Second Order Corrections to Mean Field Evolution for Weakly Interacting Bosons in The Case of Three-body Interactions
Second Order Corrections to Mean Field Evolution for Weakly Interacting Bosons in The Case of Three-body Interactions
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DOI：
10.1007/s00205-011-0453-8
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发表时间：
2012-02-01
影响因子：
2.5
通讯作者：
Chen, Xuwen
中科院分区：
文献类型：
--
作者：
Chen, Xuwen
本文研究了N个弱相互作用玻色子的哈密顿演化。假设三重碰撞，它的平均场近似由五次Hartree方程给出。我们构造了一个二阶修正的平均场近似使用核k（t，x，y），并推导出一个发展方程的k。我们证明了所得到的校正演化方程的全局存在性，并建立了一个先验估计比较近似的精确哈密顿演化。我们的误差估计是全球性的和统一的时间。与RODNIANSKI和SCHLEIN（Commun Math Phys 291：31-61，2009）以及GRILLAKIS，MACHEDON和MARGETIS（Commun Math Phys 294：273-301，2010; Adv Math 288：1788-1815，2011）的工作相比，其中误差估计随时间增长，我们的近似跟踪所有时间的精确动力学，误差为O（1/root N）。
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 RODNIANSKI AND SCHLEIN (Commun Math Phys 291:31-61, 2009), and GRILLAKIS, MACHEDON AND MARGETIS (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 O(1/root N).