We derive time-averaged \(L^1\) estimates on Littlewood–Paley decompositions for linear advection–diffusion equations. For wave numbers close to the dissipative cutoff, these estimates are consistent with Batchelor’s predictions on the variance spectrum in passive scalar turbulent mixing.

中文翻译：

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关于无标量输运方程的Littlewood-Paley谱

对于线性对流扩散方程，我们导出了Littlewood-Paley分解的时间平均\（L ^ 1 \）估计。对于接近耗散截止的波数，这些估计与Batchelor对被动标量湍流混合中方差谱的预测一致。