A necessary and sufficient condition for convergence of the Lax–Oleinik semigroup for reversible Hamiltonians on Rn

Qihuai Liu; Kaizhi Wang; Lin Wang; Jun Yan

doi:10.1016/j.jde.2016.08.001

A necessary and sufficient condition for convergence of the Lax–Oleinik semigroup for reversible Hamiltonians on Rⁿ

Qihuai Liu^*, Kaizhi Wang, Lin Wang, Jun Yan

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

This paper is devoted to the study of the convergence of the Lax–Oleinik semigroup associated with reversible Hamiltonians H(x,p) on Rⁿ. We provide a necessary and sufficient condition for the convergence of the semigroup. We also give an example to show that for irreversible Hamiltonians on Rⁿ, even if the Hamiltonian is integrable and the initial data is Lipschitz continuous and bounded, the corresponding Lax–Oleinik semigroup may not converge.

Original language	English
Pages (from-to)	5289-5305
Number of pages	17
Journal	Journal of Differential Equations
Volume	261
Issue number	10
DOIs	https://doi.org/10.1016/j.jde.2016.08.001
Publication status	Published - 15 Nov 2016
Externally published	Yes

Keywords

Convergence
Hamilton–Jacobi equations
Lax–Oleinik semigroup
Non-compact manifold
Reversible Hamiltonians

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10.1016/j.jde.2016.08.001

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Liu, Q., Wang, K., Wang, L., & Yan, J. (2016). A necessary and sufficient condition for convergence of the Lax–Oleinik semigroup for reversible Hamiltonians on Rⁿ. Journal of Differential Equations, 261(10), 5289-5305. https://doi.org/10.1016/j.jde.2016.08.001

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title = "A necessary and sufficient condition for convergence of the Lax–Oleinik semigroup for reversible Hamiltonians on Rn",

abstract = "This paper is devoted to the study of the convergence of the Lax–Oleinik semigroup associated with reversible Hamiltonians H(x,p) on Rn. We provide a necessary and sufficient condition for the convergence of the semigroup. We also give an example to show that for irreversible Hamiltonians on Rn, even if the Hamiltonian is integrable and the initial data is Lipschitz continuous and bounded, the corresponding Lax–Oleinik semigroup may not converge.",

keywords = "Convergence, Hamilton–Jacobi equations, Lax–Oleinik semigroup, Non-compact manifold, Reversible Hamiltonians",

author = "Qihuai Liu and Kaizhi Wang and Lin Wang and Jun Yan",

note = "Publisher Copyright: {\textcopyright} 2016 Elsevier Inc.",

year = "2016",

month = nov,

day = "15",

doi = "10.1016/j.jde.2016.08.001",

language = "English",

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journal = "Journal of Differential Equations",

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T1 - A necessary and sufficient condition for convergence of the Lax–Oleinik semigroup for reversible Hamiltonians on Rn

AU - Liu, Qihuai

AU - Wang, Kaizhi

AU - Wang, Lin

AU - Yan, Jun

PY - 2016/11/15

Y1 - 2016/11/15

N2 - This paper is devoted to the study of the convergence of the Lax–Oleinik semigroup associated with reversible Hamiltonians H(x,p) on Rn. We provide a necessary and sufficient condition for the convergence of the semigroup. We also give an example to show that for irreversible Hamiltonians on Rn, even if the Hamiltonian is integrable and the initial data is Lipschitz continuous and bounded, the corresponding Lax–Oleinik semigroup may not converge.

AB - This paper is devoted to the study of the convergence of the Lax–Oleinik semigroup associated with reversible Hamiltonians H(x,p) on Rn. We provide a necessary and sufficient condition for the convergence of the semigroup. We also give an example to show that for irreversible Hamiltonians on Rn, even if the Hamiltonian is integrable and the initial data is Lipschitz continuous and bounded, the corresponding Lax–Oleinik semigroup may not converge.

KW - Convergence

KW - Hamilton–Jacobi equations

KW - Lax–Oleinik semigroup

KW - Non-compact manifold

KW - Reversible Hamiltonians

UR - http://www.scopus.com/inward/record.url?scp=84992154001&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2016.08.001

DO - 10.1016/j.jde.2016.08.001

M3 - Article

AN - SCOPUS:84992154001

SN - 0022-0396

VL - 261

SP - 5289

EP - 5305

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 10

ER -

A necessary and sufficient condition for convergence of the Lax–Oleinik semigroup for reversible Hamiltonians on Rⁿ

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Keywords

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