Inverse Energy Cascade and Non-uniqueness for the Advection-Diffusion Equation

We prove non-uniqueness for the advection-diffusion equation with a drift in a logarithmically near-critical class below $L_t^2{L}_x^{\mathrm{\infty }}$. The first construction is a diffusion-assisted inverse cascade of scalar energy that diverges as $t\to 0$. The second uses compressed inverse mixing, with diffusion acting perturbatively, and produces a bounded parabolic solution with the energy jump at the initial time. In both constructions the drift and scalar are smooth for positive time. We also consider the drift in $L_t^{2,\mathrm{\infty }}{L}_x^{\mathrm{\infty }}$ and solutions satisfying the energy inequality starting from almost every time. In this setting, we prove uniqueness of solutions with bounded energy, and construct non-unique solution with energy blowing up as time goes to zero.