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We prove local in time, uniform in $N$, estimates for the solutions $\phi$,$\Lambda$ and $\Gamma$ of a coupled system of Hartree-Fock-Bogoliubov type withinteraction potential $v_N(x-y) =N^{3 \beta} v(N^{\beta}(x-y))$, with $\beta<1$ and $v$ a Schwartz function (satisfying additional technical requirements).The initial conditions are general functions in a Sobolev-type space, and theexpected correlations in $\Lambda$ develop dynamically in time. As shown in ourprevious work, as well as the work of J. Chong, (both in the case $\beta<2/3$),using the conserved quantities of the system of equations, this type of localin time estimates can be extended globally. Also, they can be used to deriveFock space estimates for the approximation of the exact evolution of a Bosonicsystem by states of the form $e^{\sqrt N A(\phi)}e^{B(k)} \Omega$. This will beaddressed in detail in future work.

Uniform in $N$ estimates for a Bosonic system of Hartree-Fock-Bogoliubov type