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

Chi-Hin Chan; Magdalena Czubak

doi:10.4310/DPDE.2013.v10.n1.a3

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

Chi-Hin Chan, Magdalena Czubak

Department of Applied Mathematics

Research output: Contribution to journal › Article › peer-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 language	English
Pages (from-to)	43-77
Number of pages	35
Journal	Dynamics of Partial Differential Equations
Volume	10
Issue number	1
DOIs	https://doi.org/10.4310/DPDE.2013.v10.n1.a3
State	Published - 1 Jan 2013

Keywords

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

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10.4310/DPDE.2013.v10.n1.a3

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