On global regular solutions to magnetohydrodynamics in axi-symmetric domains

We consider mhd equations in three-dimensional axially symmetric domains under the Navier boundary conditions for both velocity and magnetic fields. We prove the existence of global, regular axi-symmetric solutions and examine their stability in the class of general solutions to the mhd system. As a consequence, we show the existence of global, regular solutions to the mhd system which are close in suitable norms to axi-symmetric solutions.

Our main goal (cf. Theorem 3) is to prove the existence of global, regular solutions to problem (1.1)-(1.4) without any assumptions on smallness of the initial and the external data, however, fulfilling certain geometrical constraints which we will describe in the subsequent paragraphs. Our proof is based on stability reasoning: we construct a special solution and examine its stability in the class of solutions to (1.1). As a by-product, we obtain a solution (1.1). This method has been recently utilized in e.g. [1,2].
The first step of our work is a construction of a special, axially symmetric solution (v s , H s ). By axially symmetric, we mean is global and regular, that is we prove the following theorem: Then, there exists a weak solution to (1.10) and a constant A 3 (see Lemma 3.1) such that then there exists an axially symmetric solution to problem (1.10) and a constant A 6 (see Lemma 3.2), which depends on the initial and external data but neither on T and k, such that The above theorem would hold even if we take different boundary conditions for v s and H s (for a discussion about possible choices, we refer the reader to [3,4] and the references therein). The crucial point is that (1.10) is constructed through revolving Ω ϕ0 around x 3 -axis [see (1.6)], thus any solution to 2d problem in a bounded, smooth domain (for such solutions see e.g. [5,6]), which is separated from the axis of rotation, would be a good candidate for a special solution (v s , H s ). This idea seems to work for even more general MHD models (fractional diffusion, partial diffusion, etc.), which have been studied e.g. in [7][8][9][10][11][12][13][14]. However, there are two conditions that must be met first: (a) the domain cannot contain the axis of rotation (in cited papers the whole space is considered), (b) the global estimates cannot depend on time. Although in standard approach the energy estimates do not depend on time, yet they enforce exponential decay of the external force. The method we use does not lead to exponential decay of the external data (for similar ideas see e.g. [15][16][17][18]).
In the second step, we investigate the stability of solutions to (1.10) in the class of strong solutions to (1.1). To this aim, we introduce Then the pair (u, K) satisfies n × rot u = 0, n · u = 0 on S kT := S × (kT, (k + 1)T ), (1.11) This time, we no longer require (1.8); therefore, we expect that for the small initial and external data u(0), K(0) and g the solution (u, K) will remain small. This would imply that for v(0), H(0) and f being close in suitable norms to v s (0), H s (0) and f s , respectively, there exists a global, unique, regular solution to (1. and rot u(0), rot K(0) 2 L2(Ω) ≤ γ, where γ is sufficiently small number, then there exists a unique solution (u, K) to (1.11) such that u, K ∈ V 1 2 (Ω kT ) [see (2.2)], k = 0, 1, 2, . . . and there are two constants B 4 and B 5 (see Lemma 4.2) such that u, K In the last step, we prove the existence of global, strong solutions to (1.  where k = 0, 1, 2, . . ..
For a brief description of past results concerning the regularity and existence of weak solutions, we refer the reader to the introduction in [1].

Auxiliary facts
From now on, we write N 0 = N ∪ {0} and Ω kT = Ω × (kT, (k + 1)T ). We will also frequently use −Δ = rot rot −∇ div, which suggests the following "integration by parts" formula All constants are generic (i.e. they may vary from line to line) and are denoted by c. Additionally, if a constant depends on the domain (e.g. in embedding inequalities), we write c(Ω).
Below, we introduce functions spaces and recall some technical lemmas.

Function spaces
, p > 1, we denote the Sobolev spaces equipped with the following norms Let us now fix ϕ 0 ∈ [0, 2π] and define ∇ = (∂ r , ∂ z ). Since the distance between Ω ϕ0 (cf. (1.6)) and the axis of symmetry of Ω is always positive, we may write We also note that for ψ ∈ {v s , H s }, we have where X is either a Lebesgue or a Sobolev space. This inequality follows immediately from the definition of v s , H s and the geometry of Ω. More importantly, it justifies that whenever we use an embedding inequality for ψ we may take n = 2.
For function spaces defined above the following embedding will turn very useful. Namely, if u ∈ V 1 2 (Ω kT ), then u ∈ L 10 (Ω kT ) (see [22,Lemma 3.7]) and We will also use the interpolation inequality

Auxiliary results
Below, we gather crucial tools for establishing a-priori estimates for the solutions to problems (1.1), (1.10) and (1.11). Then For the proof of the above Lemma, we refer the reader to Lemma 2.1 and problem (2.7) in [6].  If F ∈ L s (Ω T ), where 1 < s < ∞, then there exists a unique solution such that v ∈ W 2,1 s (Ω T ) and . Lemma 2.4. (cf. Lemma 3.14 in [20]) Consider the following initial-boundary value problem and

The existence and properties of solutions to (1.10)
Using energy methods, we prove a priori estimates for a solution (v s , H s ) to (1.10). Therefore, the existence of the solution will follow from the Faedo-Galerkin method. We start with the basic global energy estimate.
Proof. Multiplying (1.10) 1,3 by v s and H s , respectively, integrating over Ω, using (1.10) 2,4 and the boundary conditions (1.10) 5,6 , we obtain Integrating the above inequality with respect to time from t = kT to t ∈ (kT, (k Iterating the above procedure, we get which proves (3.2) 1 . To conclude (3.2) 2 , we integrate (3.3) with respect to time and use the above inequality. This ends the proof.
In the below lemma, we establish higher-order estimates for weak solutions to (1.10).

Lemma 3.2. Let the assumptions of Lemma 3.1 hold. Assume that T > 0 is so large that
Then
Proof. We begin with multiplying (1.10) 1,3 by rot 2 v s and rot 2 H s , respectively, integrating the result over Ω and using the boundary conditions (1.10) 5,6 Utilizing the Hölder and Young inequalities and Lemma 2.5, we get Using Lemma 2.2 and taking 1 , . . ., 6 sufficiently small, we obtain In light of Lemma 2.1, we have Integrating with respect to time from t = kT to t ∈ (kT ; (k + 1)T ) yields rot vs(t), rot Hs(t) 2 Setting t = (k + 1)T and using that T ≥ Iterating the above procedure yields which proves (3.4) 1 .
To prove (3.4) 2 , we integrate (3.5) with respect to time from t = kT to t ∈ (kT, (k + 1)T ] and use Lemma 2.1. Then This completes the proof.
Using the Hölder inequality, we get By (2.6), we have The above estimate with the estimates from Lemmas 3.1, 3.2 along with Lemma 2.1 ends this remark.
Proof of Theorem 1. The proof is straightforward and follows from the energy estimates (see Lemmas 3.1, 3.2) and the Galerkin method. As the basis functions, we can take the eigenfunctions of the Laplacian equipped with the Navier boundary conditions (cf. Sections 2.3 and 3.2 in [6] and Section 3 in [23]).

Stability of solutions to (1.10)
In this section, we examine the stability of solutions to (1.10) Proof. We multiply (1.11) 1,3 by u and K, respectively, integrate the result over Ω and use the boundary conditions (1.11) 5,6 We easily note that J 1 , J 3 , J 8 , J 10 vanish. We also have J 4 = −J 11  (Ω) Choosing 2 , 5 , 7 , 9 and 12 sufficiently small and using Lemma 2.1 to estimate L 6 -norms, we get Applying Lemma 2.1 for the second term on the left-hand side, we obtain Integration with respect to time from t = kT to t ∈ (kT, (k + 1)T ) yields By the Hölder inequality, Sobolev embedding and Lemmas 2.2 and 3.2, we have We take t = (k + 1)T and use that T ≥ 2c(Ω) Iterating the above inequality, we get which proves (4.1) 1 . Next, we integrate (4.2) with respect to time from t = kT to t = (k + 1)T which proves (4.1) 2 . Using the above inequality in (4.4) ends the proof.
For I 2 , we have . Finally, for terms in I 3 , we have For I 32 , we repeat the estimate we derived for I 12 The term I 33 can be estimated in the same way as I 13 For I 34 , we have I34 ≤ rot 2 K L 2 (Ω) ∇u L 2 (Ω) K L∞(Ω) ≤ c(Ω) rot 2 K L 2 (Ω) rot u L 2 (Ω) K , Ω rot u 6 L 2 (Ω) + rot K 6 L 2 (Ω) .
The last two terms in I 3 are very similar to I 12 and I 16 , respectively, thus By the Hölder and Young inequalities, we obtain d dt U, N

Proof of Theorem 3
The proof follows immediately form Lemmas 3.2, 4.2 and Theorem 2.