Second Order Corrections to Mean Field Evolution for Weakly Interacting Bosons in The Case of Three-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 ev...
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| Published in | Archive for rational mechanics and analysis Vol. 203; no. 2; pp. 455 - 497 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer-Verlag
01.02.2012
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0003-9527 1432-0673 |
| DOI | 10.1007/s00205-011-0453-8 |
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| Abstract | 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 |
|---|---|
| AbstractList | 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 / N ) . [PUBLICATION ABSTRACT] 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 |
| Author | Chen, Xuwen |
| Author_xml | – sequence: 1 givenname: Xuwen surname: Chen fullname: Chen, Xuwen email: chenxuwen@math.umd.edu organization: Department of Mathematics, University of Maryland |
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| Snippet | In this paper, we consider the Hamiltonian evolution of
N
weakly interacting bosons. Assuming triple collisions, its mean field approximation is given by a... In this paper, we consider the Hamiltonian evolution of N weakly interacting bosons. Assuming triple collisions, its mean field approximation is given by a... |
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| Title | Second Order Corrections to Mean Field Evolution for Weakly Interacting Bosons in The Case of Three-body Interactions |
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