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

Friedlander, Susan; Vicol, Vlad

doi:10.1016/j.anihpc.2011.01.002

Cited by 100 publications

(147 citation statements)

References 30 publications

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“…One handy tool which allows to treat more general type of vector fields b is a simple two-dimensional integration by parts argument (2.8). Another tool is the John-Nirenberg inequality for BMO functions, which was first employed by Friedlander and Vicol [8], and also by Seregin, Silvestre, Sverak, Zlatos [25] to treat the linear heat equation…”

Section: Introductionmentioning

confidence: 99%

A Liouville theorem for the axially-symmetric Navier–Stokes equations

Lei¹,

Zhang²

2011

Journal of Functional Analysis

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Let v(x, t) = v r e r + v θ e θ + v z e z be a solution to the three-dimensional incompressible axiallysymmetric Navier-Stokes equations. Denote by b = v r e r + v z e z the radial-axial vector field. Under a general scaling invariant condition on b, we prove that the quantity Γ = rv θ is Hölder continuous at r = 0, t = 0. As an application, we prove that the ancient weak solutions of axi-symmetric Navier-Stokes equations must be zero (which was raised by Koch, Nadirashvili, Seregin and Sverak (2009) in [15] and Seregin and Sverak (2009) in [26] as a conjecture) under the condition that b ∈ L ∞ ([0, T ], BMO −1 ). As another application, we prove that if b ∈ L ∞ ([0, T ], BMO −1 ), then v is regular.

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Section: Introductionmentioning

confidence: 99%

A Liouville theorem for the axially-symmetric Navier–Stokes equations

Lei¹,

Zhang²

2011

Journal of Functional Analysis

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“…The magneto-geostrophic (MG) equation [39,38,28]. This is a three-dimensional active scalar equation, with symbol given by m(ξ) = ξ 2 ξ 3 |ξ| 2 + ξ 1 ξ for all ξ ∈ Z 3 * with ξ 3 = 0, and by m(ξ 1 , ξ 2 , 0) = 0.…”

Section: The Incompressible Porous Media (Ipm) Equation With Velocitymentioning

confidence: 99%

Hölder Continuous Solutions of Active Scalar Equations

2015

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We consider active scalar equations ∂tθ + ∇ · (u θ) = 0, where u = T [θ] is a divergence-free velocity field, and T is a Fourier multiplier operator with symbol m. We prove that when m is not an odd function of frequency, there are nontrivial, compactly supported solutions weak solutions, with Hölder regularity C 1/9− t,x . In fact, every integral conserving scalar field can be approximated in D ′ by such solutions, and these weak solutions may be obtained from arbitrary initial data. We also show that when the multiplier m is odd, weak limits of solutions are solutions, so that the h-principle for odd active scalars may not be expected.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Friedlander

Suen

2019

J. Math. Fluid Mech.

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This paper considers a family of non-diffusive active scalar equations where a viscosity type parameter enters the equations via the constitutive law that relates the drift velocity with the scalar field. The resulting operator is smooth when the viscosity is present but singular when the viscosity is zero. We obtain Gevrey-class local well-posedness results and convergence of solutions as the viscosity vanishes. We apply our results to two examples that are derived from physical systems: firstly a model for magnetostrophic turbulence in the Earth's fluid core and secondly flow in a porous media with an "effective viscosity".Mathematics Subject Classification (2010). 76D03, 35Q35, 76W05.

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

Cited by 100 publications

References 30 publications

A Liouville theorem for the axially-symmetric Navier–Stokes equations

A Liouville theorem for the axially-symmetric Navier–Stokes equations

Hölder Continuous Solutions of Active Scalar Equations

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

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