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 second third author globally time. The proof is based on interaction Morawetz-type estimates Strichartz (including some new end-point results) for equation {1i?t??x??y+1NVN(x?y)}?(t,x,y)=F mixed coordinates such as Lp(dt)Lq(dx)L2(dy), Lp(dt)Lq(dy)L2(dx), Lp(dt)Lq(d(x?y))L2(d(x+y)). main technical ingredient a dispersive estimate coordinates, which may be interest its own right.