PERIODIC STRONG SOLUTIONS OF THE MAGNETOHYDRODYNAMIC TYPE EQUATIONS

We obtain, using the spectral Galerkin method together with compactness arguments, existence and uniqueness of periodic strong solutions for the magnetohydrodynamic type equations. AMS Subject Classification : 35G25


Introduction
In several situations the motion of incompressible electrical conducting fluid can be modelled by the magnetohydrodynamics equation , which correspond to the Navier-Stokes equations coupled with the Maxwell equations.In presence of a free motion of heavy ions, not directly due to the electrical field (see Schlüter [21], and Pikelner [15]), the magnetohydrodynamics equation can be reduced to Here, u and h are respectively the unknown velocity and magnetic fields; p * is the unknown hydrostatic pressure; w is an unknown function related to the motion of heavy ions (in such way that the density of electric current, j 0 , generated by this motion satisfies the relation rot j 0 = −σ∇w); ρ m is the density of mass of the fluid (assumed to be a positive constant); µ > 0 is the constant magnetic permeability of the medium; σ > 0 is the constant electric conductivity; η > 0 is the constant viscosity of the fluid; f is an given external force field.
We append to equation (1.1) the following boundary conditions In this paper, we will consider the problem of the existence and uniqueness of the periodic strong solutions in a bounded domain Ω ⊂ IR N , N = 3 or 4; the given external force f be periodic in t with some period τ .Then we will prove the existence and uniqueness of periodic strong solution (u, h) of the magnetohydrodynamic type equations (1.1) with the same period τ u (x, t + τ ) = u (x, t) ; h (x, t + τ ) = h (x, t) (1.3)The initial value problem associated to the system (1.1) has been studied by several authors.Lassner [13], by using the semigroup results of Kato and Fujita [9], proved the existence and uniqueness of strong solutions.Boldrini and Rojas-Medar , [5], [18] improved this results to global solutions by using the spectral Galerkin method.Damásio and Rojas-Medar [8] studied the regularity of weak solutions, Notte-Cuello and Rojas-Medar [16] using an iterative approach to show the existence and uniqueness of strong solutions.The initial value problem in a time dependent domains was studied by Rojas-Medar and Beltrán-Barrios [17] and Berselli and Ferreira [4].
The periodic problem to the classical Navier-Stokes equations, was studied by Serrin [19] using the perturbation method and recently by Kato [12] using the spectral Galerkin method.In this work we follow [12].
Finally, we would like to say that, as it usual in this context, to simplicity the notation in the expressions we will denote by C, C 1 , . . .,generic positive constants depending only on the fixed data of the problem.

Preliminaries and Results
We begin by recalling certain definitions and facts to be used later in this paper.
The L 2 (Ω)-product and norm are denoted by (, ) and | |, respectively; the If B is a Banach space, we denote L q (0, T ; B) the Banach space of the B-valued functions defined in the interval (0, T) that are L q -integrable in the sense of Bochner. Let Let P be the orthogonal projection from (L 2 (Ω)) N onto H obtained by the usual Helmholtz decomposition.Then, the operator A : H → H given by A = −P ∆ with domain In order to obtain regularity properties of the Stokes operator we will assume that Ω is of class C 1,1 [2].This assumption implies, in particular, that when Au ∈ L 2 (Ω), then u ∈ H 2 (Ω) and u H 2 and |Au| are equivalent norms.
Now, let us introduce some functions spaces consisting of τ -periodic functions.For k ≥ 0, k ∈ IN , we denote by Then, let us define the norm We denote for 1 ≤ p ≤ ∞, the spaces where Similarly, we denote by In particular, H k (τ ; B) = W k,2 (τ ; B), when B is a Hilbert space.
The following results will be using in this paper.
Lemma 1.3.(Simon [10]) Let X, B and Y Banach spaces such that X → B → Y , where the first embedding is compact and the second is continuous.Then, if T > 0 is finite, we have that the following embedding is compact Ours results are the following.
The idea of the proof is use the spectral Galerkin method together with compactness arguments.The principal problem is obtain the uniform boundedness of certain norms of u n (t) and h n (t) in some point t * .This difficult was early treated by Heywood [11] to prove the regularity of the classical solutions for Navier-Stokes equations.

Approximate problem and a priori estimates
By using the operator P , the periodic problem (1.1)-(1.3) is formulated as a system of ordinary differential equations Where, We consider V n = span{w 1 (x), w 2 (x), ..., w n (x)} and the approximations u n (t) = n j=1 c jn (t)w j (x) and h n (t) = n j=1 d jn (t)w j (x), of u and h, respectively satisfying the following system of ordinary differential equations To show that the system (3.1) has an unique τ −periodic solution, we consider the following linearized problem: It is well know that the linearized system (3.2) has an unique τ −periodic solution (u n (t), h n (t)) ∈ (C 1 (τ ; V n )) 2 (see for instance, [1], [6]).On the other hand, it is easily checked that the map: By using the Leray-Schauder principle is sufficient to show the boundedness sup where C is a positive constant independent of λ , for all solutions of (1. Adding the above inequalities and using (2.3), we get and |Ω| ≡ the volume of Ω.By using the Young inequality, we obtain where C 4 = C 2 3 α 2 /ν and M is defined as in theorem 2.4.Moreover, since (u n , h n ) are τ −periodic functions, we have and consequently, from (3.3), we obtain It follows by Mean Value Theorem for integral, that there exists Now, using the Lemma 2.2, with θ = 0, β = 1/2, we get and consequently where K 1 is independent of λ and n.Thus, we have proved the existence of the solution ( Lemma 2.1.Let (u n (t), h n (t)) be the solution of (3.1) given above.Suppose that M < min{K −2 2 , K −3 3 , 1} where Then, we have Proof.Taking A 2γ u n and A 2γ h n as test functions in (3.1) i and (3.1) ii respectively, we get Now, we estimate the right-hand sides of the above equalities as follows: here we use the Hölder's inequality, where we use the Giga-Miyakawa estimate with θ = γ and ρ = 2γ+1 2 .Now, adding the above estimates, we get By using the Lemma 2.2 with θ = γ and β = 1 2 , we obtain from (2.5) and (3.5) Consequently, if M < 1, we have Thus, where We will prove by contradiction that T * = ∞.In fact, if T * (t * < T * ) is finite it should follow that Therefore, for such a value t = T * , the estimates of the right-hand side of (3.8) are Consequently, the above estimate and (3.8) implies where On the solutions of the magnetohydrodynamic type equations 209 Thus, in a neighborhood of t = T * it follows which implies T * = ∞.Therefore, this given since u n (t) and h n (t) are periodics.

Estimates of derivatives of higher order
To show the convergence of the approximate solutions we shall derive estimates of derivatives of higher order.By Lemma 3.1, if M is sufficiently small the approximate solutions satisfy , where C(M ) denotes a constant depending on M and independent of n.
Then, we have and sup where C(M 0 , M ) and C(M 0, M 1 , M ) denote constants depending on M 0 , M 1 , M and independent of n.Proof.From (3.1) and using (2.1), we have Adding the above inequalities and using the estimate (4.1), we have Recalling the periodicity of ∇u n (t) and ∇h n (t), we deduce from the above inequality τ 0 Newly, applying the Mean Value Theorem for integrals, we have that there exists t * ∈ [0, τ ] such that By using the Lemma 2.2 , with θ = 1 2 , β = 1, we have On the solutions of the magnetohydrodynamic type equations 211 Now, integrating the inequality (4.2) from t * to t + τ (t ∈ [0, τ ]), we deduce easily where C(M 0 , M ) is independent of n.
By other hand, from equations (3.1) we have or, equivalently Now, we estimate the right-hand sides of the above inequalities, we have where we use the Hölder' inequality, the estimate (2.3) and Sobolev's embedding with r = 2N N −2 and γ = 1.Thus, by using the above estimates and the equalities (4.4), we get Integrating from 0 to τ , we have and using the inequality (2.2) and (2.3) with r = 3 and β = 1, we infer of the above inequality for any φ, v, b ∈ V. Analogously, If, we use the estimate (4.8) for second and third terms in equality (4.7) i and the estimate (4.9) in the last term of this equality, we get Lemma 3.2.Let (u n (t), h n (t)) be the approximate solution of (3.1) given above.Then, we have Proof.From the equalities (3.1), we obtain easily Now, we use the Proposition 2.1 with θ = γ, δ = 0 and ρ = 1, to obtain where we use the estimate (4.22) , here we set φ = u n or h n and v = u n or h n .
The inequalities (4.20) and (4.21) together with (4.22) imply Using the Young inequality to the third first terms and the fact that C(M ) is small, we obtain We will show that Once these later convergencies are established, it is a standard procedure take the limit along the previous subsequences in (3.1) to conclude that (u, h) is a periodic strong solution of (1.1)-(1.3).
Thus, we have in (5.
[3]eplacing P u n • ∇u n by λP u n • ∇u n , P h n • ∇h n by λP h n • ∇h n , P u n • ∇h n by λP u n • ∇h n and P h n • ∇u n by λ P h n • ∇u n (0 ≤ λ ≤ 1) (see[3]).Then, multiplying (3.2) i and(3.2)iiby e jn (t) and g jn (t) respectively, and adding in j, we obtain → u t , h n t → h t , w * in L ∞ (τ ; V ).and the functions u(t) and h(t) satisfies