A partial uniqueness result and an asymptotically sharp nonuniqueness result for the Zhikov problem on the torus

We consider the stationary diffusion equation - div ( ∇ u + b u ) = f in d -dimensional torus T d , where f ∈ H - 1 is a given forcing and b ∈ L p is a divergence-free drift. Zhikov (Funkts Anal Prilozhen, 38(3):15–28, 2004) considered this equation in the case of a bounded, Lipschitz domain Ω ⊂ R d , and proved existence of solutions for b ∈ L 2 d / ( d + 2 ) , uniqueness for b ∈ L 2 , and has provided a point-singularity counterexample that shows nonuniqueness for b ∈ L 3 / 2 - and d = 3 , 4 , 5 . We apply a duality method and a DiPerna–Lions-type estimate to show uniqueness of the solutions constructed by Zhikov for b ∈ W 1 , 1 . We use a Nash iteration to demonstrate sharpness of this result, and also show that solutions in H 1 ∩ L p / ( p - 1 ) are flexible for b ∈ L p , p ∈ [ 1 , 2 ( d - 1 ) / ( d + 1 ) ) ; namely we show that the set of b ∈ L p for which nonuniqueness in the class H 1 ∩ L p / ( p - 1 ) occurs is dense in the divergence-free subspace of L p .