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THE GLOBAL L2 STABILITY OF SOLUTIONS TO THREE DIMENSIONAL MHD EQUATIONS

In this paper, we mainly study the global L-2 stability for large solutions to the MHD equations in three-dimensional bounded or unbounded domains. Under suitable conditions of the large solutions, it is shown that the large solutions are stable. And we obtain the equivalent condition of this stability condition. Moreover, the global existence and the stability of two-dimensional MHD equations under three-dimensional perturbations are also established.


Introduction
Magneto-hydrodynamic (MHD) equations are closely related to many branches of physics, chemistry, metallurgy, nuclear energy and space flight. In this paper, we mainly consider the following MHD equations: where Ω is a domain in R 3 , unknown vector function u = u(x, t) is the velocity of fluid, B = B(x, t) and p = p(x, t) are magnetic field and pressure function respectively. f = f (x, t) is the given forcing function, u 0 = u 0 (x), B 0 = B 0 (x) are the initial velocity and the initial magnetic field, ε = 1 Re , μ = 1 Rm , Re and Rm are Reynolds of fluid and magnetic field, respectively. The MHD equations are similar to the Navier-Stokes equations in the structure. In particular, if B = 0, then equations (1.1) transform to the impressible Navier-Stokes equations. So we can use the similar method of Navier-Stokes equations to solve the problem of MHD equations. But there are unknown magnetic function B, more nonlinear terms and strong coupled terms in the MHD equations. Therefore, it is more difficult to study the MHD equations. In 1994, Ponce, Racke, Sideris and Titi [5] proved the global stability of large solutions to the 3D Navier-Stokes equations. They proved the global stability of the strong solutions to Navier-Stokes on bounded or unbounded domains under suitable conditions. Moreover, they also established the global existence and the stability of two-dimensional MHD equations under three-dimensional perturbations. Decaying qualities of solutions about the Navier-Stokes equations can be seen in [1-3, 6, 12-14, 23], and references therein. Existence, uniqueness and regularity of solutions about Navier-Stokes are related to [4,7,8] and references therein.
Many mathematical studies were made on the MHD equations. Duvant and Lions [21] proved the existence and uniqueness of the global strong solutions of two dimensional MHD equations with initial boundary problem. They also proved the existence of the global weak solutions and existence and uniqueness of the locally strong solutions of three dimensional MHD equations with ε = 1, μ = 1 in (1.1). Based on above arguments, Sermange and Teman [22] studied the large time behaviors of solutions of MHD equations. Kozono [23] proved the existence of the global weak solutions to the two dimensional MHD equations on Ω t × (0, T ) (Ω is a bounded domain with smooth boundary in R 2 ) for ε = 1, μ = 1. He also proved the existence of the local classical solutions in Hölder space. Wu [24] considered the convergence rate of a class of MHD equations by using the method of vanishing viscosity. He and Xin [16] got the regularity conditions of MHD equations (ε = 0, μ = 0). Miao, Yuan and Zhang [25] studied the well-posedness problem of the MHD equations in the BMO −1 space. Some results about the MHD equations can be seen in [15,17,19,20] and references therein.
We will extend the arguments of global stability of strong solutions to Navier-Stokes equations studied in [5] to that of MHD equations.
Before we state the main results, we give some notations. Throughout this paper we will always assume that Ω ⊂ R 3 is a domain with boundary ∂Ω uniformly of class C 3 . H denotes a complete space of {u ∈ C ∞ 0 (Ω), ∇ · u = 0} in L 2 (Ω). Let P : L 2 (Ω)| → H be the Helmholtz projection operator. After applying P on the either sides of the equations in (1.1), we obtain (for simplicity, we choose ε = μ = 1 and where H 2 (Ω) and H 1 0 (Ω) denote the usual Sobolev spaces defined over Ω. In this paper, for Ω bounded or unbounded, we assume that the following Poincare inequality is satisfied for all g ∈ H 1 0 (Ω), where C is a constant. The main results will be given in Section 2 and the corresponding proof will be presented in Section 3.

The Main Results
The main results of this paper are as follows: then there exists a unique global strong solution of (1.1) with the initial data (u 0 , B 0 ) and P f.
(ii) Let Ω ⊂ R 3 be a general domain and P f 1 ∈ L 1 L 2 ([0, ∞), H). If there exists a δ > 0 such that u 0 ∈ V, B 0 ∈ V and P f ∈ L 1 L 2 ([0, ∞), H) and then there exists a unique global strong solution of (1.1) with the initial data (u 0 , B 0 ) and P f, Theorem 2 Assume that Ω is R 3 or a three dimensional domain for which (1.3) holds, with f 1 ∈ L 2 ([0, ∞), L ∞ (Ω)). For global strong solutions (v, N ) of (1.1) in the class described in Theorem 1, the condition then there exists a unique global strong solution of (1.1) with the initial data (u 0 ,

Proof of Theorems
In this section we will give the proofs of Theorems 1-3. Before we give the usual Young inequality as follows : for a > 0, b > 0 and ε > 0, one has Multiplying Aw on the both sides of equation (3.2), and integrating on Ω, we obtain Multiplying AE(t) on the both sides of (3.3), and integrating on Ω, we get Adding (3.4) to (3.5), we obtain In order to estimate I 1 −I 13 , we need to collect a few interpolation inequalities for functions g ∈ D(A). First, under the assumption that ∂Ω is uniformly C 3 , it was shown in Lemma 1 of [9] that ∂ 2 ij g ≤ C( Ag + ∇g ). Since D(A) ⊂ P L 2 (Ω), we may integrate by parts and use Cauchy-Schwarz inequality to get Therefore, in general, we have the following inequality Now in the case where (1.3) holds, it is clear from the argument of (3.7) that Using Fourier analysis, we easily know this inequality is true for R 3 as well. If g ∈ D(A), one has [12] which holds for any Ω ⊂ R 3 . For the domain satisfying (1.3), using Gagliardo-Nirenberg inequality, (3.10) and (3.11), we obtain Thus, using Calderon extension theorem [10] and then (3.8), we obtain for all of the domain under consideration the following inequality is true Moreover, in the case Ω = R 3 , we obtain from Gagliardo-Nirenberg inequality and (3.10) (3.14) In the general case, we can apply the Calderon extension theorem and (3.8) to get which, exactly as above, improves to (3.14) Hence Ω is such that (3.12) and (3.14) hold. I 1 − I 13 are estimated as follows: Choosing ε sufficiently small (for example ε = 0.01), substituting (3.16)-(3.28) into (3.6), we obtain d dt Defining h(t) =: ∇w(t) 2 + ∇E(t) 2 , we have proved with help of (3.6), (3.16)-(3.28) that holds with positive constants C, C 0 . Let is true, then we infer that holds for 0 ≤ s ≤ t 1 , and some t 1 > 0. Thus, (3.30) implies that for s ≤ t 1 , the following inequality holds Consequently, by a generalized Gronwall inequality, we obtain On the other hand, the equations with the solution (v, N ) as follows: Multiplying Av on the both sides of the first equation of (3.35), integrating on Ω, we obtain Multiplying AN on the both sides of the second equation of (3.35), integrating on Ω, we get Adding (3.36) to (3.37) leads to . J 1 − J 3 are estimated as follows: Choosing ε sufficiently small (for example ε = 0.01), we obtain By Gronwall inequality, we have Therefore, ∇v(t) + ∇N (t) is uniformly bounded. Since where the following equality holds Multiplying w on the both sides of (3.3), integrating on Ω, we have where the following equality holds (3.45) K 1 − K 5 are estimated as follows: Choosing ε sufficiently small (for example ε = 0.01), substituting (3.47)-(3.51) into (3.46), we obtain By Gronwall inequality, we obtain for all t > 0. Concerning (3.6), when Ω is the general domain, I 1 − I 13 can be estimated as follows: ≤ C ε ∇v 4 ∇w 2 + ε Aw 2 + C ε ∇v 2 ∇w w + ε Aw 2 ≤ C ε ∇v 4 ∇w 2 + C ε 2 ( ∇v 4 ∇w 2 + w 2 ) + 2ε Aw 2 ≤ C( ∇v 4 ∇w 2 + w 2 ) + 2ε Aw 2 , (3.55) Choosing ε sufficient small (for example ε = 0.01) and substituting (3.54)-(3.66) into (3.6) Choosing δ small enough and using the similar method as Theorem 1 (i), we can get the proof of Theorem 1 (ii). This completes the proof of Theorem 1. 2 Proof of Theorem 2 Now we turn to the proof of Theorem 2. we assume that ∞ 0 ∇v(t) 4 + ∇N (t) 4 dt < ∞. The same argument used to Theorem 1, leads to (3.38). Integrating (3.38) from 0 to t, we obtain Noticing that (v, N ) satisfies (1.2) and by the usual energy estimate, we know that Taking ε = 1/4 and integrating from 0 to t, we get For p ≥ 6, 2 q + 3 p = 1, thanks to Gagliardo-Nirenberg inequality, (3.11) and Hölder inequality, where α = 4 q , therefore 2 ≤ αq( 1 2 + 3 p ) ≤ 4. Using (3.73) and the condition of Theorem 2, we have

Furthermore, by the inner interpolation inequality, we have
Using (3.72), we obtain which completes the proof for p ≥ 6. For p < 6, 2 q + 3 p = 1, using (3.71) and (3.73), we have Conversely, assume v ∈ L q ([0, ∞), L p (Ω)) for some p, q satisfying 2 q + 3 p = 1. We will prove that Firstly, applying the derivatives x i on both sides of the following equation then applying Helmholtz projection operator P , multiplying ∂ i v on both sides of the equation and integrating on Ω, we obtain Adding (3.79) to (3.80) reaches to (3.81) Integrating by parts and Hölder inequality, we have Similarly, we have

85)
and Choosing ε sufficiently small, substituting the estimates above into (3.81) then adding them from 1 to 3 about i, we get By Gronwall inequality, we obtain This completes the proof of Theorem 2.
By the smoothing properties of the MHD, we know that v(t) ∈ L 2 (R 2 ), N (t) ∈ L 2 (R 2 ) for all t > 0. we also know Moreover, using this estimate and the decay of solutions in [11], We are going to construct a global solution of the form u =ṽ + w, B =Ñ + E wherẽ v = (v 1 , v 2 , 0),Ñ = (N 1 , N 2 , 0), and w is small in V (R 3 ), E is small in H 1 0 (R 3 ). The equations for w and E are given by We claim that for w(t) ∈ V (R 3 ), (w · ∇)w, (w · ∇)ṽ, (ṽ · ∇)w and (f −f 1 ) lie in L 2 (R 3 ). Hence, for positive times in (3.92), it makes sense to apply the three-dimensional Helmholtz projection to them. An application of Hölder's inequality is enough to bound the following terms: The other terms are estimated by considering the variables x 1 , x 2 separately from x 3 : Using the Gagliardo-Nirenberg inequality in two dimensions, we have Integrating with respect to x 3 and use Cauchy-Schwarz inequality, we obtain (w · ∇)ṽ ≤ ∇w The corresponding estimates of E(t) can be done similarly like that of w(t).