pu(t) is defined for a.e. ttt,
* P∂tuP \partial_t uP∂tu is Bochner integrable.

This is the minimal hypothesis for time differentiation to commute with a bounded linear operator.

===== We need: =====
* u(t,⋅)∈H2(R3)u(t,\cdot) \in H^2(\mathbb{R}^3)u(t,⋅)∈H2(R3) (or W2,pW^{2,p}W2,p with 1<p<∞1<p<\infty1<p<∞),
* PPP bounded on the target space (true on LpL^pLp, HsH^sHs).

This ensures:
* Δu\Delta uΔu exists as an L2L^2L2 (or LpL^pLp) function,
* both P(Δu)P(\Delta u)P(Δu) and Δ(Pu)\Delta(Pu)Δ(Pu) are meaningful.

Thus:

u∈Lloc1((0,T);H2(R3))u \in L^1_{\text{loc}}\big((0,T); H^2(\mathbb{R}^3)\big)u∈Lloc1((0,T);H2(R3))
is enough.

==== ### ====

The Leray projector PPP is a Fourier multiplier with symbol

P(ξ)=I−ξ⊗ξ∣ξ∣2.\mathbb{P}(\xi) = I - \frac{\xi\otimes\xi}{|\xi|^2}.P(ξ)=I−∣ξ∣2ξ⊗ξ.
===== Δu^(ξ)=−∣ξ∣2u^(ξ).\widehat{\Delta u}(\xi) = -|\xi|^2 \widehat{u}(\xi).Δu(ξ)=−∣ξ∣2u(ξ). =====
===== P(Δu)^(ξ)=P(ξ)(−∣ξ∣2u^(ξ))=−∣ξ∣2P(ξ)u^(ξ)=Δ(Pu)^(ξ).\widehat{P(\Delta u)}(\xi) =====
= \mathbb{P}(\xi)\big(-|\xi|^2 \widehat{u}(\xi)\big)
= -|\xi|^2 \mathbb{P}(\xi)\widehat{u}(\xi)
= \widehat{\Delta(Pu)}(\xi).P(Δu)(ξ)=P(ξ)(−∣ξ∣2u(ξ))=−∣ξ∣2P(ξ)u(ξ)=Δ(Pu)(ξ).
Hence:

PΔu=ΔPu,P\Delta u = \Delta P u,PΔu=ΔPu,
provided both sides lie in the same space (this is exactly where “sufficient regularity” is required).

No functional-analysis tricks here: this is a symbol-level identity.

==== This is much simpler and purely functional-analytic. ====

===== If P:X→XP : X \to XP:X→X is a bounded linear operator on a Banach space XXX, then: =====

: 

===== Take test function φ∈Cc∞(0,T)\varphi \in C_c^\infty(0,T)φ∈Cc∞(0,T): =====

∫0TPu(t)φ′(t) dt=P(∫0Tu(t)φ′(t) dt)=−P(∫0T(∂tu)(t)φ(t) dt)=−∫0TP(∂tu)(t)φ(t) dt.\int_0^T Pu(t)\varphi'(t)\,dt
= P\left(\int_0^T u(t)\varphi'(t)\,dt\right)
= -P\left(\int_0^T (\partial_t u)(t)\varphi(t)\,dt\right)
= -\int_0^T P(\partial_t u)(t)\varphi(t)\,dt.∫0TPu(t)φ′(t)dt=P(∫0Tu(t)φ′(t)dt)=−P(∫0T(∂tu)(t)φ(t)dt)=−∫0TP(∂tu)(t)φ(t)dt.
This is exactly the distributional definition of ∂t(Pu)=P(∂tu)\partial_t(Pu)=P(\partial_t u)∂t(Pu)=P(∂tu).

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==== - In weak solutions, commutation with ∂t\partial_t∂t is valid (distributionally). ====
* Commutation with Δ\DeltaΔ holds after mollification or for t>0t>0t>0 where smoothing gives H2H^2H2-regularity.
* In the mild solution framework, all commutations are justified a posteriori for t>0t>0t>0 by heat semigroup smoothing.

So the correct logic is:
# Prove existence in a mild/weak sense.
# Use heat smoothing to upgrade regularity for t>0t>0t>0.
# Once u(t)∈H2u(t)\in H^2u(t)∈H2, all commutations become rigorously valid.

==== : ====

If you want, next I can:
* rewrite your entire Navier–Stokes derivation with each commutation tagged with the exact regularity used, or
* show how to avoid commutation altogether by working directly with the mild formulation.

Just say which.