Second order corrections to mean field evolution for weakly interacting Bosons in the case of 3-body interactions

In this paper, we consider the Hamiltonian evolution of N weakly interacting Bosons. Assuming triple collisions, its mean field approximation is given by a quintic Hartree equation. We construct a second order correction to the mean field approximation using a kernel k(t,x,y) and derive an evolution equation for k. We show the global existence for the resulting evolution equation for the correction and establish an apriori estimate comparing the approximation to the exact Hamiltonian evolution. Our error estimate is global and uniform in time. Comparing with the work in [22,12,13] where the error estimate grows in time, our approximation tracks the exact dynamics for all time with an error of the order O(1/$\sqrt{N}$).