论文标题
弱KAM解决方案的全球行为，用于精确的符号扭曲图
Global Behaviors of weak KAM Solutions for exact symplectic Twist Maps
论文作者
论文摘要
我们研究了弱KAM解决方案的几种全局行为$ u_c（x，t）$由$ c \ in H^1（\ Mathbb t，\ Mathbb r）$参数化。对于确切的符号扭曲地图的悬挂式汉密尔顿$ h（x，p，t）$，我们可以找到由$ c（c（x，t）$ commotertions $ c（c（x，t）$）由$ c（σ）\ in h^1（\ mathbb t，\ mathbb t，\ mathbb r）$ comminetry和$ c（σ）$ contine -c（σ） \ partial_tu_c+h（x，x，\ partial_x u_c+c，t）=α（c），\ quad \ text {a.e. \}（x，x，x，x，t）\ in \ mathbb t^2，\]，因此弱kam解决方案的顺序r）} $是$ 1/2- $hölder的参数$σ\ in \ mathbb {r} $的连续性。此外，对于每个通用特征（无论是规则或单数）求解\ [\ left \ {\ begin {Aligned}＆\ dot {x}（x}（s）\ in \ text {co} \ big [\ partial_ph \ big（x（s），c+d^+u_c \ big（x（s），s+t \ big），s+t \ big）\ big] \ big]，＆\\＆x（0）= x_0，\ x_0，\ quad（x_0，x_0，x_0，t） \]我们通过唯一识别的旋转号$ω（c）\在h_1（\ mathbb t，\ mathbb r）$中对其进行评估。该特性导致相位空间中的一定拓扑阻塞，并引起局部横向轨迹现象。此外，我们讨论了这一点适用于高维情况。
We investigated several global behaviors of the weak KAM solutions $u_c(x,t)$ parametrized by $c\in H^1(\mathbb T,\mathbb R)$. For the suspended Hamiltonian $H(x,p,t)$ of the exact symplectic twist map, we could find a family of weak KAM solutions $u_c(x,t)$ parametrized by $c(σ)\in H^1(\mathbb T,\mathbb R)$ with $c(σ)$ continuous and monotonic and \[ \partial_tu_c+H(x,\partial_x u_c+c,t)=α(c),\quad \text{a.e.\ } (x,t)\in\mathbb T^2, \] such that sequence of weak KAM solutions $\{u_c\}_{c\in H^1(\mathbb T,\mathbb R)}$ is $1/2-$Hölder continuity of parameter $σ\in \mathbb{R}$. Moreover, for each generalized characteristic (no matter regular or singular) solving \[ \left\{ \begin{aligned} &\dot{x}(s)\in \text{co} \Big[\partial_pH\Big(x(s),c+D^+u_c\big(x(s),s+t\big),s+t\Big)\Big], & \\ &x(0)=x_0,\quad (x_0,t)\in\mathbb T^2,& \end{aligned} \right. \] we evaluate it by a uniquely identified rotational number $ω(c)\in H_1(\mathbb T,\mathbb R)$. This property leads to a certain topological obstruction in the phase space and causes local transitive phenomenon of trajectories. Besides, we discussed this applies to high-dimensional cases.
