\(L^2\)-Critical Nonuniqueness for the 2D Navier-Stokes Equations

In this paper, we consider the 2D incompressible Navier-Stokes equations on the torus. It is well known that for any \(L^2\) divergence-free initial data, there exists a global smooth solution that is unique in the class of \(C_t L^2\) weak solutions. We show that such uniqueness would fail in the class \(C_t L^p\) if \( p<2\). The non-unique solutions we constructed are almost \(L^2\)-critical in the sense that (i) they are uniformly continuous in \(L^p\) for every \(p<2\); (ii) the kinetic energy agrees with any given smooth positive profile except on a set of arbitrarily small measure in time.