ASYMPTOTIC BEHAVIOR OF WEAK AND STRONG SOLUTIONS OF THE MAGNETOHYDRODYNAMIC EQUATIONS

. We prove some results on the stability of slow stationary solutions of the MHD equations in two- and three-dimensional bounded domains for external force ﬁelds that are asymptotically autonomous. Our results show that weak solutions are asymptotically stable in time in the L 2 -norm. Further, assuming certain regularity hypotheses on the problem data, strong solutions are asymptotically stable in the H 1 and H 2 -norms.


1.
Introduction. In several situations the motion of incompressible electrical conducting fluid can be modeled by the magnetohydrodynamic (MHD) equations, which correspond to the Navier-Stokes equations coupled with the Maxwell equations. In the presence of a free motion of heavy ions, not directly due to the electrical field (see Schlüter [19] and Pikelner [15]), the MHD equations can be reduced to together with the following boundary and initial conditions: u(x, t) = 0, h(x, t) = 0, on ∂Ω × (0, T ), u(x, 0) = u 0 (x), h(x, 0) = h 0 (x), in Ω. (2) In the previous expressions, u and h are respectively the unknown velocity and magnetic field; p * is the unknown hydrostatic pressure; w is an unknown function related to the heavy ions (in such a way that the density of electric current, j 0 , generated by this motion satisfies the relation rotj 0 = −σ∇ω), ρ 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 and f is a given external force field.
Due to its importance, the MHD system has been discussed in a broad variety of studies encompassing subjects such as the existence of weak solutions and strong solutions, uniqueness and regularity criteria. See e.g. [6], [13], [12], [14], [7], [17], [18] and the references therein.
In the present work we discuss the stability of stationary solutions of the MHD equations in two-and three-dimensional bounded domains with respect to both initial conditions and external forcing variations. Under certain regularity hypotheses on the problem data, we establish in Theorem 3.2 the aforementioned stability in the L 2 -norm for weak slow flow stationary solutions. Additionally, in Theorem 4.4 and Theorem 5.2 we discuss respectively the H 1 -stability and the H 2 -stability for strong solutions. We note that, for a fixed given external force field, our results in particular imply the asymptotic stability of such stationary solutions.
The issue of stability of solutions is an important one, since solutions of any dynamical system are thought to be physically reasonable only if they are stable. There exists a number of ways in which stability can be examined. In past years, many efforts have been made to study the asymptotic behavior of classical Navier-Stokes equations. We refer the reader to Heywood and Rannacher [11], Beirão da Veiga [2], Qu and Wang [16], Zhang [21] and the references therein.
A few of the references mentioned above, e.g. [8], are closely related to the contents of this paper. In effect, it was shown in [8] that, under condition (52) stated in Section 5 below, the strong solution of the two dimensional Navier-Stokes equation is asymptotically stable in a bounded domain of R 2 . In this paper we establish the corresponding result for the magnetohydrodynamic equations, assuming instead condition (34) below, which is weaker than the used in [8], both in two dimensional and three dimensional domains. Further, under hypotheses (52) we will show that stability actually holds in the H 2 -norm. Thus, our results improve the existing ones even for Navier-Stokes equations.
If X is a Banach space, we denote by L q (0, T ; X) the Banach space of the Xvalued functions defined in the interval [0, T ] that are L q -integrable in the sense of Bochner. In addition, vector spaces will be denoted by boldface letters.
We also consider the following spaces of divergence free functions: Throughout the paper, the Helmholtz projection P is the orthogonal projection from L 2 (Ω) into H and A = −P ∆ with D(A) = V ∩ H 2 (Ω) is the usual Stokes operator. We observe that, by the regularity of the Stokes operator, it is usually assumed that Ω is of class C 3 in order to apply Cattabriga's results [5]. However, we use the stronger results of Amrouche and Girault [1], which imply, in particular, that when Au ∈ L 2 (Ω), then u ∈ H 2 (Ω) and u H 2 (Ω) and Au are equivalent norms when Ω is of Class C 1,1 . For ease of reference, we also recall the following inequalities which are consequences of the Sobolev and Hölder inequalities: (b) If each integral makes sense, for p, q, r > 0 and 1/p + 1/q + 1/r = 1, we have We also need the following regularity result for the Stokes problem (see Temam [20]) Proposition 1.
2.1. Mathematical setting of the problem. By applying the Helmholtz operator P to both sides of the first equation in problem (1), and by taking into account the previous considerations, one obtains the operational form of the problem: Here we have set α = ρ/µ, ν = η/µ and γ = 1/(µσ). The associated variational formulation is the following: to find (u, h) in suitable functional spaces such that u(0) = u 0 , h(0) = h 0 and, for every (v, b) ∈ V × V , the following holds: The corresponding stationary system in operational form is In this last system we considered a time-independent external force field f ∞ , possibly different from the previous f , because we want to check also the stability associated to changes in the external force field. This last problem, in its associated variational formulation becomes: find (u ∞ , h ∞ ) ∈ V × V such that, for every (v, b) ∈ V × V , the following holds: We call such pair (u ∞ , h ∞ ) a weak solution of the stationary problem (9) (or (8)).
By using the Galerkin method, it is possible to show the following result on existence of weak solutions of (9) (see Chizhonkov [6]): Under smallness conditions, we also have uniqueness of such solutions: Proposition 3. (Uniqueness) Any stationary weak solution satisfying the condi- where 0 < C L is the constant appearing in (3), is unique.
Proof. (of Proposition 3) Let (u 1 ∞ , h 1 ∞ ) be a slow-flow solution of (9), that is, a weak solution satisfying (11) and (12) , and let (u 2 ∞ , h 2 ∞ ) be another tentative weak solution of (9). By setting u = u 1 We take v = u and b = h in the above equalities and obtain By Lemma 2.1, we have By adding equalities (13) and (14), using Young's inequality, and the last estimates, we obtain which, together with hypothesis (11) and (12), implies that ∇u = 0 and ∇h = 0.
Remark 1. Since, by (10), u L 3 (Ω) ≤ C ∇u and h L 3 (Ω) ≤ C ∇h , conditions (11) and (12) can be interpreted either as saying that ν, γ are suficiently large or that f ∞ V * is sufficiently small. In these cases, we say that the associated (u ∞ , h ∞ ) is a stationary slow flow solution.
Next, we show that the regularity of the weak solutions of the boundary value problem (9) correlates with that of f ∞ , i.e., the more regular f ∞ is, the more regular the indicated solutions will be. To this end, we note that, by putting the nonlinearities on the right-hand side of (9), the stationary problem is equivalent to the following two coupled Stokes problems. The first Stokes problem is: where Proposition 4. Under the assumptions of Proposition 2 and the condition f ∞ ∈ L 2 (Ω), we have u ∞ , h ∞ ∈ H 2 (Ω) ∩ V . Moreover, the following inequality holds: where Ψ is a continuous and nondecreasing function of its argument such that Ψ(0) = 0.
By following the previous estimates and using Proposition 2, we obtain (17), which completes the proof.
In the remainder of this work, (u ∞ , h ∞ ) will denote a stationary slow-flow solution of the type discussed in this Section, i.e., (u ∞ , h ∞ ) satisfies the conclusions of Proposition 2 and 3.
Further, we assume that (u ∞ , h ∞ ) is a weak slow-flow solution of (9), i.e., (11) and (12) hold. Then there exists a positive constant β 0 > 0 such that, for every β ∈ (0, β 0 ], we have Then By taking v = w in (25), we obtain (27) On the other hand, taking b = z in (26), we obtain To estimate the terms on the right-hand side of the above expressions, we first note that Similarly, The above equalities together with (27) and (28) imply the following differential identity The terms on the right-hand side of (29) can be estimated as follows. From Lemma 2.1 and the Young inequality, we have Similarly, Using the above in equality (29), we obtain Now we observe that (11) and (12) imply that Therefore, by (30), we have Now, using the embedding H 1 0 (Ω) → L 2 (Ω), we have 1 2 for every β ∈ (0, β 0 ], where with C e1 equal to the embedding constant of H 1 0 (Ω) → L 2 (Ω). By integrating this inequality, we obtain the desired decay property (23).
In the next section we will also need the following estimate.
We can now prove the following stability result.
Proof. We must show that lim t→+∞ α u(t) − u ∞ 2 + h(t) − h ∞ 2 = 0, that is, given any > 0 (< 1, without loss of generality), there exists a T such that To find a T , let us start by considering δ > 0 (which will be chosen later in function of ). Since lim t→∞ f (t) − f ∞ V * = 0, we can choose a T δ such that Now, by (23) with a fixed β ∈ (0, β 0 ], we have We now choose δ so that δ 2 α 2 /β < /3, i.e., δ < (β /(3α 2 )) 1/2 , which yields the corresponding T δ . Then, from the last estimate we see that it suffices to choose T sufficiently large so that, for t > T , we have These conditions are satisfied with Remark 2. When n = 2, there exists a unique global weak solution (u, h) of (7) satisfying the initial condition (u 0 , h 0 ) ∈ H × H. Moreover, it is not difficult to check that u t , h t ∈ L 2 (0, T ; V * ) (see [17]). Thus the estimates stated in Proposition 5 and therefore the conclusions of Theorem 3.2, hold. In particular, the above implies that any stationary slow-flow solution is weakly asymptotically stable. When n = 3 and u, h ∈ L s (0, T ; L r (Ω)) with 2/s + 3/r ≤ 1 and r > 3, it can be shown that the solution satisfies u t , h t ∈ L 2 (0, T ; V * ) and is, furthermore, unique (see [7]). In this setting, the estimates stated in Proposition 5 and therefore the conclusions of Theorem 3.2 hold. Moreover, as before, any stationary slow-flow solution is weakly asymptotically stable.
The following is an immediate corollary of the theorem The following conditions on the initial data will remain in force throughout this section: Under assumptions (34), the existence and uniqueness of a local solution was established in [4], as follows: The following global existence theorem was proved in [18]:  In what follows, we will work under the assumption that it does, i.e., we henceforth assume that there exist constants M 2 and T , where 0 < T ≤ ∞ is as in Theorem 4.2, such that We note that it is also possible to carry out the following discussion without making the above assumption, namely by repeating the preceding smallness condition in three dimensions whenever needed. This approach, however, complicates the exposition and we avoid it.
Clearly, by Theorem 4.3, condition (35) holds without additional hypotheses in the two-dimensional case.

Thus
Ce −κt Using the above in (41), we conclude that Finally, the second and third terms on the right-hand side of the last inequality can be estimated using (33), which yields the estimate (36).
In the next section we will need the following estimates.
Using the preceding statements, we obtain the following estimates: Proposition 9. Assume that (35), (52) and the uniqueness condition in the stationary system hold and let (u, h) be a strong solution as in Theorem 5.1. Then Finally, the stated estimates for α w t (0) and z t (0) are consequences of (50) and (51), respectively.
Arguing as before, the above inequality yields estimate (60).
The following is an immediate corollary of the theorem.

Corollary 3.
If we set f (t) = f ∞ in Theorem 5.2, then the convergence rate there is exponential in the H 2 -norm.