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

Abstract

We consider the stationary diffusion equation \(-\mathrm {div} (\nabla u + bu )=f\) in d-dimensional torus \(\mathbb {T}^d\), where \(f\in H^{-1}\) is a given forcing and \(b\in L^p\) 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 \(\Omega \subset \mathbb {R}^d\), and proved existence of solutions for \(b\in L^{2d/(d+2)}\), uniqueness for \(b\in L^2\), and has provided a point-singularity counterexample that shows nonuniqueness for \(b\in L^{3/2-}\) 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 \(b\in W^{1,1}\). We use a Nash iteration to demonstrate sharpness of this result, and also show that solutions in \(H^1\cap L^{p/(p-1)}\) are flexible for \(b\in L^p\), \(p\in [1,2(d-1)/(d+1))\); namely we show that the set of \(b\in L^p\) for which nonuniqueness in the class \(H^1\cap L^{p/(p-1)}\) occurs is dense in the divergence-free subspace of \(L^p\).