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

==== 1. Fix a Banach pair X,YX,YX,Y (e.g. X=Hs,  Y=Hs−1X=H^s,\; Y=H^{s-1}X=Hs,Y=Hs−1 with s>3/2s>3/2s>3/2). ====
# Run the Picard contraction in C([0,T];X)C([0,T];X)C([0,T];X) to show the iterates u(n)u^{(n)}u(n) converge in the sup-norm on [0,T][0,T][0,T] (this is the standard local existence proof). This gives strong convergence in the space where multiplication is continuous.
# Use continuity of BBB and the Bochner integral continuity to pass the limit inside the integral. Conclude the limit uuu satisfies the Duhamel formula.
# If you still want to keep a generalized LLL, prove the continuity properties of LLL with respect to the L1L^1L1/Bochner topology (this is hard and essentially equivalent to building a well-behaved generalized-function algebra).