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

=== If you want the “antiderivative” definition to be safe, add these explicit statements to your setup and then the argument becomes valid: ===
* (T1) Choose a Banach space XXX into which the Picard iterates embed, and state the norm ∥⋅∥X\|\cdot\|_X∥⋅∥X.
* (T2) Prove for each nnn that s↦Gt−sP∇⋅(u(n)⊗u(n))(s)s\mapsto G_{t-s}P\nabla\cdot(u^{(n)}\otimes u^{(n)})(s)s↦Gt−sP∇⋅(u(n)⊗u(n))(s) is continuous in ∥⋅∥X\|\cdot\|_X∥⋅∥X on (0,t](0,t](0,t] and Bochner integrable on [0,t][0,t][0,t].
* (T3) Prove uniform-in-nnn integrability / estimates so you can pass the limit inside the integral (e.g. dominated convergence in the Bochner sense, or show the sequence is Cauchy in the Bochner integral norm).
* (T4) Conclude the limit uuu is continuous in the chosen topology; if you also require classical pointwise continuity, prove embedding theorems that map XXX-continuity to pointwise continuity.
* (T5) Only then, define the Duhamel term for uuu as the Bochner integral of the limiting integrand (or as the limit of the integrals of approximants).

If you cannot or do not want to choose such an XXX, you must instead prove strong properties of your linear limit operator LLL: namely that LLL preserves uniform convergence and continuity and commutes with the primitive construction — but this is a heavy additional axiom.