Non-uniqueness of parabolic solutions for advection-diffusion equation

TL;DR

This paper constructs divergence-free velocity fields in specific Lebesgue spaces that lead to non-uniqueness of solutions for the advection-diffusion equation, highlighting the importance of velocity field regularity for solution uniqueness.

Contribution

It provides explicit examples of non-unique solutions for the advection-diffusion equation in the subcritical Lebesgue space regime, using a stochastic Lagrangian approach, and clarifies the necessity of time integrability.

Findings

Non-uniqueness occurs for velocity fields in L^p with p<2.

Constructs divergence-free velocity fields causing non-uniqueness.

Highlights the necessity of time integrability for solution uniqueness.

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.