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

作者：Liang Zhenguo^*; Yan Jun

来源：Journal of Dynamics and Differential Equations, 2009, 21(2): 353-370.

DOI：10.1007/s10884-009-9131-z

• Strong stability
• KAM tori
• Viscosity solution
• H-J equation

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 tau. If the invariant torus is represented as G = U(x is an element of T)(n) (x, P(0) D nu(x, P(0))) in the cotangent bundle T*T(n), then we can show(1) that for any viscosity solution u (x, P), which satisfies the H-J Eq. (1.1),
parallel to Du(x, P) - D nu(x, P(0))parallel to(infinity) <= C parallel to P - P(0)parallel to(1/tau 1),
when parallel to P - P(0)parallel to is small enough.

• 出版日期2009-6
• 单位复旦大学