Global estimates for the Hartree-Fock-Bogoliubov equations

We prove that certain Sobolev-type norms, slightly stronger than those given by energy conservation, stay bounded uniformly in time and N. This allows one to extend the local existence results of the second and third author globally in time. The proof is based on interaction Morawetz-type estimates and Strichartz estimates (including some new end-point results) for the equation {1/i partial derivative(t) - Delta(x) - Delta(y) + 1/NVN(x - y)}Lambda(t, x, y) = F in mixed coordinates such as L-p(dt)L-q(dx)L-2(dy), L-p(dt)L-q(dy)L-2(dx), L-p(dt)L-q(d(x-y))L-2(d(x+y)). The main new technical ingredient is a dispersive estimate in mixed coordinates, which may be of interest in its own right.

Automated summary

This paper proves that a certain Sobolev-type norm, slightly stronger than those given by energy conservation, remains uniformly bounded in time and N, enabling the extension of local existence results globally in time. The proof is based on interaction Morawetz-type estimates and Strichartz estimates, introducing a new dispersive estimate in mixed coordinates which may have independent interest.