Time periodic solutions to the three--dimensional equations of compressible magnetohydrodynamic flows

In this paper, the compressible magnetohydrodynamic system with some smallness and symmetry assumptions on the time periodic external force is considered in $\mathbb{R}^3$. Based on the uniform estimates and the topological degree theory, we prove the existence of a time periodic solution in a bounded domain. Then by a limiting process, the result in the whole space $\mathbb{R}^3$ is obtained.

(1) Here x ∈ R 3 , ρ(x, t), u(x, t) = (u 1 , u 2 , u 3 )(x, t) and H(x, t) = (H 1 , H 2 , H 3 )(x, t) represent the density, the velocity and the magnetic field respectively, and the pressure P (ρ) = ρ γ with γ > 1. The constants µ, λ are the viscosity coefficients with the usual physical conditions µ > 0, λ + 2 3 µ ≥ 0; the constants ν > 0 is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field. In addition, f (x, t) is a given external force, which is assumed to be periodic in time with period T . The aim of this paper is to show the problem (1) admits a time periodic solution around the constant state (ρ, 0, 0) in R 3 , which has the same period as f . Based on the uniform estimates and the topological degree theory, we are able to establish the existence of a time periodic solution in a sequence of bounded domain. And then the existence in the whole space is derived by several uniform bounds and the limiting process on the approximate solutions.
Our main result in this paper is stated as follows.
Theorem 1.1. Let the time periodic external force f (x, t) ∈ L 2 (0, T ; L for some small constant h > 0, then the problem (1) admits a time periodic solution (ρ, u, H) ∈ S δ0 with the same period as the external force f (x, t), and S δ0 is defined later.
There are extensive studies on the existence and uniqueness of solutions of the magnetohydrodynamic equations for the physical importance, complexity and wide range of applications, cf. [3,4,5,6,7,8,9,11,12,16,17,18,21,22,24,25,26] and references therein. However, comparatively less works are received to the periodic solutions case. In particular, as for the incompressible case, in [21], the authors obtained the existence and uniqueness of the periodic strong solutions in a bounded domain Ω ⊂ R n , n = 3 or 4 by the spectral Galerkin method and compactness arguments. On the other hand, for the compressible case, Yan and Li [25] proved the existence of time periodic solutions on a bounded domain of R 3 in the framework of weak solution for large time periodic external forces. For the problem in an unbounded domain, Tan and Wang [22] studied the existence, uniqueness and timeasymptotic stability of time periodic solutions in the whole space R n . It was shown in [22] that only when the space dimension n ≥ 5, there exists a time periodic solution for a sufficiently small time periodic external force f (x, t) ∈ C(0, T ; H N −1 ∩ L 1 ), N ≥ n+2. However, how to handle the case when n ≤ 4, especially the physical case when n = 3, there is no results available. Based on this, in this paper we will consider the existence of a periodic solution under some smallness and structure conditions on the time periodic external force in the whole space R 3 .
When there is no electromagnetic field, system (1) reduces to the compressible Navier-Stokes equations, cf. [1,2,10,13,14,15,19,20,23] and references therein. Here we only mention some of them related to our paper. In [19], Ma, Ukai and Yang combined the linear decay analysis and the contraction mapping theorem to obtain the existence and stability of time periodic solutions when the space dimension n ≥ 5. For recent works, in [14], Jin and Yang considered the existence of time periodic solutions to the whole space R 3 through the topological degree theory. Also, by the spectral properties, the author in [15] obtained a time periodic solution for sufficiently small and symmetry condition on the time periodic external force when the space dimension is greater than or equal to 3. Inspired by [14], in the present paper, we improve the result from [22] in the following sense: we change the space dimension from n ≥ 5 to n = 3.
The rest of the paper is organized as follows. We will reformulate the problem and give some preliminaries lemmas for later use in Section 2. In Section 3, we will obtain the existence of periodic solutions for (3) by uniform estimates and topological degree theory in a bounded domain. And the proof of the main result will be given in the last section.
Notations. Throughout this paper, for simplicity, we will omit the variables t, x of functions if it does not cause any confusion. C denotes a generic positive constant which may vary in different estimates. Moreover, we use H s to denote the usual L 2 −Sobolev spaces with normal · H s and L p , 1 ≤ p ≤ ∞ to denote the usual L p spaces with norm · L p . Finally, denote the t-anisotropic Sobolev spaces as And for 0 < α < 1, denote C α, α 2 ((0, T ) × Ω L ) be the set of all functions u such that 2. Preliminaries. In order to prove the existence of periodic solutions, we first substitute = ρ −ρ, then the system (1) can be rewritten as To solve the time periodic problem for (2) in R 3 , we now concern the following regularized problem in a bounded domain where is a time periodic function and an odd function on the space variable x with periodic boundary, satisfying dt + 2 f 2 is time periodic functions with periodic boundary condition; ||| < δ}, for some positive constant δ and with the norm ||| · ||| defined as Then we have the following proposition for the problem (3), the proof of this proposition will be given in the end of next section.
for some small constant h * > 0, then the problem (3) admits a solution ( L , u L , H L ) ∈ S L δ0 , here δ 0 is a small constant independent of L and . Several elementary inequalities are needed later, cf. [14], here the coefficients independent of the domain play an important role in passing the limit of the approximate solutions in the last section. In particular, if q = p * = N p N −p , then Assume that Ω ⊂ R 3 is a bounded domain, and ∂Ω is locally Lipschitz where C is independent of Ω. Moreover, the above inequalities also hold in R 3 if u(x) → 0 as |x| → ∞.

Existence in bounded domain.
In this section, we are devoted to obtaining the existence of time periodic solutions for the regularized problem (3) in a bounded domain of Proposition 1. The proof is a combination of the uniform estimates and the topological degree theory. To do this, we begin with the introduction of a completely continuous operator F to the linear parabolic problem (4).

3.1.
Introduction of an operator F. For any τ ∈ [0, 1], we define an operator with δ being suitably small, here ( , u, H) is the solution of the following system with periodic boundary where In what follows, we will concentrate on the properties of the operator F. To show F is well defined, we shall establish the following lemma first. Lemma 3.1. Assume that δ is sufficiently small, then for any (ρ, v, B) ∈ S L δ , τ ∈ [0, 1], the problem (4) admits a unique time periodic solution ( , u, H) ∈ S L .
Proof. By Lemma 2.2, we have Then from the smallness of δ, we obtain which implies that 1 2ρ Now, we begin with the following initial value problem of (4) in Ω L , that is with periodic boundary condition and the initial data 0 (x) is an even function with Ω 0 dx = 0, u 0 , H 0 are odd functions. Obviously, the solution ( , u, H) has the same properties as the initial data ( 0 , u 0 , H 0 ).
Multiplying (8) 2 by u and the integration over Ω L yields In view of (5), then for appropriately small δ, we see that Summing up the above two inequalities, we obtain On the other hand, multiplying (8) 2 by µ∆u + (µ + λ)∇ div u, we deduce that Applying ∇ to (8) 1 , (8) 3 and multiplying the resultant identities by (2µ+λ) P (ρ) ρ 2 ∇ , ∇H respectively, we have From the above two estimates, we arrive at This together with (9) and the poincáre inequality imply Then by Duhamel's principle, the solution to the system (4) can be written in a mild form as and it satisfies that where we have used the time periodic property of W and F , also we have That is, is a time periodic solution of (4) with time period T. Moreover, by the classical theory of parabolic equations, we have that for any (ρ, v, B) ∈ S L δ , τ ∈ [0, 1], the problem (4) admits a time periodic solution ( , u, H) ∈ S L . To prove the uniqueness, assume that there exists two solutions Similar to the first half of this proof, Let is also the solution of (4), by the uniqueness, we easily obtain ( ( , t)). This completes the proof of Lemma 3.1.
Now, in the following lemma, we see that the operator F is completely continuous. The proof is similar to Lemma 2.2 and Lemma 2.3 in [13].
Lemma 3.2. Assume that δ is sufficiently small, then the operator F is compact and continuous.
3.2. Uniform estimates. The task of this subsection is to give several uniform estimates for ( , u, H) of the following system. This is crucial for the proof of Proposition 1 and the main theorem.
where τ ∈ (0, 1]. Since when τ = 0, similar to the proof given in the section 3 of [14], it is easy to obtain that F ((ρ, v, B), 0) = 0. By elaborate calculation, the uniform estimates for ( , u, H) independent of L and are derived as follows .
where C is a constant independent of L and .
Proof. By multiplying (11) 2 with u and integrating over Ω L , we obtain where we have used the (11) 1 and the periodic boundary conditions. Then multiplying (11) 3 by H, integrating over Ω L , we can arrive at Similarly, multiplying (11) 1 by P (ρ + τ ) and integrating over Ω L by part to conclude that 1 τ Summing up the above identities, we derive from Lemma 2.1, Lemma 2.2, the Hölder and Young inequalities that This gives the estimate (12) and completes the proof of the lemma.
where C is a constant independent of L and .

TIME PERIODIC SOLUTIONS TO MHD 1857
The above two estimates imply that 1 2 which yields (13). This completes the proof of the lemma.
where C is a constant independent of L and .
Proof. Multiplying (11) 2 , (11) 3 by ∆u, ∆H respectively, and then integrating the resulting equalities over Ω L give that 1 2 For the first term at right-hand side of the above equality, due to ∆ div u = div ∆u, it holds that

HONG CAI AND ZHONG TAN
Then applying ∇ to (11) 1 , and multiplying the resulted equation by ∇ , one can get that Plugging it into the first estimate yields where we have used the fact that (P (ρ + τ ) − P (ρ)) ∼ τ . Hence, (14) follows from the above inequality immediately. This completes the proof of the lemma.
Lemma 3.6. Under the same conditions in Lemma 3.3, we have where C, d 0 , η 1 , η 2 , C η1 , C η2 are constants independent of L and . Moreover, d 0 can be chosen to be suitably large, η 1 , η 2 can be chosen to be arbitrarily small. And the constants C η1 , C η2 are depending on η 1 , η 2 respectively.
Proof. Applying ∂ t to (11) 1 − (11) 3 , and then multiplying the resultant equations by P (ρ) ρ t , u t , H t respectively, integrating by part over Ω L to have Now, applying the operator ∆ to (11) 1 and (11) 3 , then multiplying the resultant identities by P (ρ) ρ ∆ , ∆H respectively, it holds that Then multiply (11) 2 by ∇∆ div u and integrating it over Ω L lead tō Summing up the above two estimates, we obtain Note that ∇∆u 2 L 2 ≤ C( div ∆u 2 L 2 + curl∆u 2 L 2 ). We apply the operator curl to (11) Multiplying the above equation by curl∆u, then the integration over Ω L yields Multiplying (16) by a suitably large constant d 0 and combining it with (17), (18) yield (15). This completes the proof of the lemma. Now, we will use the equations (11) 1 , (11) 2 to give the uniform estimates for t , ∇ t , ∇ and ∆ . The proof is similar to Lemma 3.5 given in [14].
where C is a constant independent of L and . and where d 1 and C are constants independent of L and , and d 1 can be chosen to be suitably large.
Proof. Applying ∂ t to (11) 3 , and multiplying the resulting identity by ∆H t , we see that Then it follows from the above inequality that the estimate (23) holds.
On the other hand, multiplying (11) 2 by ∆u t and integrating by parts, we find (17) and (19), by choosing d 1 suitably large, yields the desired estimate (24). This completes the proof of the lemma.