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

===== - No need for pointwise continuity of ∇⋅(u⊗u)\nabla\cdot(u\otimes u)∇⋅(u⊗u) at s=ts=ts=t. The integrand may blow up in operator norm as s→t−s\to t^-s→t− (because of the (t−s)−1/2(t-s)^{-1/2}(t−s)−1/2 factor), but this is an integrable singularity; the Bochner integral still exists. Thus the Duhamel integral ∫0tGt−sB(u(s),u(s)) ds\int_0^t G_{t-s} B(u(s),u(s))\,ds∫0tGt−sB(u(s),u(s))ds is well-defined as a Bochner integral in HsH^sHs. =====
* If you prefer genuine continuity of the integrand: impose stronger time-regularity on uuu. For instance u∈Cα([0,T];Hs)u\in C^\alpha([0,T];H^s)u∈Cα([0,T];Hs) for some α>0\alpha>0α>0 often yields better behavior of B(u,u)B(u,u)B(u,u) in time; or if you require u∈C([0,T];Hs+1)u\in C([0,T];H^{s+1})u∈C([0,T];Hs+1) then ∇⋅(u⊗u)∈C([0,T];Hs)\nabla\cdot(u\otimes u)\in C([0,T];H^{s})∇⋅(u⊗u)∈C([0,T];Hs), and semigroup smoothing is no longer needed to obtain continuity in HsH^sHs.
* How this fits into Picard: the contraction estimates use exactly the integrability estimate above. You do not need continuity of the integrand; you only need the operator norm estimates to show T\mathcal TT maps a ball in C([0,T];Hs)C([0,T];H^s)C([0,T];Hs) into itself and is Lipschitz for small TTT. The Bochner-integral formulation + the estimate ∫0t(t−s)−1/2 ds=2t\int_0^t (t-s)^{-1/2}\,ds = 2\sqrt{t}∫0t(t−s)−1/2ds=2t provides the required smallness factor in time.