Linear transport equations for vector fields with subexponentially integrable divergence

Abstract

We face the well-posedness of linear transport Cauchy problems $$\{\begin{array}{ll}\frac{\partial u}{\partial t}+b\cdot \nabla u+cu=f& (0,T)\times {\mathbb{R}}^n\\ u(0,\cdot )=u_0\in L^{\infty }& {\mathbb{R}}^n\end{array}$$ under borderline integrability assumptions on the divergence of the velocity field $b$. For $W_{loc}^{1,1}$ vector fields $b$ satisfying $\frac{|b(x,t)|}{1+|x|}\in L^1(0,T;L^1)+L^1(0,T;L^{\infty })$ and $$\mathrm{div}b\in L^1(0,T;L^{\infty })+L^1\left(0,T;\mathrm{Exp}\left(\frac{L}{\log L}\right)\right),$$ 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 $$\mathrm{div}b\in \mathrm{Exp}\left(\frac{L}{{\log }^{\gamma }L}\right),$$ 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.