Abstract:We prove the absence of anomalous dissipation for passive scalars driven by some random autonomous divergence-free vector fields in $\mathbb T^d$. In dimension $d=2$ we just need continuity almost surely and a mild nondegeneracy condition on the randomness. In dimension $d\geq 3$ we assume a special geometric structure and almost sure Hölder regularity with a Hölder exponent bigger than $\frac{1}{8}$. No regularity is assumed on the passive scalar except for boundedness in the initial data. The proof relies on dimension-theoretic arguments, as opposed to commutator estimates. A consequence of these results is that the same assumptions prevent (almost surely) many other expected properties of turbulent flows, such as anomalous regularization, the Yaglom-Obukhov-Corrsin law, and Richardson diffusion.
From: Daniel W. Boutros [view email]
[v1]
Thu, 12 Mar 2026 02:44:36 UTC (21 KB)
[v2]
Wed, 1 Apr 2026 23:00:35 UTC (26 KB)
[v3]
Thu, 16 Jul 2026 20:05:52 UTC (27 KB)
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