Weak and parabolic solutions of advection-diffusion equations with rough velocity field

TL;DR

This paper investigates the existence, regularity, and uniqueness of weak and parabolic solutions to the advection-diffusion equation with rough, divergence-free velocity fields on the torus, highlighting recent ill-posedness results.

Contribution

It provides a comprehensive overview of current results, includes original proofs, and discusses open problems related to the equation's well-posedness in various integrability regimes.

Findings

Existence and uniqueness results for distributional solutions.

Regularity properties depending on integrability conditions.

Recent ill-posedness results via convex integration schemes.

Abstract

We study the Cauchy problem for the advection-diffusion equation $\partial_t u + \mathrm{div} (u b ) = \Delta u$ associated with a merely integrable divergence-free vector field $b$ defined on the torus. We discuss existence, regularity and uniqueness results for distributional and parabolic solutions, in different regimes of integrability both for the vector field and for the initial datum. We offer an up-to-date picture of the available results scattered in the literature, and we include some original proofs. We also propose some open problems, motivated by very recent results which show ill-posedness of the equation in certain regimes of integrability via convex integration schemes.