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

===== Expand iteratively: =====

u(1)(t)=Gtu0−∫0tGt−sP∇ ⁣⋅(Gsu0⊗Gsu0) ds,u(2)(t)=Gtu0−∫0tGt−sP∇ ⁣⋅(u(1)⊗u(1)) ds,\begin{aligned}
u^{(1)}(t) &= G_t u_0 - \int_0^t G_{t-s} P\nabla\!\cdot \big(G_s u_0\otimes G_s u_0\big)\,ds,\\
u^{(2)}(t) &= G_t u_0 - \int_0^t G_{t-s} P\nabla\!\cdot\big(u^{(1)}\otimes u^{(1)}\big)\,ds,
\end{aligned}u(1)(t)u(2)(t)=Gtu0−∫0tGt−sP∇⋅(Gsu0⊗Gsu0)ds,=Gtu0−∫0tGt−sP∇⋅(u(1)⊗u(1))ds,
and so on. Each u(n)u^{(n)}u(n) can be written as a finite sum of iterated convolution integrals of the heat kernel with polynomial nonlinearities in u0u_0u0. In other words there is a formal expansion in iterated Duhamel integrals (a Dyson/Picard series). Classically this series may converge only for small ttt; but under (P2)/(P1) we can form the termwise limit in our extended sense.

Consider the sequence {u(n)(t)}n≥0\{u^{(n)}(t)\}_{n\ge0}{u(n)(t)}n≥0. By (P1) the linear functional LLL is defined on sequences of functions and therefore Ln→∞u(n)(t)L_{n\to\infty} u^{(n)}(t)Ln→∞u(n)(t) exists in the extended algebra (we take LLL applied pointwise for each (x,t)(x,t)(x,t) or in a suitable linear topology; the premise asserts this is globally defined). Define

u(t):=Ln→∞u(n)(t).u(t) := L_{n\to\infty} u^{(n)}(t).u(t):=Ln→∞u(n)(t).
Use of premises: Existence of this limit and the ability to pass linear operations and (by P2) derivatives/integrals through LLL are direct uses of (P1),(P2).