Non-uniqueness of the Leray-Hopf solutions in the hyperbolic setting

Chi-Hin Chan, Magdalena Czubak

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

The Leray-Hopf solutions to the Navier-Stokes equation are known to be unique on ℝ2. We show the uniqueness of the Leray-Hopf solutions breaks down on ℍ2(-a2), the two dimensional hyperbolic space with constant sectional curvature -a2. We also obtain a corresponding result on a more general negatively curved manifold for a modified geometric version of the Navier-Stokes equation. Finally, as a corollary we also show a lack of the Liouville theorem in the hyperbolic setting both in two and three dimensions.

Original languageEnglish
Pages (from-to)43-77
Number of pages35
JournalDynamics of Partial Differential Equations
Volume10
Issue number1
DOIs
StatePublished - 1 Jan 2013

Keywords

  • Hyperbolic space
  • Leray-hopf
  • Liouville theorem
  • Navier-stokes
  • Non-uniqueness

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