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

=== Because you choose the phase space HsH^{s}Hs and you assume the force fff belongs to this space. ===
This is part of the well-posedness hypothesis, not something you derive later.

In the classical Navier–Stokes semigroup/Picard framework, you begin with:
* initial data u0∈Hsu_0 \in H^{s}u0∈Hs with s>3/2s > 3/2s>3/2,
* external force f∈L1(0,T;Hs)f \in L^1(0,T; H^{s})f∈L1(0,T;Hs) or sometimes f∈L∞(0,T;Hs−1)f \in L^\infty(0,T; H^{s-1})f∈L∞(0,T;Hs−1).

These are assumptions of the problem formulation.

The reason is simple:

: 

If fff had infinite HsH^{s}Hs-norm, the nonlinear and linear operators in the mild formulation would not be defined.

This is the same reason an ODE theory requires the right-hand side to be continuous or Lipschitz: the equation has meaning only if the terms lie in the ambient Banach space.