Extreme temporal intermittency in the linear Sobolev transport: almost smooth nonunique solutions

TL;DR

This paper demonstrates sharp nonuniqueness of solutions for linear transport equations with divergence-free vector fields of high Sobolev regularity, highlighting the limits of the DiPerna-Lions theory and extending to transport-diffusion equations.

Contribution

It constructs divergence-free vector fields with high Sobolev regularity that admit nonunique solutions, revealing the sharpness of the time-integrability condition in the DiPerna-Lions theory.

Findings

Nonuniqueness of weak solutions in Sobolev regularity settings.

Sharpness of the time-integrability assumption in the DiPerna-Lions theory.

Extension of nonuniqueness results to transport-diffusion equations with high-order diffusion operators.

Abstract

In this paper, we revisit the notion of temporal intermittency to obtain sharp nonuniqueness results for linear transport equations. We construct divergence-free vector fields with sharp Sobolev regularity $L^1_t W^{1,p}$ for all $p<\infty$ in space dimensions $d\geq 2$ whose transport equations admit nonunique weak solutions belonging to $L^p_tC^k$ for all $p<\infty$ and $k\in \mathbb{N}$. In particular, our result shows that the time-integrability assumption in the uniqueness of the DiPerna-Lions theory is sharp. The same result also holds for transport-diffusion equations with diffusion operators of arbitrarily large order in any dimensions $d \geq 2$.