Global well-posedness in homogeneous $H^1$

Let $(u_0,f,T)$ be $H^1$ data and let an $H^1$ mild solution be a solution with $u\in L^\infty_tH^1_x\cap L^2_tH^2_x$ and the corresponding pressure and Duhamel formulation. Global well-posedness in homogeneous $H^1$. Let $(u_0,0,T)$ be a homogeneous $H^1$ set of data. Then there exists an $H^1$ mild solution $(u,p,u_0,0,T)$ with the indicated data. This is the mild-solution version of homogeneous global regularity; its resolution is not supplied.