Pathological solutions of Navier-Stokes equations on $\mathbb{T}^2$ with gradients in Hardy spaces

For an arbitrary smooth initial datum, we construct multiple nonzero solutions to the $2$d Navier-Stokes equations, with their gradients in the Hardy space $\mathcal{H}^p$ with any $p \in (0,1)$. Thus, in terms of the path space $C(\mathcal{H}^p)$ for vorticity, $p=1$ is the threshold value distinguishing between non-uniqueness and uniqueness regimes. In order to obtain our result, we develop the needed theory of Hardy spaces on periodic domains.