Non-uniqueness of parabolic solutions for advection-diffusion equation

Abstract

We present a novel example of a divergence–free velocity field $b\in L^{\infty }\left(\right(0,1);L^p(T^2\left)\right)$ for $p<2$ arbitrary but fixed which leads to non-unique solutions of advection–diffusion in the class $L_{t,x}^{\infty }\cap L_t^2{H}_x^1$ while satisfying the local energy inequality. This result complements the known uniqueness result of bounded solutions for divergence-free and $L_{t,x}^2$ 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 $b\in L^p\left(\right(0,1);L^{\infty }(T^2\left)\right)$, 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 $d\ge 3$ divergence–free velocity fields $b\in L^{\infty }\left(\right(0,1);L^p(T^d\left)\right)$ with $p<\frac{2d}{d+2}$ could lead to non–unique solutions in the space $L_t^{\infty }{L}_x^{\frac{2d}{d-2}}\cap L_t^2{H}_x^1$. Our proof is based on a stochastic Lagrangian approach and does not rely on convex integration.