Linear transport equations for vector fields with subexponentially integrable divergence

Albert Clop, Renjin Jiang, Joan Orobitg, Juan Eugenio Mateu Bennassar

Research output: Contribution to journalArticleResearchpeer-review

3 Citations (Scopus)

Abstract

© 2016, Springer-Verlag Berlin Heidelberg. We face the well-posedness of linear transport Cauchy problems (Formula presented.) under borderline integrability assumptions on the divergence of the velocity field b. For (Formula presented.) vector fields b satisfying (Formula presented.) and (Formula presented.), we prove existence and uniqueness of weak solutions. Moreover, optimality is shown in the following way: for every (Formula presented.) , we construct an example of a bounded autonomous velocity field b with (Formula presented.) 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.
Original languageEnglish
Article number21
Pages (from-to)1-30
JournalCalculus of Variations and Partial Differential Equations
Volume55
DOIs
Publication statusPublished - 1 Feb 2016

Keywords

  • Primary 35F05
  • Secondary 35F10

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