Weak Sard characterization of dissipation anomalies for passive scalar flows

Let $v\in C^0(\mathbb T^2,\mathbb R^2)$ be an autonomous, average-free, divergence-free vector field, and let its stream function be the scalar function whose rotated gradient gives $v$. The weak Sard property is the property defined in the source for this stream function.

Weak Sard characterization. The vector field $v$ allows dissipation anomalies if and only if its stream function violates the weak Sard property.

This conjecture proposes an exact deterministic characterization of dissipation anomalies in terms of the weak Sard property. The surrounding discussion explains that the weak Sard property is related to the DiPerna–Lions renormalization property and uniqueness of regular Lagrangian flows, while the corresponding implication for random continuous divergence-free vector fields follows from a result of Alberti, Bianchini, and Crippa. The converse deterministic characterization remains open in the source.