Linear transport equations for vector fields with subexponentially integrable divergence

TL;DR

This paper establishes well-posedness for linear transport equations with vector fields having borderline integrability conditions on divergence, and demonstrates the optimality of these conditions through counterexamples.

Contribution

It proves existence and uniqueness of solutions under near-critical divergence integrability assumptions and constructs examples showing these conditions are sharp.

Findings

Existence and uniqueness of solutions under specific borderline conditions.

Counterexamples showing failure of uniqueness when divergence exceeds these conditions.

Analysis of stability and extensions to bounded variation (BV) vector fields.

Abstract

We face the well-posedness of linear transport Cauchy problems $$\begin{cases}\dfrac{\partial u}{\partial t} + b\cdot\nabla u + c\,u = f&(0,T)\times{\mathbb R}^n\\u(0,\cdot)=u_0\in L^\infty&{\mathbb R}^n\end{cases}$$ under borderline integrability assumptions on the divergence of the velocity field $b$. For $W^{1,1}_{loc}$ vector fields $b$ satisfying $\frac{|b(x,t)|}{1+|x|}\in L^1(0,T; L^1)+L^1(0,T; L^\infty)$ and $$\operatorname{div} b\in L^1(0,T;L^\infty) + L^1\left(0,T; \operatorname{Exp}\left(\frac{L}{\log L}\right)\right),$$ we prove existence and uniqueness of weak solutions. Moreover, optimality is shown in the following way: for every $\gamma>1$, we construct an example of a bounded autonomous velocity field $b$ with $$\operatorname{div} b\in \operatorname{Exp}\left(\frac{L}{\log^\gamma L}\right) ,$$ for which the associate Cauchy problem for the transport equation admits infinitely many solutions. Stability questions and further extensions to the $BV$ setting are also addressed.