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

==== 1. Equip EEE (or at least the subspace in which all Picard iterates live) with a Banach-norm ∥⋅∥X\|\cdot\|_X∥⋅∥X. For example, choose an XXX like L2L^2L2, HkH^kHk, or CαC^\alphaCα if you embed classical functions there; if EEE is larger you still need a topology that makes the heat semigroup and Leray projector continuous operators Gt,P:  X→XG_t,P:\;X\to XGt,P:X→X. ====
# Show the integrand s↦Gt−sP∇ ⁣⋅(u⊗u)(s)s\mapsto G_{t-s}P\nabla\!\cdot( u\otimes u)(s)s↦Gt−sP∇⋅(u⊗u)(s) is strongly continuous (i.e. continuous in the ∥⋅∥X\|\cdot\|_X∥⋅∥X-norm) on [0,t][0,t][0,t] or at least on (0,t](0,t](0,t] with an integrable singularity at 000. For Picard iterates this is standard because each iterate is smooth for s>0s>0s>0 and heat-kernel estimates give control near 000.
# Then define the integral as the Bochner integral in XXX. Bochner integrals are limits of Riemann sums in the Banach norm and satisfy linearity, additivity, and the usual fundamental theorem relationships when the integrand is continuous. This gives a well-defined vector-valued integral in XXX.
# Remarks: Bochner integration also gives estimates (norm bounds) which you can use in contraction/Grönwall arguments. It avoids having to construct antiderivatives in EEE.

What you must assume/prove: existence of a Banach topology on the target space that makes the mapping continuous and the operators bounded; and that the extended multiplication u⊗uu\otimes uu⊗u maps the function class into that Banach space.