Navier-Stokes, admit space of divergence free or no?

Question

Consider the problem of Navier-Stokes $$\frac{du}{dt}-\mu \Delta u+u \cdot \nabla u+\nabla p=f$$ such that $u(0)(x)=u_0(x)$ for $x \in \mathbb{R}^N$ in a adequate space (for example $L^p(\mathbb{R}^N)$) and $\nabla \cdot u=0$ (divergent of $u$ vanishes). Applying the projector of Leray $\mathbb{P}$ above and using the properties of $\mathbb{P}$ we have the following equivalent system: $$\frac{du}{dt}-\mu \Delta u+\mathbb{P}(u \cdot \nabla u)=\mathbb{P}(f),$$ such that $u(0)(x)=u_0(x)$. The definition of mild solution proposed by many researches is concerning the integral formulation $$u(t)=G(t)u_0+\int_0^t \mathbb{P}G(t-s)(u\cdot \nabla u)(s)ds$$, where $G(t)$ is the heat operator (kernel of Weierstrass). I don't understand why to the prove of the existence of solution many researches consider the space of divergence free (for example $\{u \in L^p(\mathbb{R^N}): \ \nabla \cdot u=0\}$) and others no. I think so that may be some concern with to use the problem before apply the Leray projector or after. Can someone help me?