We study the mixing and dissipation properties of the advection-diffusion equation with diffusivity $0 < \kappa \ll 1$ and advection by a class of random velocity fields on $\mathbb T^d$, $d=\{2,3\}$, including solutions of the 2D Navier-Stokes equations forced by sufficiently regular-in-space, non-degenerate white-in-time noise. We prove that the solution almost surely mixes exponentially fast uniformly in the diffusivity $\kappa$. Namely, that there is a deterministic, exponential rate (independent of $\kappa$) such that all mean-zero $H^1$ initial data decays exponentially fast in $H^{-1}$ at this rate with probability one. This implies almost-sure enhanced dissipation in $L^2$. Specifically that there is a deterministic, uniform-in-$\kappa$, exponential decay in $L^2$ after time $t \gtrsim |\log \kappa|$. Both the $O(|\log \kappa|)$ time-scale and the uniform-in-$\kappa$ exponential mixing are optimal for Lipschitz velocity fields and, to our knowledge, are the first rigorous examples of velocity fields satisfying these properties (deterministic or stochastic). This work is also a major step in our program on scalar mixing and Lagrangian chaos necessary for a rigorous proof of the Batchelor power spectrum of passive scalar turbulence.

中文翻译：

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通过随机 Navier-Stokes 对平流扩散的几乎肯定增强的耗散和均匀扩散指数混合

我们研究了扩散系数为 $0 < \kappa \ll 1$ 的对流-扩散方程的混合和耗散特性，并且在 $\mathbb T^d$, $d=\{2,3\ }$，包括二维 Navier-Stokes 方程的解，该方程由足够规则的空间非退化时间白噪声强制。我们证明该解决方案几乎肯定会在扩散系数 $\kappa$ 中以指数方式快速均匀混合。即，存在确定性的指数速率（与 $\kappa$ 无关），使得所有均值为零的 $H^1$ 初始数据在 $H^{-1}$ 中以该速率呈指数快速衰减，概率为 1。这意味着在 $L^2$ 中几乎肯定会增强耗散。具体来说，在时间 $t \gtrsim |\log \kappa|$ 之后，$L^2$ 中存在确定性的、均匀的 $\kappa$ 指数衰减。$O(|\log \kappa|)$ 时间尺度和均匀在 $\kappa$ 指数混合都是 Lipschitz 速度场的最佳选择，据我们所知，这是第一个满足这些速度场的严格例子属性（确定性或随机性）。这项工作也是我们关于标量混合和拉格朗日混沌程序的重要一步，这对于严格证明被动标量湍流的 Batchelor 功率谱是必不可少的。