CONVERGENCE RESULTS FOR MHD SYSTEM

A magnetohydrodynamic system is investigated in both cases of the periodic domain T3 and the whole space R3. Existence and uniqueness of strong solution are proved. Asymptotic behavior of the solution when the Rossby number ε goes to zero is studied. The proofs use the spectral properties of the penalization operator and involve Friedrich’s method, Schochet’s methods, and product laws in Sobolev spaces of sufficiently large exponents.


Introduction and summary of results
In this paper, we study the existence, the uniqueness, and the asymptotic behavior of strong solutions of the following MHD system for a 3D incompressible ideal fluid: where the velocity field u, the induced magnetic perturbation b, and the pressure p are unknown functions of time t and space variables x = (x 1 ,x 2 ,x 3 ), e 2 and e 3 are, respectively, the second and the third vectors of the Cartesian coordinate system, Ω is either the torus T 3 or the whole space R 3 and ε is a small positive parameter destined to go to zero.
The above system is a particular modelization of the MHD flow in the earth's core which is believed to support a self-excited dynamo process generating the earth's magnetic field.
In a more general case, throughout their paper [7], Desjardins et al. have modelized this process by the following general formulation: where E, ε, Λ, and θ represent, respectively, the Ekman number, the Rossby number, the Elsasser number and the magnetic Reynolds number, e is a fixed vector axe of rotation of the earth, and e is the time-independent component of the earth's magnetic field B defined in [7] by (1. 3) Here we take E = ε 2 , Λ = 1, θ = ε, e = −e 3 , and e = −e 2 .We believe that our choice belongs to the parameter range relevant to the earth's core as it satisfies the following ordering (see [7,8]): Physically, according to [8], letting the Rossby number ε go to zero and consequently the Ekman number means that inertial forces and viscous attraction-repulsion modelized, respectively, by ∂ t u + u • ∇u and εΔu are of neglected effects compared to the Coriolis one due to rotation of the earth in the direction e 3 with the high speed 1/ε.The aim of this study is to find the limit system, when ε goes to zero, which physically corresponds to the equations characterizing the magnetostrophic equilibrium.For more physical details, we refer the readers to [7,8].We denote by P the L 2 orthogonal projection on divergence-free vector fields.Applying P to (1.1), one can see that U := (u,b) is a solution of the following abstract system: where, if we set U = (u,b), the quadratic term Q is defined by the viscous term is and the linear perturbation L ε is given by where A is a linear operator, a ε 2 (D) is elliptic, and L ε = (1/ε)L is a skew-symmetric linear operator.This skew-symmetry is an important property for the existence result since the perturbation disappears in the energy estimate.
Singular limits in systems such as ( ε ) have been studied by several authors.In the hyperbolic case, Babin et al. [1] studied the incompressible rotating Euler equation on the torus.Using the method introduced by Schochet in [18,19], Gallagher studied, respectively, in [9,10] this problem in its abstract parabolic and hyperbolic form.In the case of the incompressible rotating Navier-Stokes equation on the torus, it is shown in [1,11] that the solutions converge to a solution of a certain diffusion equation.Moreover, for a special initial condition, there exists a sequence of solutions convergent to a solution of a two-dimensional Navier-Stokes equation.MHD systems were investigated, respectively, with the choice of parameters E = ε 2 , Λ = 1, θ = ε, and e = e = −e 3 in [2] and E = ε 2 , Λ = ε, θ = 1, and e = e = −e 3 in [3].We also refer to the results proved in [6,[12][13][14]16].
We specify in this paper to choose e to be equal to −e 2 .Physically, this choice deals with the effect of the longitudinal component of the time-independent magnetic field B on the dynamo process.The mathematical study in this case requires to use in a precise way the structure and the spectral properties of the penalization operator to establish the convergence results.Such structure, namely, the e 2 dependence, allows us to put the system in the appropriate form before applying Schochet's method.
We begin by establishing an existence result which follows directly from the Friedrichs method and the energy estimate.Precisely, we will prove in Section 2 local existence of strong solutions on uniform time, namely, solutions given in the following theorem.
Theorem 1.1.Let s > 3/2 + 2 be an integer and U 0 = (u 0 ,b 0 ) ∈ H s (Ω) such that div u 0 = div b 0 = 0.Then, there exist T > 0 and a constant C > 0 such that, for all ε > 0, there exists a unique solution U ε ∈ Ꮿ 0 ([0,T],H s (Ω)) ∩ L 2 ([0,T],H s+1 (Ω)) of the system ( ε ) satisfying, for all t ∈ [0,T], (1.8) Once the existence result is established, we turn to the asymptotic behavior of the strong solution of ( ε ) when ε goes to zero.Since ∂ t U ε is not a priori bounded in ε, the classical proofs used, for example, in [15,20] and based on taking the limit directly in the system no longer work.
In the case of the torus T 3 , this difficulty will be avoided by writing the velocity field u ε in the following form: where (1.10) The asymptotic behavior of each term of u ε will be investigated separately.For the oscillating part u ε osc , spectral properties of L ε and Lebesgue's convergence theorem allow to conclude in the case of low frequencies.For high frequencies, the nonlinear part will be investigated term by term; the convergence result is due to the energy estimate and the smallness of the nonlinearity.To study the nonoscillating component u ε , we average the first MHD equation in the second space variable to obtain a two-dimensional Navier-Stokes equation singularly perturbed by the well-known linear operator L(u) := P(u × e 3 ).Following the method introduced by Schochet in [19], we filter the system by the associated group ᏸ(t) in order to look for the limit system (in the sense of distributions) satisfied by the possible limit v of the filtered solution v ε := ᏸ(−t/ε)u ε .
For the induction equation, we prove that the diffusion process vanishes as the Rossby number goes to zero.The magnetic perturbation b ε conserves its initial average on the torus, but its oscillating component b ε osc has the same behavior as u ε osc .More precisely, we prove the following convergence results.Theorem 1.2.Let s > 3/2 + 2 be an integer and U 0 = (u 0 ,b 0 ) ∈ H s (T 3 ) such that div u 0 = div b 0 = 0. Let U ε = (u ε ,b ε ) be the family of solutions of ( ε ) given by Theorem 1.1.Then ) Moreover, for all s < s, the family v ε := ᏸ(−t/ε)u ε converges strongly in Ꮿ 0 ([0,T],H s (T 3 )) to the solution v of the following two-dimensional limit system: where, Theorem 1.2 deserves a few comments that will be summarized in the following remark.
Remark 1.3.(1) Following the notation of Theorem 1.2, we note that ᏸ(t/ε)v ε is the noncompact part of u ε .
(2) Results obtained about the magnetic perturbation are in accordance with physicist's suggestions.In fact, according to [8], the creation and diffusion of the earth's magnetic field needed by a self-exited dynamo process necessitates a Reynolds number greater than the unity.
(3) An explicit computation of the resonance set of the limit quadratic term Q 0 (v,v) seems not to be an easy matter.For this reason, we will just give a particular example to show that this set is not empty and then the considered torus is a resonant one.
In the case of the whole space R 3 , we mention that the plane {ξ ∈ R 3 , ξ 2 = 0}, where the resonance phenomenon is supposed to take place, is zero Lebesgue measure.That is why the following theorem holds.
Theorem 1.4.Let s > 3/2 + 2 be an integer and This paper is organized as follow.In the next section, we present the proof of the existence result (Theorem 1.1).Section 3 is devoted to the proof of the convergence results on the torus (Theorem 1.2).Finally, we turn to the convergence results in the case of the whole space ( Theorem 1.4).

Existence result
In this section, we prove Theorem 1.1.Notice that this proof is similar to that of [2, 3, Theorem 1.2].We include it here for the convenience of the reader.
We begin by observing that by classical energy methods one can prove global existence of the so-called "Leray's solutions" for the system ( ε ) and derive the following L 2 -energy estimate: To study the existence and the regularity of strong solutions, we first approximate the nonlinear part of MHD system by a family of nicer nonlinearities, for which we can apply the classical theory of ODE in order to construct approximate solutions as in [4], for example.Next, we obtain uniform estimates on the approximate solutions, by using the conservation laws.Finally, we use these estimates to pass to the limit in the approximate equation.Precisely, we introduce, for a strictly positive integer n, the Friedrichs operator J n defined by We consider the following approximate magnetohydrodynamic system (MHD n ): 3) The above system is an ODE and can be written in the following abstract form: where U n = (u n ,b n ) and the expression of F n is given by the (MHD n ) system.Since F n is a continuous function from H σ (Ω) into H σ (Ω) for all σ ∈ R, then (MHD n ) has a unique maximal solution U n in the space Ꮿ 1 ([0,T * n (ε)[,H s (Ω)).Since J 2 n = J n and div u n = div b n = 0, it follows by uniqueness that The following product law will be useful throughout this paper (see [3]).
To continue the proof of Theorem 1.1, we take the scalar product in H s (Ω) and we use the above lemma to obtain, for all Using (2.7) and Gronwall's lemma, we obtain, for all t ∈ [0,T(n,ε)[, Moreover, for all t ∈ [0,T], Using Ascoli's theorem, the Cantor diagonal process, and the estimate (2.11), for n tends to infinity, we obtain a solution that satisfies the following estimate: (2.12) This regularity implies in a standard way the uniqueness.It remains to prove the global existence when the initial data is small enough.We now assume that U 0 H s (Ω) ≤ cε, (c = 1/C), and we set

Study of the periodic case
In this section, we will prove Theorem 1.2 stated in the introduction.The assertion (1.11) is obvious and follows directly from the energy estimate.For the equality (1.12), it suffices to integrate the second MHD equation in the space variable x over the torus T 3 .To prove (1.13), we start by writing u osc H σ (T 3 ) in terms of the Fourier expansion Then we have just to estimate | u ε (t,k)| for k 2 = 0. To do so, we observe that for k 2 = 0, u ε (•,k) is a solution of an ordinary differential equation.First of all, we rewrite the system ( ε ) in the following form: where (3. 3) The energy estimate implies that the family (F ε ) is bounded in L 2 ([0,T],H s−1 (T 3 )).If we apply the operator "curl" to the first equation, by the partial Fourier transform with respect to the space variable, we get where We recall that the eigenvalues of M(k) are ±i|k|, 0 and the corresponding eigenvectors are ν(k) ± and k/|k| which is not divergence-free.In this eigenbase, the velocity field u ε and the magnetic perturbation b ε can be written as follow: In terms of variables u ε + , u ε − , b ε + , and b ε − , (3.4) splits into four equations which can be summarized in where With a direct computation, one can prove the following lemma. where Moreover, if we denote by ω ± j (k) the real part of λ ± j (k), it holds that By Duhamel's formula, we have Let ω(k) := Min(ω ± j (k)).Using the Cauchy-Schwarz inequality, we obtain, for k 2 = 0, For the case of low frequencies, since (F ε ) is bounded in L 2 ([0,T],H s−1 ), then for all δ ∈]0, T] we have Now, we have to study the high frequencies.Let n ∈ N * and k ∈ Z 3 with |k| ≥ n.For t ∈ [0,T], the MHD system gives First, the energy estimate and the condition s > 3/2 + 1 imply that On the other hand, we remark that It remains to study the following cubic term: As in [2], we begin by mentioning that for |k| ≥ n we have The Cauchy-Schwarz inequality yields where Using the energy estimate (1.8), we obtain where The Gronwall lemma implies that, for all t ∈ [0,T], By interpolation argument, we complete the proof of (1.13).Now we turn to study the average case (k 2 = 0).We integrate the first equation of MHD system with respect to the second space variable x 2 .We obtain an average equation, in which the term P(curl(b) × e 2 ) disappears and the penalization operator becomes L(v) = P(v × e 3 ).If we denote g ε = εΔu ε + P(b ε • ∇b ε ), we obtain (3.27) The energy estimate implies that Using the decomposition u ε = u ε + u ε osc , by a direct Fourier computation, one proves that the attraction between "slow" waves u ε and "fast" ones u ε osc has no resultant; that is u ε • ∇u ε osc = u ε osc • ∇u ε = 0.However, "fast" waves interfere to create additional ones given by The idea is to filter the average system (3.29) by the group ᏸ(t) associated to L = P(u × e 3 ).Although the operator L is well known in the literature, we recall in the following lemma some of its properties which are useful in our purpose.
Lemma 3.2.The wave equation has a global solution denoted by u(t) = ᏸ(t)u 0 , satisfying for all s ∈ R and for all u 0 in H s (T 2 ), where ( Denote v ε (t) := ᏸ(−t/ε)u ε (t).System (3.29) becomes where K ε = ᏸ(−t/ε)G ε .System (3.34) can be rewritten as follows: where the filtered quadratic form is given by Let Q 0 = lim ε→0 Q ε in the sense of distributions.We obtain formally the following Eulertype limit system: for which we establish the following theorem.
Theorem 3.3.Let s > 2/2 + 2 be an integer and U 0 = (u 0 ,b 0 ) ∈ H s (T 3 ) a pair of divergencefree fields.Then, there exist T > 0 and a unique solution v of (ᏸ) in the space Ꮿ 0 ([0,T], Proof.We have to estimate the term , where v n is the solution of the approximate limit system.Since lim ε→0 t , it holds as in the proof of the existence result that (3.37) Remark 3.4.There are many papers dealing with the possible relation between the life span of the solution of the considered system (such as ( ε )) and the life span of the solution of the corresponding limit system (such as (ᏸ)), especially how to recuperate global well-posedness or uniform local well-posedness of the considered system from the limit one, at least for ε small enough (see, e.g., [5,9,10,17]).However, since we will focus on justifying the convergence result, any uniform time of existence of both ( ε ) and (ᏸ), denoted also by T, do the job.
Let us now turn to the proof of the convergence result.Denote W ε = v ε − v.We obtain where By the nonstationary phases theorem, we get where and, for any vector X, The right-hand side of the above system will be composed of two terms.The first is K ε , which tends to zero in L 2 ([0,T],H s−2 (T 2 )) when ε → 0. The second is the oscillating term R ε osc , which converges weakly to zero but not strongly.The method we use here to deal with R ε osc is inspired from the ideas introduced by Schochet in [19] and applied by Gallagher in [9,10].The idea consists in dividing R ε osc into high-frequency term R ε,N osc and low-frequency term R ε osc,N defined, respectively, for any arbitrary cutoff integer N ≥ 1, by In a first time, we consider the case of high frequencies.We recall the following lemma (see [10]).
This lemma leads to the following proposition.
To absorb the low-frequency term, we adopt the following change of function: where It is easy to see that ϕ ε N satisfies the following equation: where We note that the equation satisfied by ϕ ε N has the advantage that the low-frequency terms have disappeared up to an ε.To show that low-frequency terms are ᏻ(ε), we recall the following product law (see [3]).Lemma 3.7.Let σ > 2/2 be an integer.A constant C exists such that for all f ∈ H σ (T 2 ) and (3.48) The following proposition holds.
Proposition 3.8.There exists a constant C N (T) which depends only on T and N such that Proof.Recall that all the functions considered here are truncated in low frequencies.Hence, the result is simply due to the fact that v ∈ Ꮿ 0 ([0,T],H s (T 2 )), ∂ t v ∈ Ꮿ 0 ([0,T], H s−2 (T 2 )) and Lemma 3.7 (see, e.g., [10]).By the scalar product in H s−2 (T 2 ), (3.46) gives 1 2 (3.50)By Lemma 3.7, we have 1 2 (3.51) Integrating this inequality and using Proposition 3.8, we obtain where Then, for all 0 ≤ t < T , we have  Then, the resonance set is not empty and the unit torus T 2 is a resonant one.

Case of the whole space R 3
Theorem 1.4 and its proof are similar to those given in [2].We give them here for the sake of completeness.then one infers that for any function f , Consequently, it suffices to estimate | u ε (t,ξ)| for ξ 2 = 0.
Observe that the same computation used in the periodic case still holds in the case of the whole space, just change k by ξ.Precisely, as in the periodic case, one just have to change k by ξ, apply the Duhamel formula and the Cauchy-Schwarz inequality to obtain, for ξ 3 = 0, 3) It easily follows that