On the supercritically diffusive magnetogeostrophic equations

Abstract

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 (−∆) γ with 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 γ goes 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 ∈ (0, 1).

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

“…Details of the singular behaviour of the Fourier multiplier symbols for the operator M 0 in certain regions of Fourier space are given in [24]. More general issues concerning the ill-posedness and wellposedness of the unforced MG 0 equation can be found in [21], [24], [25]. In particular, it is to be noted that the MG 0 with κ > 0 is the so-called critical MG equation in the sense of the delicate balance between the nonlinear term and the dissipative term.…”

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

“…Contrary to the previous case, when assumption A5 2 is in force, the operator ∂ x T 0 becomes a zero order operator with ∂ x T 0 : L 2 → L 2 being bounded. Following the idea given in [11], we show that under the assumptions A1-A2 and A5 2 , the equation (4.18) is locally wellposed in Sobolev space H s for s > d 2 + 1, thereby proving Theorem 2.4. Before we give the proof of Theorem 2.4, we recall the following proposition from [11]: Proposition 4.6.…”

“…and the details follow from Theorem A1 in [11]. This shows that there exists a unique solution θ n ∈ L ∞ (0, T ; H s ) of (4.25).…”

“…Since we assume that s > d 2 + 1, this also implies that the strong convergence occurs in a Hölder space relative to x as n → ∞, hence the limiting function θ is a solution of (4.18). Uniqueness of θ follows by the same argument given in [11] and we omit the details. It finishes the proof of Theorem 2.4.…”

Growth in the Muskat problem

“…Equation (1.1) is a system of hyperbolic active scalar equations. Some other equation in this family are the famous surface quasi-geostrophic equation [25,69,24,70,7,17,28], the magnetogeostrophic equation [78,54,52,55,53], the Stokes system [71,3] or the 2D Euler equation in vorticity formulation [73,72]. The Muskat problem studies the particular type of solution where there are two different immiscible fluids, a fluid on top with label + and a fluid below with label −, with properties given by (ρ + , µ + ) and (ρ − , µ − ) (or a fluid with (ρ − , µ − ) and a dry zone with ρ + = µ + = 0) separated by a moving interface, parametrized as Γ(t) = {(x, y) ∈ R 2 , (x, y) = (z 1 (α, t), z 2 (α, t)), α ∈ R}, (1.2) for certain functions z i : R × R + → R. We observe that, in this paper, unless otherwise stated we will assume µ + = µ − .…”