$$L^2$$-Critical Nonuniqueness for the 2D Navier-Stokes Equations

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