Strong Stability of KAM Tori: from the Point of View of Viscosity Solutions of H-J Equations
Strong Stability of KAM Tori: from the Point of View of Viscosity Solutions of H-J Equations
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DOI：
10.1007/s10884-009-9131-z
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发表时间：
2009-03
影响因子：
1.3
通讯作者：
Zhenguo Liang;Jun Yan
中科院分区：
文献类型：
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作者：
Zhenguo Liang;Jun Yan
众所周知，KAM环面可以看作是光滑黏度解的图。Salamon和Zehnder （Comment Math Helv 64:84 - 132,1989）证明了具有相关HamiltonianH的近似可积正定lagrange系统存在具有规定丢番图频率的不变环面，其丢番图指数为τ。如果不变环面用协切束表示，那么我们可以证明，对于满足H-J方程（1.1）的任意粘度解u(x，P)，当足够小时$$\|Du(x,P)-Dv(x,P_0)\|_{\infty} \le C\|P-P_0\|^{\frac{1}{\tau+1}},$$。
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 HamiltonianH, whose Diophantine index isτ. If the invariant torus is represented asin the cotangent bundle, then we can show that for any viscosity solutionu(x,P), which satisfies the H-J Eq. (1.1),$$\|Du(x,P)-Dv(x,P_0)\|_{\infty} \le C\|P-P_0\|^{\frac{1}{\tau+1}},$$whenis small enough.