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

This paper is devoted to the study of the convergence of the Lax–Oleinik semigroup associated with reversible Hamiltonians H(x,p)H(x,p) on RnRn. 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 RnRn, even if the Hamiltonian is integrable and the initial data is Lipschitz continuous and bounded, the corresponding Lax–Oleinik semigroup may not converge.