Viscous stability of quasi-periodic tori

Abstract

AbstractThis paper is devoted to the study of$P$-regularity of viscosity solutions$u(x,P)$,$P\in {\Bbb R}^n$, of a smooth Tonelli Lagrangian$L:T {\Bbb T}^n \rightarrow {\Bbb R}$characterized by the cell equation$H(x,P+D_xu(x,P))=\overline {H}(P)$, where$H: T^* {\Bbb T}^n\rightarrow {\Bbb R}$denotes the Hamiltonian associated with$L$and$\overline {H}$is the effective Hamiltonian. We show that if$P_0$corresponds to a quasi-periodic invariant torus with a non-resonant frequency, then$D_xu(x,P)$is uniformly Hölder continuous in$P$at$P_0$with Hölder exponent arbitrarily close to$1$, and if both$H$and the torus are real analytic and the frequency vector of the torus is Diophantine, then$D_xu(x,P)$is uniformly Lipschitz continuous in$P$at$P_0$, i.e., there is a constant$C\gt 0$such that$\|D_xu(\cdot ,P)-D_xu(\cdot ,P_0)\|_{\infty }\le C\|P-P_0\|$for$\|P-P_0\|\ll 1$. Similar P-regularity of the Peierls barriers associated with$L(x,v)- \langle P,v \rangle $is also obtained.