Global estimates for the Hartree-Fock-Bogoliubov equations

Abstract

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 $\{\frac{1}{i}{\partial }_t-{\Delta }_x-{\Delta }_y+\frac{1}{N}{V}_N(x-y)\}\Lambda (t,x,y)=F$ 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.