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

==== Let XXX be a Banach space of vector fields on R3\mathbb{R}^3R3 such that: ====
* The heat operator generates a strongly continuous semigroup Gt:=eνtΔG_t:=e^{\nu t\Delta}Gt:=eνtΔ on XXX.
* The Leray projector PPP is a bounded linear operator on XXX (or at least maps the relevant subspace into XXX).
* For the candidate solution uuu we have u∈L∞(0,T;X)u\in L^\infty(0,T;X)u∈L∞(0,T;X) and the nonlinear term F(t):=P∇ ⁣⋅(u⊗u)(t)F(t):=P\nabla\!\cdot (u\otimes u)(t)F(t):=P∇⋅(u⊗u)(t) belongs to L1(0,T;X)L^1(0,T;X)L1(0,T;X) (Bochner integrable in XXX).

These are standard hypotheses: they hold for classical local existence frameworks (e.g. X=HsX=H^sX=Hs with s>3/2s>3/2s>3/2) and also for many weak solution contexts (with appropriate choices of XXX). The important point: we do not assume ∂tu\partial_t u∂tu or Δu\Delta uΔu exist classically.