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

==== 1. u0∈Hs, s>3/2u_0\in H^s,\ s>3/2u0∈Hs, s>3/2 ⇒ local mild solution u∈C([0,T];Hs)u\in C([0,T];H^s)u∈C([0,T];Hs). ====
# - (one of 1–4 above) ⇒ uniform control of ∥u(t)∥Hs\|u(t)\|_{H^s}∥u(t)∥Hs for all ttt.
# By the blowup alternative, uniform control ⇒ T∗=∞T^*=\inftyT∗=∞.
# Smoothing for t>0t>0t>0 ⇒ u(⋅,t)∈C∞u(\cdot,t)\in C^\inftyu(⋅,t)∈C∞ for every t>0t>0t>0.
# Therefore global classical solution — i.e. no finite-time blowup.

If you want, I can now produce a one-page checklist showing exactly which estimates (with explicit norms) you need to prove at each step for Path A or Path B, or else draft a short proof-of-concept showing how to obtain the Picard local existence with explicit constants so you can see where the a-priori bound must enter. Which follow-up do you prefer?