Solutions to a Class of Forced Drift-Diffusion Equations with Applications to the Magneto-Geostrophic Equations

Abstract

We prove the global existence of classical solutions to a class of forced drift-diffusion equations with L 2 initial data and divergence free drift velocity {u ν }ν ≥ 0 ⊂ L ∞ t BM O −1 x , and we obtain strong convergence of solutions as the viscosity ν vanishes. We then apply our results to a family of active scalar equations which includes the three dimensional magneto-geostrophic {MG ν } ν≥0 equation that has been proposed by Moffatt in the context of magnetostrophic turbulence in the Earth's fluid core. We prove the existence of a compact global attractor {A ν } ν≥0 in L 2 (T 3 ) for the MG ν equations including the critical equation where ν = 0. Furthermore, we obtain the upper semicontinuity of the global attractor as ν vanishes.Mathematics Subject Classification (2010). 76D03, 35Q35, 76W05.Keywords. active scalar equations, vanishing viscosity limit, global attractor.The unknowns are u(t, x) the velocity, b(t, x) the magnetic field (both vector valued) and θ(t, x) the scalar (temperature field of the fluid). P is the sum of the fluid and magnetic pressures, and the Cartesian unit vectors are given by e 1 , e 2 and e 3 . The physical forces governing this system are the Coriolis force, the Lorentz force and gravity acting via buoyancy, while the equation for the temperature is driven by a smooth function S(x) that represents the external forcing of the MHD system. The non-dimensional parameters in (1.1)-(1.4) are R 0 the Rossby number, R m the magnetic Reynolds number, ν a (non-dimensional) viscosity and κ a (non-dimensional) thermal diffusivity. Moffatt and Loper argue that for the geophysical context they are modelling, all these parameters are

Wellposedness and Convergence of Solutions to a Class of Forced Non-diffusive Equations with Applications

“…For sufficiently smooth initial data θ 0 and forcing term S, we aim at showing that (θ ν − θ 0 )(·, t) H s → 0 as ν → 0 for s > d 2 + 1 and t ∈ [0, T ]. Such result is parallel to the one proved in [14], in which the authors proved that if θ ν , θ 0 are C ∞ smooth classical solutions of the diffusive system (4.16) for ν > 0 and ν = 0 respectively with initial datum θ 0 ∈ L 2 and forcing term S ∈ C ∞ , then (θ ν − θ 0 )(·, t) H s → 0 as ν → 0 for s ≥ 0 and t > 0.…”

“…Remark 5.1. In the diffusive system (4.16) studied in [14] there is no smoothing assumption imposed on {T ν ij } ν≥0 when ν > 0. The main reason for the difference is that the diffusive term κ∆θ ν present in (4.16) is sufficient to smooth out the solution θ ν for all ν ≥ 0.…”

“…The nonlinear equation (1.2) with u related to θ via (1.6) is called the magnetogeostrophic (MG) equation and its mathematical properties have been studied in a series of papers including [11], [12], [13], [14], [15], [16]. In the magnetostrophic turbulence model the parameters ν, the nondimensional viscosity, and κ, the nondimensional thermal diffusivity, are extremely small.…”

On the non-diffusive Magneto-Geostrophic equation

“…In the present paper we focus our attention in the inviscid case (ǫ ν = 0). The mathematical properties of the model under the presence of viscosity have been addressed in a recent sequence of different articles [10], [11] and [12].…”

On a class of forced active scalar equations with small diffusive parameters

“…2 Such mean zero assumption is common in many physical models which include SQG equation and MG equation; see [16] and [29] for example.…”