On the Global Stability of the Wave-map Equation in Kerr Spaces with Small Angular Momentum

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Abstract

This paper is motivated by the problem of the nonlinear stability of the Kerr solution for axially symmetric perturbations. We consider a model problem concerning the axially symmetric perturbations of a wave map Φ defined from a fixed Kerr solution K(M, a) , 0 ≤ a≤ M, with values in the two dimensional hyperbolic space H2. A particular such wave map is given by the complex Ernst potential associated to the axial Killing vector-field Z of K(M, a). We conjecture that this stationary solution is stable, under small axially symmetric perturbations, in the domain of outer communication (DOC) of K(M, a) , for all 0 ≤ a< M and we provide preliminary support for its validity, by deriving convincing stability estimates for the linearized system.

Original languageEnglish (US)
Article number1
JournalAnnals of PDE
Volume1
Issue number1
DOIs
StatePublished - Dec 1 2015

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics
  • Geometry and Topology
  • Mathematical Physics
  • Physics and Astronomy(all)

Keywords

  • Axially symmetric
  • Kerr
  • Morawetz
  • Nonlinear
  • Red shift
  • Stability
  • Trapping

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