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

=== Assistant: Good point — you should not assume the full classical regularity up front. ===
Good point — you should not assume the full classical regularity up front. There are two standard, logically clean ways to proceed; both avoid assuming classical smoothness and instead derive it from weaker hypotheses. I’ll give both approaches and then a step-by-step derivation of the Duhamel (mild) formula that does not presuppose classical solutions.

Two clean routes (pick one depending on your setting):
# Abstract semigroup / fixed-point route (constructive). Pick a Banach space XXX (e.g. HsH^sHs with s>32s>\tfrac32s>23 or CbαC_b^\alphaCbα or an LpL^pLp–based space) where the heat semigroup eνtΔe^{\nu t\Delta}eνtΔ is a strongly continuous semigroup and where the nonlinear map u↦P∇ ⁣⋅(u⊗u)u\mapsto P\nabla\!\cdot(u\otimes u)u↦P∇⋅(u⊗u) is well defined and locally Lipschitz. Use the Duhamel formula as the definition of a mild solution and apply Picard contraction to produce a solution u∈C([0,T];X)u\in C([0,T];X)u∈C([0,T];X). No classical regularity is assumed — it follows (for t>0t>0t>0) from semigroup smoothing and bootstrapping.
# Weak (distributional / Leray) route (a posteriori equivalence). Start from a weak (Leray) solution satisfying the standard weak integral identity. Under mild extra temporal regularity (e.g. weak continuity into L2L^2L2 and P∇ ⁣⋅(u⊗u)∈L1(0,T;X)P\nabla\!\cdot(u\otimes u)\in L^1(0,T;X)P∇⋅(u⊗u)∈L1(0,T;X) for a suitable XXX) you can transform the weak formulation into the mild form by testing against eigenfunctions or by applying the semigroup to the distributional equation. Then prove smoothing for t>0t>0t>0.

Below I prove the Duhamel formula in a way that does not assume classical regularity: I will state minimal hypotheses, then derive the mild form step by step, and finally sketch how regularity follows.

==== Let XXX be a Banach space of vector fields on R3\mathbb{R}^3R3 such that: ====
* The heat operator generates a strongly continuous semigroup Gt:=eνtΔG_t:=e^{\nu t\Delta}Gt:=eνtΔ on XXX.
* The Leray projector PPP is a bounded linear operator on XXX (or at least maps the relevant subspace into XXX).
* For the candidate solution uuu we have u∈L∞(0,T;X)u\in L^\infty(0,T;X)u∈L∞(0,T;X) and the nonlinear term F(t):=P∇ ⁣⋅(u⊗u)(t)F(t):=P\nabla\!\cdot (u\otimes u)(t)F(t):=P∇⋅(u⊗u)(t) belongs to L1(0,T;X)L^1(0,T;X)L1(0,T;X) (Bochner integrable in XXX).

These are standard hypotheses: they hold for classical local existence frameworks (e.g. X=HsX=H^sX=Hs with s>3/2s>3/2s>3/2) and also for many weak solution contexts (with appropriate choices of XXX). The important point: we do not assume ∂tu\partial_t u∂tu or Δu\Delta uΔu exist classically.

==== ### ====

Formally apply Leray projection to the Navier–Stokes PDE to eliminate pressure:

∂tu−νΔu+P∇ ⁣⋅(u⊗u)=0.\partial_t u - \nu\Delta u + P\nabla\!\cdot(u\otimes u)=0.∂tu−νΔu+P∇⋅(u⊗u)=0.
Under our minimal hypotheses this equality is to be interpreted in the Bochner / distributional sense: there exists an XXX-valued distribution (or Bochner integrable function) t↦F(t)∈L1(0,T;X)t\mapsto F(t)\in L^1(0,T;X)t↦F(t)∈L1(0,T;X) with

∂tu(t)=νΔu(t)−F(t)\partial_t u(t) = \nu\Delta u(t) - F(t)∂tu(t)=νΔu(t)−F(t)
in the sense of distributions in time with values in XXX. Concretely this means: for every test scalar function ϕ∈Cc∞(0,T)\phi\in C_c^\infty(0,T)ϕ∈Cc∞(0,T) and every bounded linear functional ℓ∈X∗\ell\in X^*ℓ∈X∗,

∫0T⟨u(t),ℓ⟩ϕ′(t) dt=−∫0T⟨νΔu(t)−F(t),ℓ⟩ϕ(t) dt\int_0^T \langle u(t),\ell\rangle \phi'(t)\,dt
= -\int_0^T \langle \nu\Delta u(t) - F(t),\ell\rangle \phi(t)\,dt∫0T⟨u(t),ℓ⟩ϕ′(t)dt=−∫0T⟨νΔu(t)−F(t),ℓ⟩ϕ(t)dt
(where ⟨⋅,ℓ⟩\langle\cdot,\ell\rangle⟨⋅,ℓ⟩ denotes the duality pairing). This is the standard weak formulation of an evolution equation in a Banach space.

===== Define v(t):=G−tu(t)v(t):=G_{-t}u(t)v(t):=G−tu(t), i.e. =====

v(t):=e−νtΔu(t).v(t) := e^{-\nu t\Delta}u(t).v(t):=e−νtΔu(t).
Here G−tG_{-t}G−t is the inverse semigroup operator (well defined as a bounded operator on the range of GtG_tGt); to be precise, work with the strongly continuous semigroup GtG_tGt and use the identity below in integrated form rather than applying a formal inverse when not bounded — a cleaner route is the integral identity that follows.

Differentiate vvv in the weak (distributional) sense. Using the product rule in Bochner/distribution sense and the generator property of the semigroup, we obtain

ddtv(t)=ddt(e−νtΔu(t))=−νΔe−νtΔu(t)+e−νtΔ∂tu(t).\frac{d}{dt}v(t) = \frac{d}{dt}\big(e^{-\nu t\Delta}u(t)\big)
= -\nu\Delta e^{-\nu t\Delta}u(t) + e^{-\nu t\Delta}\partial_t u(t).dtdv(t)=dtd(e−νtΔu(t))=−νΔe−νtΔu(t)+e−νtΔ∂tu(t).
Substitute ∂tu(t)=νΔu(t)−F(t)\partial_t u(t)=\nu\Delta u(t)-F(t)∂tu(t)=νΔu(t)−F(t) (distributionally):

ddtv(t)=−νΔe−νtΔu(t)+e−νtΔ(νΔu(t)−F(t)).\frac{d}{dt}v(t) = -\nu\Delta e^{-\nu t\Delta}u(t) + e^{-\nu t\Delta}\big(\nu\Delta u(t)-F(t)\big).dtdv(t)=−νΔe−νtΔu(t)+e−νtΔ(νΔu(t)−F(t)).
The νΔ\nu\DeltaνΔ terms cancel (this is the algebraic magic of the conjugation), leaving

ddtv(t)=−e−νtΔF(t),\frac{d}{dt} v(t) = - e^{-\nu t\Delta} F(t),dtdv(t)=−e−νtΔF(t),
where the equality holds in the Bochner / distributional sense (the right-hand side is in L1(0,T;X)L^1(0,T;X)L1(0,T;X) because F∈L1(0,T;X)F\in L^1(0,T;X)F∈L1(0,T;X) and the semigroup is bounded on XXX).

===== Now integrate in time from 000 to ttt. Since the right-hand side is Bochner integrable in XXX, the fundamental theorem of calculus in Banach spaces (for absolutely continuous / distributional derivatives) gives =====

v(t)−v(0)=−∫0te−νsΔF(s) dsv(t)-v(0) = -\int_0^t e^{-\nu s\Delta} F(s)\,dsv(t)−v(0)=−∫0te−νsΔF(s)ds
in XXX. But v(0)=u(0)=u0v(0)=u(0)=u_0v(0)=u(0)=u0. Rewriting v(t)=e−νtΔu(t)v(t)=e^{-\nu t\Delta}u(t)v(t)=e−νtΔu(t) and applying eνtΔe^{\nu t\Delta}eνtΔ to both sides yields

u(t)−eνtΔu0=−∫0teν(t−s)ΔF(s) ds.u(t) - e^{\nu t\Delta}u_0
= -\int_0^t e^{\nu (t-s)\Delta} F(s)\,ds.u(t)−eνtΔu0=−∫0teν(t−s)ΔF(s)ds.
Replace F(s)=P∇ ⁣⋅(u⊗u)(s)F(s)=P\nabla\!\cdot(u\otimes u)(s)F(s)=P∇⋅(u⊗u)(s). This is precisely the Duhamel (mild) formula:

 u(t)=eνtΔu0−∫0teν(t−s)ΔP∇ ⁣⋅(u⊗u)(s) ds .\boxed{\,u(t)=e^{\nu t\Delta}u_0 - \int_0^t e^{\nu (t-s)\Delta} P\nabla\!\cdot (u\otimes u)(s)\,ds\, }.u(t)=eνtΔu0−∫0teν(t−s)ΔP∇⋅(u⊗u)(s)ds.
Remarks on justification of each operation:
* The identity ∂tu=νΔu−F\partial_t u = \nu\Delta u - F∂tu=νΔu−F was used only in distributional / Bochner sense.
* Conjugation by the semigroup is valid because e−νtΔe^{-\nu t\Delta}e−νtΔ is strongly continuous and bounded on the range we use, and because all multiplications were by linear bounded operators.
* Integration under the Bochner / strong topology is justified since s↦e−νsΔF(s)s\mapsto e^{-\nu s\Delta}F(s)s↦e−νsΔF(s) is Bochner integrable in XXX (bounded operator applied to an L1L^1L1 function). Thus the derivation requires no classical ∂tu\partial_t u∂tu or pointwise derivatives — only that the equation holds distributionally/Bochner-wise and the nonlinear term is Bochner integrable.

==== It is common and cleaner to define a mild solution as any u∈C([0,T];X)u\in C([0,T];X)u∈C([0,T];X) satisfying the integral identity above (with the Bochner integral). Then one proves existence by fixed point: ====
* Consider the operator T\mathcal{T}T on C([0,T];X)C([0,T];X)C([0,T];X), (Tv)(t):=eνtΔu0−∫0teν(t−s)ΔP∇ ⁣⋅(v⊗v)(s) ds.(\mathcal{T}v)(t) := e^{\nu t\Delta}u_0 - \int_0^t e^{\nu(t-s)\Delta} P\nabla\!\cdot (v\otimes v)(s)\,ds.(Tv)(t):=eνtΔu0−∫0teν(t−s)ΔP∇⋅(v⊗v)(s)ds.
* Prove T\mathcal{T}T maps a ball in C([0,T];X)C([0,T];X)C([0,T];X) to itself and is a contraction for TTT small enough (requires local Lipschitz of the nonlinearity in the XXX-norm and heat kernel bounds).
* Apply Banach fixed point to obtain a unique mild solution u∈C([0,T];X)u\in C([0,T];X)u∈C([0,T];X) — no classical regularity assumed.

This constructive route is how local-in-time existence (and uniqueness) of strong solutions is usually obtained.

==== Once a mild solution u∈C([0,T];X)u\in C([0,T];X)u∈C([0,T];X) is constructed, classical smoothness for positive times follows from smoothing properties of the heat semigroup and standard bootstrap: ====
# The heat semigroup eνtΔe^{\nu t\Delta}eνtΔ is smoothing: for any t>0t>0t>0, eνtΔe^{\nu t\Delta}eνtΔ maps XXX into C∞C^\inftyC∞ (or into arbitrarily high Sobolev spaces) and satisfies norm bounds like ∥∇keνtΔf∥Y≤Ct−k/2∥f∥X\|\nabla^k e^{\nu t\Delta} f\|_{Y} \le C t^{-k/2}\|f\|_X∥∇keνtΔf∥Y≤Ct−k/2∥f∥X for suitable YYY.
# In the mild formula each term contains a factor eν(t−s)Δe^{\nu(t-s)\Delta}eν(t−s)Δ with t−s>0t-s>0t−s>0 except possibly at the endpoint s=ts=ts=t. Heat kernel estimates plus the integrability of the nonlinear term allow you to show the integral term is smooth in space for every t>0t>0t>0.
# Therefore u(t)u(t)u(t) is smooth in space for each t>0t>0t>0. Time regularity follows by differentiating the mild formula (the integral is differentiable for t>0t>0t>0 under the established bounds) and bootstrapping the PDE to raise regularity in time and space.

Thus classical smoothness for t>0t>0t>0 is a consequence, not an assumption.

==== 1. Choose a Banach space XXX appropriate for your initial data u0u_0u0. (For a constructive local existence result take X=HsX=H^sX=Hs, s>32s>\tfrac32s>23.) ====
# Define mild solutions via the Duhamel integral in XXX (Bochner integral). Prove local existence & uniqueness by fixed point — no classical regularity assumed.
# Prove that the mild solution satisfies the PDE in the distributional sense (differentiate in time in the Bochner sense or apply semigroup conjugation as above).
# Use semigroup smoothing + bootstrapping to upgrade regularity for t>0t>0t>0 to classical C∞C^\inftyC∞ in space (and as much time regularity as needed).
# If you start from a weak solution and want to prove it satisfies the mild form, use the conjugation argument above (start from the weak/Bochner equation and integrate); no classical differentiability is required beyond the Bochner/distributional hypothesis.

If you want, I can now:
* produce the same derivation written in a fully rigorous Banach-space theorem style (state the exact theorem with hypotheses on XXX, u0u_0u0, and the nonlinear map, then give the proof), or
* show the detailed heat-kernel estimates needed to justify the bootstrapping (norm bounds, smoothing rates, estimates near s=0s=0s=0), which are the technical core for proving that mild ⇒\Rightarrow⇒ classical for t>0t>0t>0.

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