A Global Existence Result for the Anisotropic Rotating Magnetohydrodynamical Systems
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A Global Existence Result for the Anisotropic Rotating Magnetohydrodynamical Systems Van-Sang Ngo1
Received: 4 September 2015 / Accepted: 22 December 2016 © Springer Science+Business Media Dordrecht 2017
Abstract In this article, we study an anisotropic rotating system arising in magnetohydrodynamics (MHD) in the whole space R3 , in the case where there are no diffusivity in the vertical direction and a vanishing diffusivity in the horizontal direction (when the rotation goes to infinity). We first prove the local existence and uniqueness of a strong solution and then, using Strichartz-type estimates, we prove that this solution exists globally in time for large initial data, when the rotation is fast enough. Keywords MHD systems · Rotating fluids · Global existence · Strichartz estimates Mathematics Subject Classification 76D03 · 76D05 · 76U05 · 76W05
1 Introduction The fluid core of the Earth is often considered as an enormous dynamo, which generates the Earth’s magnetic field due to the motion of the liquid iron. In a moving conductive fluid, magnetic fields can induce currents, which create forces on the fluid, and also change the magnetic field itself. The set of equations which then describe the MHD phenomena are a combination of the Navier–Stokes equations with Maxwell’s equations. In this paper, we consider a MHD model, which describes the motion of an incompressible conducting fluid of density ρ, kinematic viscosity ν, conductivity σ , magnetic diffusivity η and permeability μ0 . We suppose that the fluid is fast rotating with angular velocity Ω0 around the axis e3 . We also suppose that the fluid moves with a typical velocity U in a domain of typical size L and generates a typical magnetic field B. We introduce following dimensionless parameters E = νΩ0−1 L−2 ;
ε = U Ω0−1 L−1 ;
−1 Λ = B 2 ρ −1 Ω0−1 μ−1 0 η ;
θ = U Lη−1 ,
which describe the Ekman, Rossby, Elsasser and Reynolds numbers respectively.
B V.-S. Ngo
[email protected]
1
Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS, Université de Rouen, 76801 Saint-Etienne du Rouvray Cedex, France
V.-S. Ngo
In geophysics, since the Earth is fast rotating, the Earth’s core is believed to be in the asymptotic regime of small Ekman numbers (E ∼ 10−15 ) and small Rossby numbers (ε ∼ 10−7 ). In [14], using these considerations, Desjardins, Dormy and Grenier introduced the following MHD system ⎧ u ∧ e3 Λ ∇p E Λθ ⎪ ⎪ − u + = − (curl b) ∧ e3 + (curl b) ∧ b ∂t u + u · ∇u + ⎪ ⎪ ε ε ε ε ε ⎪ ⎪ ⎪ ⎪ ⎨ curl (u ∧ e3 ) b + ∂t b + u · ∇b = b · ∇u − (1.1) θ θ ⎪ ⎪ ⎪ ⎪ ⎪ div u = div b = 0 ⎪ ⎪ ⎪ ⎩ u|t=0 = u0 , b|t=0 = b0 , with the following asymptotic conditions ε → 0,
Λ = O(1),
εθ → 0
and
E ∼ ε2 ,
(1.2)
and where u, p and b denote the velocity, pressure and magnetic field of the fluid. In the case of large scale fluids in fast rotation, the Coriolis force is a dominant factor and leads to an anisotropy between the horizontal and the vertical directions (see the Taylor–Proudman theorem, [13, 16] or [25]). Taking into account this anisotropy,
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