Effective Lagrangian regularity and the uniqueness threshold for random Hölder velocity fields
Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7201839123
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