On the supercritically diffusive magneto-geostrophic equations

Abstract

We address the well-posedness theory for the magento-geostrophic equation, namely an active scalar equation in which the divergence-free drift velocity is one derivative more singular than the active scalar. In the presence of supercritical fractional diffusion given by (-Δ)^γ, where 0<γ<1, we discover that for γ>1/2 the equations are locally well-posed, while for γ<1/2 they are ill-posed, in the sense that there is no Lipschitz solution map. The main reason for the striking loss of regularity when γgoes below 1/2 is that the constitutive law used to obtain the velocity from the active scalar is given by an unbounded Fourier multiplier which is both even and anisotropic. Lastly, we note that the anisotropy of the constitutive law for the velocity may be explored in order to obtain an improvement in the regularity of the solutions when the initial data and the force have thin Fourier support, i.e. they are supported on a plane in frequency space. In particular, for such well-prepared data one may prove the local existence and uniqueness of solutions for all values of γ\in (0,1).

24 pages

References in corpus (4)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• 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
• On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations

Cited by in corpus (4)

• On the loss of continuity for super-critical drift-diffusion equations
• Existence, uniqueness and regularity results for the viscous magneto-geostrophic equation
• Wellposedness and convergence of solutions to a class of forced non-diffusive equations with applications
• Vanishing diffusion limits and long time behaviour of a class of forced active scalar equations