借助具有二维自治速度场的自发随机性实现反常耗散
Anomalous dissipation via spontaneous stochasticity with a two-dimensional autonomous velocity field
AI总结：
本文在被动标量背景下，构造了一个表现出反常耗散的二维无散度自治速度场，并利用涨落-耗散公式证明零噪声极限下随机轨迹方差为正，从而回答了受迫三维Navier-Stokes方程反常耗散的相关问题。
摘要：
我们在被动标量的背景下研究反常耗散，并构造了一个属于 $C^α$（其中 $α\in (0,1)$ 为任意但固定的值）的二维无散度自治速度场，该速度场表现出反常耗散。我们的证明采用了涨落-耗散公式，该公式将自发随机性与反常耗散联系起来。因此，我们通过证明在零噪声极限下随机轨迹的方差保持为正，来解决反常耗散问题。基于这一结果，我们回答了 [Bruè & De Lellis '22] 中关于受迫三维 Navier-Stokes 方程反常耗散的 Question 2.2 和 Question 2.3。
英文摘要：
We study anomalous dissipation in the context of passive scalars and we construct a two-dimensional autonomous divergence-free velocity field in $C^α$ (with $α\in (0,1)$ arbitrary but fixed) which exhibits anomalous dissipation. Our proof employs the fluctuation-dissipation formula, which links spontaneous stochasticity with anomalous dissipation. Therefore, we address the issue of anomalous dissipation by showing that the variance of stochastic trajectories, in the zero noise limit, remains positive. Based on this result, we answer Question 2.2 and Question 2.3 in [Bruè & De Lellis '22] regarding anomalous dissipation for the forced three-dimensional Navier-Stokes equations.
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