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We extend the results of the 2019 paper by the third and fourth authorglobally in time. More precisely, we prove uniform in $N$ estimates for thesolutions $\phi$, $\Lambda$ and $\Gamma$ of a coupled system ofHartree-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 technicalconditions, but is not small. The initial conditions have finite energy and the"pair correlation" part satisfies a smallness condition, but are otherwisegeneral functions in suitable Sobolev spaces, and the expected correlations in$\Lambda$ develop dynamically in time. The estimates are expected to improvethe Fock space bounds from the 2021 paper of the first and fifth author. Thiswill be addressed in a different paper.

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