Global uniform in $N$ estimates for solutions of a system of Hartree-Fock-Bogoliubov type in the case $\beta<1$

We extend the results of the 2019 paper by the third and fourth author globally in time. More precisely, we prove uniform in $N$ estimates for the solutions $\phi$, $\Lambda$ and $\Gamma$ of a coupled system of Hartree-Fock-Bogoliubov type with interaction potential $V_N(x-y)=N^{3 \beta}v(N^{\beta}(x-y))$ with $\beta<1$. The potential satisfies some technical conditions, but is not small. The initial conditions have finite energy and the "pair correlation" part satisfies a smallness condition, but are otherwise general functions in suitable Sobolev spaces, and the expected correlations in $\Lambda$ develop dynamically in time. The estimates are expected to improve the Fock space bounds from the 2021 paper of the first and fifth author. This will be addressed in a different paper.
Comment: 53 pages. Comments are welcome!