TY - JOUR
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
N1 - Publisher Copyright:
© 2016 Elsevier Inc.
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 -