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

We consider the stationary diffusion equation $-$\mathrm{d}\mathrm{i}\mathrm{v}$ (\nabla u + bu )=f$in$n$-dimensionaltorus$$\mathbb{T}$^n$,where$f\in H^{-1}$ is a given forcing and $b\in L^p$ is a divergence-free drift. Zhikov (Funkts. Anal. Prilozhen., 2004) considered this equation in the case of a bounded, Lipschitz domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, and proved existence of solutions for $b\in L^{2n\mathrm{/}(n+2)}$, uniqueness for $b\in L^2$, and has provided a point-singularity counterexample that shows nonuniqueness for $b\in L^{3\mathrm{/}2-}$ and $n=3,4,5$. We apply a duality method and a DiPerna-Lions-type estimate to show uniqueness of the solutions constructed by Zhikov for $b\in W^{1,1}$. We use a Nash iteration to demonstrate sharpness of this result, and also show that solutions in $H^1\cap L^{p\mathrm{/}(p-1)}$ are flexible for $b\in L^p$, $p\in [1,2(n-1)/(n+1))$;namelyweshowthatthesetof$b\in L^p$ for which nonuniqueness in the class $H^1\cap L^{p\mathrm{/}(p-1)}$ occurs is dense in the divergence-free subspace of $L^p$.