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

Abstract

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 $\mathrm{BMO}$. For the surface quasi-geostrophic (SQG) and incompressible porous media (IPM) equations, we obtain uniqueness in spaces logarithmically rougher than $C^{0,1}$. 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.