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Linear transport equations for vector fields with subexponentially integrable divergence

  • Albert Clop
  • Renjin Jiang
  • Joan Mateu
  • Joan Orobitg
Article

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.

Mathematics Subject Classification

Primary 35F05 Secondary 35F10 

Notes

Acknowledgments

The authors are grateful to Gianluca Crippa for interesting remarks which improved the paper. Albert Clop, Joan Mateu and Joan Orobitg were partially supported by Generalitat de Catalunya (2014SGR75) and Ministerio de Economía y Competitividad (MTM2013-44699). Albert Clop was partially supported by the Programa Ramón y Cajal. Renjin Jiang was partially supported by National Natural Science Foundation of China (NSFC 11301029). All authors were partially supported by Marie Curie Initial Training Network MAnET (FP7-607647).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Albert Clop
    • 1
  • Renjin Jiang
    • 1
    • 2
  • Joan Mateu
    • 1
  • Joan Orobitg
    • 1
  1. 1.Departament de Matemàtiques, Facultat de CiènciesUniversitat Autònoma de BarcelonaBellaterra (Barcelona)Spain
  2. 2.Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Ministry of EducationBeijing Normal UniversityBeijingChina

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