AI 中文总结
该研究针对具有多尺度有限范围分解的随机Hölder速度场，确定了其关联方程适定性的$\alpha=1/2$临界阈值，给出了阈值下的正则性估计并验证了阈值上下的适定性转变。
AI 中文摘要
我们研究与自治随机速度场相关的常微分方程、流映射及连续性方程的行为，这类速度场具有自然的多尺度有限范围分解。所考虑的速度场在空间上仅为Hölder正则——即属于$C^{\alpha-}(\mathbb{T}^d)$，其中$\alpha \in (0,1)$，因此对应的常微分方程和连续性方程先验并非适定。然而，在$\alpha = 1/2$的临界阈值之上，由于多尺度随机抵消，我们证明在速度场的零水平集之外，适定性几乎必然恢复。该阈值标志着一个真实的转变，阈值下方的例子表现出强不适定性，这一点得到了验证。我们还提供了临界阈值下方的有效正则性估计，并在相关的“刷新” regime中证明了类似结果。
英文摘要
We study the behavior of the ordinary differential equations, flow maps, and continuity equations associated to autonomous random velocity fields that admit a natural multiscale finite range decomposition. The velocity fields we consider are only Hölder regular in space---$C^{α-}(\mathbb{T}^d)$ for some $α\in (0,1)$---thus the associated ODE and continuity equation are not a priori well-posed. However, above the critical threshold of $α= 1/2$, due to multiscale stochastic cancellations, we prove well-posedness is almost surely restored away from the zero level set of the velocity field. This threshold marks a genuine transition, as demonstrated by examples lying below the threshold that exhibit robust ill-posedness. We additionally provide effective regularity estimates below the critical threshold and prove analogous results in the related "refreshing" regime.
Comments109 pages
