Existence, uniqueness and regularity results for the viscous magneto-geostrophic equation (original) (raw)

Global well-posedness for an advection–diffusion equation arising in magneto-geostrophic dynamics

Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2011

We use De Giorgi techniques to prove Hölder continuity of weak solutions to a class of drift-diffusion equations, with L 2 initial data and divergence free drift velocity that lies in L ∞ t BM O −1 x . We apply this result to prove global regularity for a family of active scalar equations which includes the advection-diffusion equation that has been proposed by Moffatt in the context of magnetostrophic turbulence in the Earth's fluid core.

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

Nonlinearity, 2011

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 (cf. ). 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.

The magneto-geostrophic equations: a survey

2013

We discuss recent results obtained by the authors in , regarding the analysis of the magneto-geostrophic equation: a model proposed by Moffatt and Loper [ML94, Mof08] to study the geodynamo and turbulence in the Earth's fluid core. We conclude this review by indicating some open problems around the MG equation, that remain to be addressed in the future.

Existence and continuity of exponential attractors of the three dimensional Navier-Stokes-$\alpha$ equations for uniformly rotating geophysical fluids

Communications in Mathematical …, 2006

Three dimensional (3D) Navier-Stokes-α equations are considered for uniformly rotating geophysical fluid flows (large Coriolis parameter f = 2Ω). The Navier-Stokes-α equations are a nonlinear dispersive regularization of usual Navier-Stokes equations obtained by Lagrangian averaging. The focus is on the existence and global regularity of solutions of the 3D rotating Navier-Stokes-α equations and the uniform convergence of these solutions to those of the original 3D rotating Navier-Stokes equations for large Coriolis parameters f as α → 0. Methods are based on fast singular oscillating limits and results are obtained for periodic boundary conditions for all domain aspect ratios, including the case of three wave resonances which yields nonlinear resonant limit α-equations for f → ∞. The existence and global regularity of solutions of resonant limit α-equations is established, uniformly in α. Bootstrapping from global regularity of the resonant limit α-equations, the existence of a regular solution of the full 3D rotating Navier-Stokes-α equations for large f for an infinite time is established. Then we prove the existence of exponential attractors of the 3D rotating Navier-Stokes-α equations (α = 0) and the convergence of the exponential attractors to those of the original 3D rotating Navier-Stokes equations (α = 0) for f large but fixed as α → 0. All the estimates are uniform in α, in contrast with previous estimates in the literature which blow up as α → 0.

Derivation of Inviscid Quasi-geostrophic Equation from Rotational Compressible Magnetohydrodynamic Flows

Journal of Nonlinear Science, 2017

In this paper, we consider the compressible models of magnetohydrodynamic flows giving rise to a variety of mathematical problems in many areas. We derive a rigorous quasi-geostrophic equation governed by magnetic field from the rotational compressible magnetohydrodynamic flows with the well-prepared initial data. It is a first derivation of quasi-geostrophic equation governed by the magnetic field, and the tool is based on the relative entropy method. This paper covers two results: the existence of the unique local strong solution of quasi-geostrophic equation with the good regularity and the derivation of a quasi-geostrophic equation.

On the critical dissipative quasi-geostrophic equation

2001

The 2D quasi-geostrophic (QG) equation is a two dimensional model of the 3D incompressible Euler equations. When dissipation is included in the model then solutions always exist if the dissipation's wave number dependence is super-linear. Below this critical power the dissipation appears to be insufficient. For instance, it is not known if the critical dissipative QG equation has global smooth solutions for arbitrary large initial data. In this paper we prove existence and uniqueness of global classical solutions of the critical dissipative QG equation for initial data that have small L ∞ norm. The importance of an L ∞ smallness condition is due to the fact that L ∞ is a conserved norm for the non-dissipative QG equation and is non-increasing on all solutions of the dissipative QG., irrespective of size.