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Linear transport equations for vector fields with subexponentially integrable divergence

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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 $$\operatorname{div} b\in L^\infty + \operatorname{Exp}\left(\frac{L}{\log^\gamma L}\right),$$ we prove existence and uniqueness of weak solutions when \(\gamma=1\). Moreover, optimality is shown by providing examples of non-uniqueness for every \(\gamma>1\). Stability questions and further extensions to the \(BV\) setting are also addressed.

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Journal
2015-02-18

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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35F05, 35F10
math.AP
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