A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity

José A. Carrillo, Lucas C.F. Ferreira, Juliana C. Precioso

Research output: Contribution to journalArticleResearchpeer-review

50 Citations (Scopus)

Abstract

We consider a one dimensional transport model with nonlocal velocity given by the Hilbert transform and develop a global well-posedness theory of probability measure solutions. Both the viscous and non-viscous cases are analyzed. Both in original and in self-similar variables, we express the corresponding equations as gradient flows with respect to a free energy functional including a singular logarithmic interaction potential. Existence, uniqueness, self-similar asymptotic behavior and inviscid limit of solutions are obtained in the space P2(R) of probability measures with finite second moments, without any smallness condition. Our results are based on the abstract gradient flow theory developed by Ambrosio etal. (2005). [2]. An important byproduct of our results is that there is a unique, up to invariance and translations, global in time self-similar solution with initial data in P2(R), which was already obtained by Deslippe etal. (2004). [17] and Biler etal. (2010) [6] by different methods. Moreover, this self-similar solution attracts all the dynamics in self-similar variables. The crucial monotonicity property of the transport between measures in one dimension allows to show that the singular logarithmic potential energy is displacement convex. We also extend the results to gradient flow equations with negative power-law locally integrable interaction potentials. © 2012 Elsevier Ltd.
Original languageEnglish
Pages (from-to)306-327
JournalAdvances in Mathematics
Volume231
Issue number1
DOIs
Publication statusPublished - 10 Sep 2012

Keywords

  • Asymptotic behavior
  • Gradients flows
  • Inviscid limit
  • Optimal transport

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