Title: Fourier mass lower bounds for Batchelor-regime passive scalars Show Chinese title

Title: 傅里叶质量在批量规则被动标量中的下界

Abstract: Batchelor predicted that a passive scalar $\psi^\nu$ with diffusivity $\nu$, advected by a smooth fluid velocity, should typically have Fourier mass distributed as $|\hat \psi^\nu|^2(k) \approx |k|^{-d}$ for $|k| \ll \nu^{-1/2}$. For a broad class of velocity fields, we give a quantitative lower bound for a version of this prediction summed over constant width annuli in Fourier space. This improves on previously known results, which require the prediction to be summed over the whole ball. Show Chinese abstract

Abstract: Batchelor预测，一个扩散系数为$\nu$的被动标量$\psi^\nu$，由平滑流体速度输运，通常其傅里叶质量分布为$|\hat \psi^\nu|^2(k) \approx |k|^{-d}$，当$|k| \ll \nu^{-1/2}$时。 对于一大类速度场，我们给出了这个预测在傅里叶空间中宽度恒定的环形区域上求和的一个定量下界。 这改进了以前的结果，以前的结果要求预测在整球上求和。