A Lagrangian Approach to Uniqueness for Active Scalars in Generalized Mean Oscillation Spaces

We develop a unified Lagrangian framework for the uniqueness of weak solutions to a broad class of two-dimensional inviscid active scalar equations in generalized mean oscillation spaces. Under common structural assumptions on the Biot--Savart kernels, we identify mean oscillation spaces that ensure an Osgood condition for the velocity, and establish uniqueness in these classes. For the two-dimensional Euler equations, we establish uniqueness for vorticities whose mean oscillation grows mildly at small scales, extending beyond BMO. For the surface quasi-geostrophic (SQG) and incompressible porous media (IPM) equations, we obtain uniqueness in spaces logarithmically rougher than C0,1. We also obtain what are (to our knowledge) the strongest known uniqueness results for the $α$-SQG equation, $0<α<1$, in spaces logarithmically rougher than critical Hölder spaces C^α. The framework also applies to various logarithmically perturbed models, yielding uniqueness in the corresponding sharp regularity classes. The proof combines a sharp regularity estimate for the velocity with a stability estimate for integral operators under measure-preserving rearrangements. This provides a single uniqueness argument for the entire family under common structural assumptions on the kernels.