A priori homogeneous periodic $H^1$ bound

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

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

This is the quantitative homogeneous periodic estimate recalled in connection with Tao's result; the source does not present it as resolved here.