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

=== 1. Write Navier–Stokes in mild (integral / Duhamel) form with the heat semigroup eνtΔe^{\nu t\Delta}eνtΔ and Leray projector PPP: ===

u(t)=eνtΔu0−∫0teν(t−s)ΔP∇ ⁣⋅(u⊗u)(s) ds.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.
# Use the extended integral and multiplication (P2,P3) to define the Picard iteration sequence in the extended space:

u(0)(t)=eνtΔu0,u(n+1)(t)=eνtΔu0−∫0teν(t−s)ΔP∇ ⁣⋅(u(n)⊗u(n))(s) ds.u^{(0)}(t)=e^{\nu t\Delta}u_0,\qquad
u^{(n+1)}(t)=e^{\nu t\Delta}u_0 - \int_0^t e^{\nu (t-s)\Delta} P\nabla\!\cdot (u^{(n)}\otimes u^{(n)})(s)\,ds.u(0)(t)=eνtΔu0,u(n+1)(t)=eνtΔu0−∫0teν(t−s)ΔP∇⋅(u(n)⊗u(n))(s)ds.
# Use termwise series expansion and the extended limit functional LLL (P1,P2) to take the limit u(t):=Ln→∞u(n)(t)u(t):=L_{n\to\infty} u^{(n)}(t)u(t):=Ln→∞u(n)(t) for each fixed ttt and show the limit satisfies the mild equation (preservation of algebraic identities ensures Duhamel equation passes to the limit).
# Show the resulting uuu is smooth for all t≥0t\ge0t≥0 because the heat semigroup regularizes and derivatives/integrals are defined for every function in the extended sense, and these operations commute with the limit by the premises.
# Show uniqueness in the extended space by the standard Grönwall / contraction argument carried formally in the extended algebra.
# Recover classical smoothness on R3×(0,∞)\mathbb{R}^3\times(0,\infty)R3×(0,∞) (i.e., classical pointwise derivatives exist and are smooth) because on the class of classical smooth functions the extensions agree with the classical operations.

Each of these steps will be made explicit and the precise use of (P1)–(P3) indicated.