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

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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.