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

We study anomalous dissipation in the context of passive scalars and we construct a two-dimensional autonomous divergence-free velocity field in \(C^\alpha\) (with \(\alpha \in (0,1)\) 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 [Bru\`{e} & De Lellis '22] regarding anomalous dissipation for the forced three-dimensional Navier-Stokes equations.