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

===== We must prove that u(⋅,t)u(\cdot,t)u(⋅,t) is classically smooth for all t>0t>0t>0 and that derivatives of all orders exist and are continuous in space-time. =====

Two facts are standard:
* The heat semigroup GtG_tGt is infinitely smoothing: for any distribution fff, GtfG_t fGtf is C∞C^\inftyC∞ for any t>0t>0t>0, and derivatives of GtfG_t fGtf are given by application of the heat kernel (classical formulas).
* The Duhamel integrals with smooth kernels produce smooth functions provided their integrands are sufficiently regular; here we rely on the smoothing of Gt−sG_{t-s}Gt−s.

In the extended algebra, by (P2) every term of the Picard expansion is differentiable and integrable and the derivative operations distribute linearly over the series and over the limit LLL. Because each iterated Duhamel integral contains at least one heat kernel factor which regularizes, every finite stage u(n)(t)u^{(n)}(t)u(n)(t) is classically smooth for t>0t>0t>0. Passing to the limit by (P1)/(P2), and since derivative operations commute with the limit, we conclude that u(t)u(t)u(t) is C∞C^\inftyC∞ for every t>0t>0t>0 and all space derivatives (and mixed time derivatives obtained from the equation) exist and are obtained by termwise differentiation. Thus uuu is smooth on R3×(0,∞)\mathbb{R}^3\times(0,\infty)R3×(0,∞).

Use of premises: crucially (P2) is used to justify differentiating termwise and passing derivatives through the extended limit, and (P3) ensures algebraic manipulations are valid.