Second Order Corrections to Mean Field Evolution for Weakly Interacting Bosons in The Case of Three-body Interactions

• Published in: Archive for rational mechanics and analysis Vol. 203; no. 2; pp. 455 - 497
• Main Author: Chen, Xuwen
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.02.2012; Springer Nature B.V
• Subjects: Classical Mechanics; Complex Systems; Fluid- and Aerodynamics; Mathematical and Computational Physics; Physics; Physics and Astronomy; Theoretical; Commun Math Phys; Metaplectic Representation; Hartree Equation; Order Correction; Endpoint Strichartz Estimate
• ISSN: 0003-9527; 1432-0673
• DOI: 10.1007/s00205-011-0453-8

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 global existence for the resulting evolution equation for the correction and establish an a priori estimate comparing the approximation to the exact Hamiltonian evolution. Our error estimate is global and uniform in time. Comparing with the work of R odnianski and S chlein (Commun Math Phys 291:31–61, 2009 ), and G rillakis , M achedon and M argetis (Commun Math Phys 294:273–301, 2010 ; Adv Math 288:1788–1815, 2011 ), where the error estimate grows in time, our approximation tracks the exact dynamics for all time with an error of the order