Linear transport equations for vector fields with subexponentially integrable divergence

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.