Non-uniqueness of parabolic solutions for advection-diffusion equation

Abstract: We present a novel example of a divergence-free velocity field $b \in L^\infty ((0,1); L^p (\mathbb{T}^2))$ for $p<2$ arbitrary but fixed which leads to non-unique solutions of advection-diffusion in the class $L^\infty_{t,x} \cap L^2_t H^1_x$ while satisfying the local energy inequality. This result complements the known uniqueness result of bounded solutions for divergence-free and $L^2_{t,x}$ integrable velocity fields. Additionally, we also prove the necessity of time integrability of the velocity field for the uniqueness result. More precisely, we construct another divergence-free velocity field $b \in L^p ((0,1); L^\infty (\mathbb{T}^2))$, for $p< 2$ fixed, but arbitrary, with non-unique aforementioned solutions. Our contribution closes the gap between the regime of uniqueness and non-uniqueness in this context. Previously, it was shown with the convex integration technique that for $d\geq 3$ divergence-free velocity fields $ b \in L^\infty((0,1);L^p (\mathbb{T}^d))$ with $p < \frac{2d}{d+2}$ could lead to non-unique solutions in the space $L^\infty_t L^{\frac{2d}{d-2}}_x \cap L^2_t H^1_x$. Our proof is based on a stochastic Lagrangian approach and does not rely on convex integration.