Anomalous dissipation via spontaneous stochasticity with a two-dimensional autonomous velocity field

TL;DR

The paper constructs a two-dimensional autonomous, divergence-free velocity field $u\in C^{\alpha}$ that induces anomalous dissipation for a passive scalar in the advection–diffusion equation, establishing a direct link via the fluctuation–dissipation formula to spontaneous stochasticity. The core idea is to design a backward stochastic flow around a fat Cantor-type dissipative set where small noise yields significantly divergent trajectories, ensuring nonvanishing dissipation in the vanishing-diffusivity limit and continuous-in-time dissipation. The authors introduce a novel criterion that reduces anomalous dissipation to finite-time stochastic separation of backward trajectories, prove stability via a pipe-geometry construction with branching and widening, and carefully manage mollification to achieve the required Hölder regularity. As a major consequence, they adapt the construction to produce anomalous dissipation for the forced 3D Navier–Stokes equations in a $(2+\tfrac12)$-dimensional setting, addressing open questions in the literature and providing new insight into energy cascade mechanisms in autonomous flows.

Abstract

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.

Figures (14)

• Figure 1: The velocity field $b_1$. The set $C_1$, in water green, is composed by $N_1 = 2 n_1 = 8$ rectangles of size $L_1 \times (A_1 + B_1)$.
• Figure 2: We draw in water green the set $C_1$ and in blue the set $C_2$. Each small blue rectangle has size $L_2 \times (A_2 + B_2)$.
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• Figure 4: The initial steps of the proof of Theorem \ref{['thm-main']}. Note that for $\omega \in \Omega_2$ the backward stochastic trajectory is only depicted until it hits the middle of the pipe.
• Figure 5: An enlarging pipe.

Theorems & Definitions (15)

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