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

2011/05/13 by Susan Friedlander, Vlad C. Vicol · 1 citation
Mathematics · #math.AP #msc:76D03 #msc:35Q35 #msc:76W05

paper · pdf · doi:10.1088/0951-7715/24/11/001

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

arxiv created 2011/05/13 · arxiv updated 2015/05/28

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 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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