A priori periodic $H^1$ bound

Let $(u,p,u_0,f,T,L)$ be a smooth periodic, normalised-pressure solution, with periodic $H^1$ data norm ${\mathcal H}^1(u_0,f,T,L)$ finite. A priori periodic $H^1$ bound. There exists a function $F:\mathbf{R}^+\times\mathbf{R}^+\times\mathbf{R}^+\to\mathbf{R}^+$ such that, whenever $0<T<T_0<\infty$ and ${\mathcal H}^1(u_0,f,T,L)\leq A<\infty$, one has

$$\|u\|_{L^\infty_tH^1_x([0,T]\times\mathbf{R}^3/L\mathbf{Z}^3)}\leq F(A,L,T_0).$$

Such a bound would quantitatively control periodic solutions on bounded time intervals and is presented as a quantitative form of global regularity; the source gives no resolution.