### 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 H^{2}. 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 language | English (US) |
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Article number | 1 |

Journal | Annals of PDE |

Volume | 1 |

Issue number | 1 |

DOIs | |

State | Published - 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