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## Linear transport equations for vector fields with subexponentially integrable divergence

1502.05303

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### 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.

### Author and article information

###### Journal
18 February 2015
2015-04-16

35F05, 35F10
math.AP