Study of Anisotropic MHD system in Anisotropic Sobolev spaces

Ameur, Jamel Ben; Selmi, Ridha

doi:10.5802/afst.1172

Cited by 7 publications

(8 citation statements)

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“…The proof is almost similar to the ones given in [2]. In fact, Lemma 1 and Lemma 4 stated there in the case of R 3 , still apply in the case of the tore T 3 and are the bases of the existence and uniqueness results.…”

Section: Q(u U ) = P(u · ∇U) − P(b · ∇B) U · ∇B − B · ∇U the Viscosupporting

confidence: 56%

“…j=1 R/ Z , e 2 and e 3 are the second and the third vector of the Cartesian coordinate system, ν and η designate respectively the dynamic viscosity and the magnetic diffusivity and ε is the Rossby number destined to tend to zero and defined as the ratio between the fluid's typical velocity to the earth rotation velocity around the axis e 3 . Here, ∆ h denotes the horizontal Laplace operator given by ∆ h = ∂ 2 1 + ∂ 2 2 .…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic Study of Mixed Rotating MHD System

Selmi¹

2010

Bulletin of the Korean Mathematical Society

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Abstract. Asymptotic behavior of three-dimensional mixed, periodic and rotating magnetohydrodynamic system is investigated as the Rossby number goes to zero. The system presents the difficulty to be singular and mixed, that is hyperbolic in the vertical direction and parabolic in the horizontal one. The divergence free condition and the spectral properties of the penalization operator are crucial in the proofs. The main tools are the energy method, the Schochet's method and products laws in anisotropic Sobolev spaces.

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Section: Q(u U ) = P(u · ∇U) − P(b · ∇B) U · ∇B − B · ∇U the Viscosupporting

confidence: 56%

Section: Introductionmentioning

confidence: 99%

Asymptotic Study of Mixed Rotating MHD System

Selmi¹

2010

Bulletin of the Korean Mathematical Society

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“…where ν, ν ′ > 0 are diffusion coefficients in the horizontal direction and where we put together the gradient terms and we still denote the obtain quantity ∇p to simplify the notation. To the best of my knowledge, there are only few known results, concerning local existence and uniqueness of solution, and global existence of solutions in the case of small initial data of systems of the same type as the MHD system, in both isotropic ( [16], [17], [35], [3]) and anisotropic cases ( [1], [4]).…”

Section: Introductionmentioning

confidence: 99%

“…The first goal of this paper is to establish Fujita-Kato type results about the local existence and uniqueness of a strong solution (global for small data) of the anisotropic system (1.3). One can already find a proof of the local existence in [4]. Here, in order to have a self-contained paper, we give another proof for the local existence result.…”

Section: Introductionmentioning

confidence: 99%

A Global Existence Result for the Anisotropic Rotating Magnetohydrodynamical Systems

Ngo

2017

Acta Appl Math

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In this article, we study an anisotropic rotating system arising in magnetohydrodynamics (MHD) in the whole space R 3 , 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.1991 Mathematics Subject Classification. 76D03; 76D05; 76U05, 76W05.

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“…We believe that our proofs are new, simpler, and shorter than those performed for other alpha-systems in phase space. In the case of geophysical magnetohydrodynamic systems, we have used frequency analysis to deal with existence, uniqueness, and convergence results as a small parameter (Rossby number) vanishes [2][3][4][5][6]. Using the Fourier transform as a principal tool, we also gave an asymptotic study and stability results for both two-dimensional Leray weak solutions [7] and three-dimensional Fujita-Kato strong solution [8,9] to the periodic Navier-Stokes equation in critical spaces as time goes to infinity.…”

Section: )mentioning

confidence: 99%