Periodic solutions for a 1D-model with nonlocal velocity via mass transport

Lucas C.F. Ferreira, Julio Cesar Valencia Guevara

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2 Scopus citations

Abstract

This paper concerns periodic solutions for a 1D-model with nonlocal velocity given by the periodic Hilbert transform. There is a rich literature showing, via numerics and rigorous analysis, that this model presents singular behavior of solutions. For instance, they can blow up by forming mass-concentration. We develop a global well-posedness theory for periodic measure initial data that allows, in particular, to analyze how the model evolves from those singularities. Our results are based on periodic mass transport theory and the abstract gradient flow theory in metric spaces developed by Ambrosio et al. (2005). A viscous version of the model is also analyzed and inviscid limit properties are obtained.

Original languageEnglish
Pages (from-to)7093-7114
Number of pages22
JournalJournal of Differential Equations
Volume260
Issue number10
DOIs
StatePublished - 15 May 2016
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2016 Elsevier Inc.

Keywords

  • Gradient flows
  • Inviscid limit
  • Nonlocal fluxes
  • Optimal transport
  • Periodic solutions

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