On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations

Details On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations

2011-05-07

arxiv.org/abs/1105.1403v2

Mathematics

Analysis of PDEs

We have modified the definition of Lipschitz well-posedness, in order to allow for a possible loss in regularity of the soluti

Scientific paper

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 well-posed. 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.

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