Strong Stability of KAM Tori: from the Point of View of Viscosity Solutions of H-J Equations

Abstract

It is well-known that a KAM torus can be considered as a graph of smooth viscosity solution. Salamon and Zehnder (Comment Math Helv 64:84–132, 1989) have proved that there exist invariant tori having prescribed Diophantine frequencies for nearly integrable and positively definite Lagrangian systems with associated Hamiltonian H, whose Diophantine index is τ. If the invariant torus is represented as ${\mathcal{G}=\bigcup\limits_{x\in \mathbb{T}^n}(x,P_0+Dv(x,P_0))}$ in the cotangent bundle ${T^{*}\mathbb{T}^n}$, then we can show that for any viscosity solution u (x, P), which satisfies the H-J Eq. (1.1),

$$\|Du(x,P)-Dv(x,P_0)\|_{\infty} \le C\|P-P_0\|^{\frac{1}{\tau+1}},$$

when ${\|P-P_0\|}$ is small enough.