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

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.

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

References in corpus (6)

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• Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics
• Inviscid models generalizing the 2D Euler and the surface quasi-geostrophic equations
• A note on the Prandtl boundary layers
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• Existence, uniqueness and regularity results for the viscous magneto-geostrophic equation
• Holder Continuous Solutions of Active Scalar Equations
• Wellposedness and convergence of solutions to a class of forced non-diffusive equations with applications
• On the second iterate for active scalar equations
• Vanishing diffusion limits and long time behaviour of a class of forced active scalar equations
• Instability of unidirectional flows for the 2D $\alpha$-Euler equations
• Suppression of blow up by a logistic source in $2$D Keller-Segel system with fractional dissipation
• Self-similar solutions for active scalar equations in Fourier-Besov-Morrey spaces
• On a class of forced active scalar equations with small diffusive parameters
• Solutions to a class of forced drift-diffusion equations with applications to the magneto-geostrophic equations