Global well-posedness from spatially smooth $H^1$ data

Let $(u_0,f,T)$ be an $H^1$ set of data such that, for every multi-index $\alpha$ and every compact $K\subset\mathbf{R}^3$,

$$\sup_{x\in K}|\nabla_x^\alpha u_0(x)|<\infty$$

and

$$\sup_{(t,x)\in[0,T]\times K}|\nabla_x^\alpha f(x)|<\infty.$$

Global well-posedness from spatially smooth $H^1$ data. There exists an $H^1$ mild solution $(u,p,u_0,f,T)$ with the indicated data. This is a local spatial smoothness condition without Schwartz decay, and the global existence assertion remains open in the source.