Abstract
We consider an active scalar equation that is motivated by a model for magneto-geostrophic dynamics and the geodynamo. We prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast, the critically diffusive equation is globally well-posed (cf Friedlander and Vicol (2011 Ann. Inst. Henri Poincaré Anal. Non Linéaire 28 283-301)). In this case we give an example of a steady state that is nonlinearly unstable, and hence produces a dynamo effect in the sense of an exponentially growing magnetic field.
Original language | English (US) |
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Pages (from-to) | 3019-3042 |
Number of pages | 24 |
Journal | Nonlinearity |
Volume | 24 |
Issue number | 11 |
DOIs | |
State | Published - Nov 2011 |
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics