  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 with
interaction 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 the
expected correlations in $\Lambda$ develop dynamically in time. As shown in our
previous 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 local
in time estimates can be extended globally. Also, they can be used to derive
Fock space estimates for the approximation of the exact evolution of a Bosonic
system by states of the form $e^{\sqrt N A(\phi)}e^{B(k)} \Omega$. This will be
addressed in detail in future work.
