Linear transport equations for vector fields with subexponentially integrable divergence

Abstract

We face the well-posedness of linear transport Cauchy problems

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial u}{\partial t} + b\cdot \nabla u + c\,u = f&{}(0,T)\times \mathbb {R}^n\\ u(0,\cdot )=u_0\in L^\infty &{}\mathbb {R}^n \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned} {\text {div }}b\in L^1(0,T; L^\infty ) + L^1\left( 0,T; {{\text {Exp}}\;}\left( \frac{L}{\log L}\right) \right) , \end{aligned}$$

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

$$\begin{aligned} {\text {div }}(b)\in {{\text {Exp}}\;}\left( \frac{L}{\log ^\gamma L}\right) \end{aligned}$$

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.