A priori global homogeneous $H^1$ bound

Let $(u,p,u_0,0,T)$ be a smooth homogeneous $H^1$ solution, with $\|u_0\|_{H^1_x(\mathbf{R}^3)}\leq A<\infty$. A priori global homogeneous $H^1$ bound. There exists a function $F:\mathbf{R}^+\to\mathbf{R}^+$ such that

$$\|u\|_{L^\infty_tH^1_x([0,T]\times\mathbf{R}^3)}\leq F(A).$$

This global-in-time quantitative bound is stronger than the corresponding bounded-time estimate and would rule out unbounded $H^1$ growth for all finite solution intervals; it remains unresolved in the source.