A GLOBAL EXISTENCE RESULT FOR THE ANISOTROPIC MAGNETOHYDRODYNAMICAL SYSTEMS

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A GLOBAL EXISTENCE RESULT FOR THE ANISOTROPIC MAGNETOHYDRODYNAMICAL SYSTEMS VAN-SANG NGO Abstract. We study an anisotropic system arising in magnetohydrodynamics (MHD) in the whole space R3, in the case where there are no diffusivity in the vertical direction and only a small diffusivity in the horizontal direction (of size ?? with 0 < ? ≤ ?0, for some ?0 > 0). We prove the local existence and uniqueness of a strong solution and then, using Strichartz-type estimates, we prove that this solution globally exists in time for large initial data, when the rotation is fast enough. 1. Introduction The fluid core of the Earth is often considered as an enormous dynamo that 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 Maxwell's equations. In this paper, we consider a MHD model which describes the motion of an incom- pressible conducting fluid of density ?, kinematic viscosity ?, conductivity ?, magnetic diffusivity ? and permeability µ0. We suppose that the fluid is fast rotating with angu- lar velocity ?0 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 magnetic field B.

• equations


• fast enough


• large initial


• global existence


• force term


• ∂tbn ?


• e3 ?

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• Équation

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• Publié par: pefav
• Nombre de lectures: 15
• Langue: English

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A GLOBAL EXISTENCE RESULT FOR THE ANISOTROPIC MAGNETOHYDRODYNAMICAL SYSTEMS

VAN-SANG NGO

Abstract.We study an anisotropic system arising in magnetohydrodynamics (MHD) in the whole spaceR 3 the case where there are no diffusivity in the vertical direction, in and only a small diffusivity in the horizontal direction (of sizeε α with 0< α≤α 0 , for someα 0 >existence and uniqueness of a strong solution0). We prove the local and then, using Strichartz-type estimates, we prove that this solution globally exists in time for large initial data, when the rotation is fast enough.

1.Introduction

The fluid core of the Earth is often considered as an enormous dynamo that 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 Maxwell’s equations. In this paper, we consider a MHD model which describes the motion of an incom- pressible conducting fluid of densityρ, kinematic viscosityν, conductivityσ, magnetic diffusivityηand permeabilityµ 0 . We suppose that the fluid is fast rotating with angu- lar velocity Ω 0 around the axise 3 . We also suppose that the fluid moves with a typical velocityUin a domain of typical sizeLand generates a magnetic fieldB introduce. We following dimensionless parameters E=νΩ 0 −1 L −2 ;ε=UΩ 0 −1 L −1 ;B 2 ρ −1 Ω 0 −1 µ 0 −1 η −1 ;θ=U Lη −1 , which describe the Ekman, Rossby, Elsasser and Reynolds numbers respectively. In geophysics, the Earth’s core is believed to be in the asymptotic regime of small Ekman number (E∼10 −15 ) and small Rossby number (ε∼10 −7  [13], using these). In considerations, Desjardins, Dormy and Grenier came to the following system (M H D)  ∂ t u+u.ru+ rp ε − ε E Δu+ u∧ ε e 3 =− ε lruc( Λ b)∧e 3 + Λθ ε (curlb)∧b   ∂ t b+u.rb=b.ru− curl (u∧e 3 ) Δb θ + θ divu= divb= 0   (u(0), b ((0)) =u 0 , b 0 ), with the following asymptotic conditions (1)ε→0,Λ =O(1), εθ→0 etE∼ε 2 .

1991Mathematics Subject Classification.76D03; 76D05; 76U05. Key words and phrases.MHD systems; Rotating fluids; Anisotropy; Strichartz estimates. 1

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VAN-SANG NGO

Here,u,pandb Sincedenote the velocity, pressure and magnetic field of the fluid. the Earth is fast rotating, these equations should also contain a large Coriolis force term. 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, [12], [17] or [27]). Taking into account this anisotropy, we suppose that the diffusion term in the vertical direction is negligible, compared with the one in the horizontal direction. Thus, we replace the “whole” Laplacian operator Δ by the “horizontal” Laplacian operator Δ h . Taking into account the classical vectorial identities, (curlb)∧b=b.rb− 2 1 r |b| 2 ,(curlb)∧e 3 =∂ 3 b− rb 3 , and curl (u∧e 3 ) =∂ 3 u(since divu0), we are led to consider the following MHD= system inR 3 :  ∂ t u− E Δ h u+u.ru−b.rb+ u∧e 3 + Λ ∂ 3 b=− rpe ε ε ε (M H D hε )   ∂ t b− θ Δ 1 h b+u.rb−b.ru+ θ 1 ∂ 3 u= 0 divu= divb= 0   (u(0), b ((0)) =u 0 , b 0 ). To the best of my knowledge, there are only few known results, concerning local existence (global in the case of small initial data) and stability of systems of the same type as the MHD system, in both isotropic case ([13], [14], [29], [3]) and anisotropic case ([1], [4]). The first goal of this paper is to establish Fujita-Kato-type results about the local existence and uniqueness of a strong solution (global in the case of small data) of the anisotropic system (M H D ε h another proof of the local existence in [4]. can find ). One However, in order to have a self-contained text, we introduce a short proof in this part. Several difficulties arise in proving such results. In the (M H D h ε ) system, beside the problem due to the lack of the vertical diffusion terms, we encounter problems dealing with the terms Λ ε ∂ 3 band 1 θ ∂ 3 u terms require more regularity in the vertical. These Λ 1 r, if = , a simpl direction. Howeve ε θ e integration by parts shows that the sum of these two terms cancels, and these terms disappear in the energy estimates. We also remark that similar difficulties arise in estimating the termsb.rbin the first equation andb.ru in the second equation. In this part, we consider the system (M H D hε ) in the general case and we suppose that Λ ε = 1 θ  order to state the . Infirst result, we introduce the anisotropic Sobolev spaces, defined by H σ,s (R 3 ) =  u∈ S 0  kuk 2 H σ,s d = ef Z R 3  1 +|ξ h | 2  σ  1 +|ξ 3 | 2  s |bu| 2 dξ <+∞  . We prove the following theorem Theorem 1.1(Local existence and uniqueness).Lets > 1 2 and ε Λ = 1 θ . Then, for any divergence free vector fieldsu 0 , b 0 inH 0,s (R 3 ), there exists a timeT >0such that the system(M H D ε )has a unique solution u, b∈L ∞  [0, T],H 0,s (R 3 )  ∩C  [0, T],H 0,s (R 3 )  withr h u,r h b∈L 2  [0, T],H 0,s (R 3 )  .

GLOBAL EXISTENCE FOR ANISOTROPIC MHD SYSTEMS 3 Moreover, there exists a constantc >0such that ifku 0 k H 0,s +kb 0 k H 0,s ≤cmin{ν, ν 0 }, then the solution exists globally in time, that is u, b∈L ∞  R + ,H 0,s (R 3 )  ∩C  R + ,H 0,s (R 3 )  andr h u,r h b∈L 2  R + ,H 0,s (R 3 )  .

The goal of the second and main part of this paper consists in proving the global existence of this strong solution for large initial data when the rotation is fast enough. Following the ideas of [9], [24] and [6], we wish to show that the strong Coriolis force implies global existence of the solutions for large initial data. Notice however that the MHD system in general is much more complex than the system of general rotating fluids ([9], [24]) or the system of primitive equations. Here, we address the particular case where the parameters are “well chosen”: Λ = 1,θ=εandE=ε 1+α , withα >0. To simplify the calculations, we choose the diffusion terms to be equal in the two equations.  ∂ t u−ε α Δ h u+u.ru−b.rb+ u∧e + ∂ 3 b =− rpe ε ε (M H D ε )   ∂ t b−ε α Δ h b+u.rb−b.ru+ ∂ 3 u ε = 0 divu= divb= 0   (u(0), b(0)) = (u 0 , b 0 ), To state the second result, we need to introduce the anisotropic Lebesgue spaces L h p L v q withp, q≥1, which are defined as L p h L v q (R 3 ) =L p (R h 2 ;L v q (R)) =u u(x h , x 3 )| q dx 3  q dx h i p <+∞ o . n ∈ S 0 :kuk L hp L q v = h Z R 2 h  Z R v | p 1 For anys > 1 2 ,η >0 and 1≤p <2, we set Y s,η,p =L h 2 L p v (R 3 )∩L p h H s v (R 3 )∩H η,s+η (R 3 ) = n u∈ S 0 :kuk Y s,η,p = sup n kuk L 2 h L vp ,kuk L p h H vs ,kuk H η,s+η o <+∞ o . We show the following global existence result.

Theorem 1.2(Global existence for large initial data).Lets > 1 2 ,η >0and1≤p <2. We can chooseα 0 >0in such a way that, for any0< α≤α 0 and for anyr 0 >0, there existsε 0 >0such that, for any0< ε≤ε 0 and for any divergence free initial data u 0 , b 0 ∈Y s,η,p satisfyingku 0 k 2 Y s,η,p +kb 0 k Y 2 s,η,p ≤r 2 0 , the system(M H D ε )has a unique, global strong solution,(u, b)∈C b (R + ,H 0,s )with(r h u,r h b)∈L 2 (R + ,H 0,s ). The plan of the paper is as follows. The next section consists in proving the local existence (global existence in the case of small data) and uniqueness of a strong solution of (M H D hε  the third section, we establish Strichartz estimates for the corresponding). In linear system. We show that with an appropriate frequency cut-off, the corresponding non linear system is globally well-posed, and we prove Theorem 1.2. The last section (appendix) is devoted for the recall of some elements of the Littlewood-Paley theory.