论文标题
增强二维哈密顿流量的耗散
Enhanced dissipation for two-dimensional Hamiltonian flows
论文作者
论文摘要
令$ h \ in c^1 \ cap w^{2，p} $是在紧凑型$ 2 $ 2二维的歧管上的自主，非恒定的哈密顿式，产生了不可压缩的速度字段$ b = \ nabla^\ perp h $。就封闭轨道$ \ \ {h = h \} $的周期$ t（h）$的属性而言，我们对$ b $的增强耗散率的急剧上限。具体而言，如果$ 0 <ν\ ll 1 $是扩散系数，则增强的耗散率最多可以是$ o（ν^{1/3}）$，一般而言，当$ h $具有隔离，非分支椭圆点时，边界会有所改善。我们的结果为标准的蜂窝流量提供了更好的绑定$ O（ν^{1/2}）$，由$ H_ \ Mathsf {C}（x）= \ sin x_1 \ sin x_2 $给出，我们还可以证明，我们还可以证明，在其混合混合率和较低的消散率上，我们还可以在其上获得新的上限。这些证明是基于使用动作角度坐标的使用以及存在$ b $生成的常规拉格朗日流的良好不变域。
Let $H\in C^1\cap W^{2,p}$ be an autonomous, non-constant Hamiltonian on a compact $2$-dimensional manifold, generating an incompressible velocity field $b=\nabla^\perp H$. We give sharp upper bounds on the enhanced dissipation rate of $b$ in terms of the properties of the period $T(h)$ of the close orbits $\{H=h\}$. Specifically, if $0<ν\ll 1$ is the diffusion coefficient, the enhanced dissipation rate can be at most $O(ν^{1/3})$ in general, the bound improves when $H$ has isolated, non-degenerate elliptic point. Our result provides the better bound $O(ν^{1/2})$ for the standard cellular flow given by $H_\mathsf{c}(x)=\sin x_1 \sin x_2$, for which we can also prove a new upper bound on its mixing mixing rate and a lower bound on its enhanced dissipation rate. The proofs are based on the use of action-angle coordinates and on the existence of a good invariant domain for the regular Lagrangian flow generated by $b$.
