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

=== Assistant: Good question — it should not appear out of nowhere. ===
Good question — it should not appear out of nowhere. Either you define mild solutions inside a space that already gives the boundedness, or you prove the boundedness as part of the existence argument. You cannot in general proceed without some control (an a-priori bound) on the solution: the Bochner integrals and estimates used in the Duhamel derivation require that the integrand and solution live in a Banach space and be bounded there (or at least locally bounded). Below I give a concise explanation of the two standard, rigorous ways to obtain the u∈L∞(0,T;X)u\in L^\infty(0,T;X)u∈L∞(0,T;X) property, with a self-contained Picard (fixed-point) estimate and an alternative energy estimate for weak/Leray solutions.