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 [Commun. Partial. Differ. Equ. 44(12), 1431–1465 (2019)] by the third and fourth author globally in time. More precisely, we prove uniform-in-N Strichartz estimates for the solutions \(\phi \) $\varphi$ , \(\Lambda \) $\mathrm{\Lambda }$ and \(\Gamma \) $\mathrm{\Gamma }$ of a coupled system of Hartree–Fock–Bogoliubov type with interaction potential \(V_N(x-y)=N^{3 \beta }v(N^{\beta }(x-y))\) $V_N(x-y)=N^{3\beta }v(N^{\beta }(x-y))$ for \(\beta <1\) $\beta <1$ . The potential v 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 \(\Lambda \) $\mathrm{\Lambda }$ develop dynamically in time. The estimates are expected to improve the Fock space bounds of [Ann. Henri Poincaré 23(2), 615–673 (2021)] by the first and fifth author. This will be addressed in a subsequent paper.