A uniqueness result for the continuity equation in two dimensions

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Global existence of weak solutions for compressible Navier--Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor (2015, background): “This can only be improved with specific additional structure, such as provided by low dimension, see [3, 11, 23, 24, 41], Hamiltonian properties [19, 44], or as a singular integral [10].”
Exponential self-similar mixing and loss of regularity for continuity equations (2014, background): “1) is not uniquely solvable in general (see [1, 2]).”
Structure of level sets and Sard-type properties of Lipschitz maps (2013, background): “6 in [1] establishes the following relation between the locality of the divergence and the weak Sard property: let b be a bounded vector field on the plane which has bounded divergence and can be represented as b = a ∇⊥f with a a bounded function and f a Lipschitz function; if f satisfies the weak S”