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Viscous stability of quasi-periodic tori

Published online by Cambridge University Press:  27 November 2012

ZHENGUO LIANG ,
JUN YAN  and
YINGFEI YI
ZHENGUO LIANG
Affiliation:
School of Mathematical Sciences and Key Lab of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, China (email: zgliang@fudan.edu.cn, yanjun@fudan.edu.cn)
JUN YAN
Affiliation:
School of Mathematical Sciences and Key Lab of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, China (email: zgliang@fudan.edu.cn, yanjun@fudan.edu.cn)
YINGFEI YI
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA College of Mathematics, Jilin University, Changchun 130012, PR China (email: yi@math.gatech.edu)
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Abstract

This 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 1 , February 2014 , pp. 185 - 210
Copyright
Copyright © 2012 Cambridge University Press 

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