Stability of axially-symmetric solutions to incompressible magnetohydrodynamics with no azimuthal velocity and with only azimuthal magnetic field

Incompressible viscous axially-symmetric magnetohydrodynamics is considered in a bounded axially-symmetric cylinder. Vanishing of the normal components, azimuthal components and also azimuthal components of rotation of the velocity and the magnetic field is assumed on the boundary. First, global existence of regular special solutions, such that the velocity is without the swirl but the magnetic field has only the swirl component, is proved. Next, the existence of global regular axially-symmetric solutions which are initially close to the special solutions and remain close to them for all time is proved. The result is shown under sufficiently small differences of the external forces. All considerations are performed step by step in time and are made by the energy method. In view of complicated calculations estimates are only derived so existence should follow from the Faedo-Galerkin method.


WOJCIECH M. ZAJA CZKOWSKI
Equations (1.1)- (1.4) are considered in a cylindrical domain Ω ⊂ R 3 with the axis of symmetry equal to the x 3 -axis. Let S be the boundary of Ω.
The boundary S is split into two parts, S = S 1 ∪ S 2 , where S 1 is parallel to the x 3 -axis and S 2 is perpendicular to it. We have that S 2 = S 2 (−a) ∪ S 2 (a), where a > 0 is given and S 2 (b) meets the x 3 -axis at x 3 = b, b ∈ {−a, a}.
Since There is a huge literature concerning the problem of global existence of regular solutions to incompressible viscous mhd. The first result on local existence, uniqueness and global existence for small data was established by Duvaut and Lions [4]. Moreover, we recall the paper of Sermange and Temam [12], where global existence of regular two-dimensional solutions is proved. Finally, we recall results on regularity of solutions to incompressible mhd equations. The two-dimensional case is treated in [2,7,19] and the three-dimensional case in [3,5,6,10].
Recently there appear papers concerning non-resistivity incompressible mhd. In [13] global small solutions to three-dimensional mhd system is proved. In [11] global small solutions of 2-D incompressible mhd is shown. Moreover, Z. Lei in [9] proved existence of long time solutions to incompressible axially-symmetric mhd system with zero diffusivity in the equation for the magnetic field. The received result is weak because the estimate increases very strongly to infinity as time passes to infinity. Probably, this feature is relevant to the Cauchy problem only.
In this paper we are interested to prove existence of incompressible axiallysymmetric mhd with the following properties: 1. We are going to show existence of global regular solutions which are estimated by a bound independent of time. This gives a possibility of showing stability, periodicity and finally existence of stationary solutions. 2. We want to show existence of global non-small solutions. This can be realized by showing existence of global regular large special solutions and subsequently proving existence of solutions which remain close to these special solutions for all time. 3. Finally, we want to consider mhd system with an external force that is nonvanishing and nondecreasing in time. Looking at the results of Lei [9] we see that to obtain the above aims we have to consider viscous and resistivity mhd. Moreover, we think that the mhd motions should be considered in a bounded axially-symmetric domain with appropriate boundary conditions. We need such boundary conditions so that the interior motion is separated from any exterior influence. The proposed boundary conditions are a little more restricted than the slip boundary conditions (see [18]). Hence much stronger restrictions are imposed on the angular component of velocity because the slip boundary conditions imply only that v ϕ,r = 1 R v ϕ on S 1 , v ϕ,z = 0 on S 2 (see [18,Ch. 4, Sect. 2, Lemma 2.1]) but we assume that v ϕ | S = 0. For the non-swirl Navier-Stokes system (v ϕ = 0) the considered boundary conditions are the same as in celebrated paper of O.A. Ladyzhenskaya [8]. The imposed boundary conditions for the magnetic field have a deep physical meaning. Vanishing of the normal component of the magnetic field H on the boundary and assumption B = µH means that the flux of the magnetic induction B through the boundary equals zero. Hence the lines of the magnetic induction are parallel to the boundary so the considered motion is separated from the exterior region.
In view of (1.13) 2 , vanishing of the angular component of the magnetic field H implies that current J| S is parallel to S. Hence there is no flux of charges through the boundary. Finally, we assume vanishing angular component of curl of H so that the angular component of current vanishes on S (see (1.13) 2 ).
Since the Navier-Stokes and magnetohydrodynamics equations have special regular global solutions it is natural to examine their stability. Thanks to this we are also able to prove existence of global regular solutions which are close in some spaces to the special solutions for all time. To prove stability we transform the considered equations to linear equations for perturbations of considered quantities with regular coefficients depending on the special solutions. Then it is possible to derive a differential inequality guaranteeing non-increasing in time of sufficiently small initial perturbations (see (4.2)). A vast literature concerning stability of special solutions to the Navier-Stokes equations is presented in [14,17]. We have to emphasize that derivation of the differential inequality (4.2) for magnetohydrodynamics equations is far from trivial. The technique presented in this paper was already developed in [16,17,14,15]. This paper is organized in the following way. In Introduction axially-symmetric magnetohydrodynamics equations are formulated (see (1.26)-(1.32)). The question of existence of solutions is presented in Remark 4. Separately equations for special solutions such that velocity is without swirl and magnetic field has only swirl component are presented (see (1.34)-(1.38)).
Moreover, the equations for perturbations are described by (1.39)-(1.48). Finally, the main results are presented at the end of Introduction. In Section 3 we prove existence of global regular special solutions (see Lemmas 3.1-3.10) and Theorem 3.11. The differential inequality necessary for showing stability is derived in Proposition 1. Then, Proposition 2 proves stability.
The aim of this paper is to prove stability of axially-symmetric solutions with non-swirl velocity and swirl magnetic field (we denote them as special solutions) in a set of general axially-symmetric solutions. Moreover, we have to prove global existence of regular non-swirl velocity and swirl magnetic field of axially-symmetric solutions bounded by constants independent of time. To examine axially-symmetric solutions we introduce the cylindrical coordinates r, ϕ, z by the relations (1.10) Next, we use the or-tho-normal basis e r = (cos ϕ, sin ϕ, 0),ē ϕ = (− sin ϕ, cos ϕ, 0),ē z = (0, 0, 1). (1.11) Then the cylindrical coordinates of v, ω = rot v, H, j = rot H, f are defined by and curl and current density have the forms Let us recall that the swirls of velocity and the magnetic field are denoted by (1.14) Finally, using the cylindrical coordinates, we describe domain Ω. Let R > 0 and a > 0 be given. Then Then the equations describing axially-symmetric incompressible viscous magnetohydrodynamics have the form where Therefore, the axially-symmetric magnetohydrodynamics motions are described by the following system of equations To prove stability of the special solutions we have to formulate equations for general and for special axially-symmetric solutions, separately. The first will be denoted by the upper index 1 and the second by 2. Since we are going to work in a class of smooth solutions we recall the following compatibility conditions at r = 0 for axially-symmetric mhd equations v ϕ (0, z, t) = ω ϕ (0, z, t) = ψ(0, z, t) = 0, It is natural for axially-symmetric motions to replace equations (1.19)-(1.24) for v ϕ , H ϕ , ω ϕ , j ϕ , ψ, φ by equations for v ϕ /r, H ϕ /r, ω ϕ /r, j ϕ /r, ψ/r, φ/r. Therefore we introduce the quantities with the lower index 1 as follows v ϕ (r, z, t) = rv 1 (r, z, t), ω ϕ (r, z, t) = rω 1 (r, z, t), ψ(r, z, t) = rψ 1 (r, z, t), H ϕ (r, z, t) = rH 1 (r, z, t), j ϕ (r, z, t) = rj 1 (r, z, t), φ(r, z, t) = rφ 1 (r, z, t). (1.26) Then the equations for general axially-symmetric mhd equations have the form Hz,r) H are reexpressed as We assume the following special solutions to axially-symmetric mhd equations Then the special solutions satisfy the system of equations To prove stability of the special solutions we introduce the notation

H1
, (1.39) The above quantities describe a distance between general and special axially-symmetric solutions. Finally, we formulate the equations for the differences (1.43) Moreover, we assume the following boundary conditions for solutions to (1.40)-(1.43), v 1 | S = 0, H 1 | S = 0, ω 1 | S = 0, j 1 | S = 0 (1.44) and also the initial conditions (1.45) Finally, coordinates of v and H are calculated from the relations and ψ 1 , φ 1 are solutions to the elliptic problems Now, we formulate the main results of this paper. From Section 3 we have Then the following estimates for solutions to problem (1.34)-(1.38) hold: for t ∈ [kT, (k + 1)T ] and any k ∈ N 0 = N ∪ {0}. The aim of this paper is to prove stability of special solutions described by problem (1.34)-(1.38) in a set of axially-symmetric solutions. Therefore, to prove stability we are looking for solutions to problem (1.39)-(1.48) which describe disturbances to the special solutions. The stability means existence of small solutions to problem (1.39)-(1.48) for all time with data sufficiently small. We proved stability in Section 4, where appropriate estimates are derived. Hence, Propositions 1 and 2 imply Let γ * > 0 be so small that Assume that Then Remark 2. Since the calculations in this paper are very complicated we restrict our considerations to derive estimates only. All estimates are proved by the energy method. This means that existence should follow from the Faedo-Galerkin method.
, Ω ⊂ R n we denote the Lebesgue space of integrable functions and by H s (Ω), s ∈ N 0 , Ω ⊂ R n , the Sobolev space of functions with the finite norm we denote a space of functions with the following finite norm Under similar proof to the proof of Lemma 3.3 we have Then there exists a solution to problem (1.48) such that φ 1 ∈ H 2 (Ω) and Let us consider the elliptic problem (1.24). Assume that j ϕ ∈ H 1 (Ω). Then there exists a solution to problem (1.24) such that φ ∈ H 3 (Ω) and 3. Special solutions. In this Section we derive estimates for special solutions and prove global in time existence of regular special solutions. For this purpose we consider system (1.35)-(1.38) with appropriate boundary and initial conditions. To simplify presentation we skip the upper index 2. Therefore, the special solutions satisfy with the boundary conditions and the initial conditions Then solutions to (3.2), (3.5 2,4 ) satisfy Proof. Multiplying (3.2) by H 1 |H 1 | σ−2 , integrating over Ω, using the boundary conditions (3.5) 2,4 , we obtain The Poincaré inequality yields Hence, inequality (3.11) implies Integrating (3.12) with respect to time from kT to t ∈ (kT, (k + 1)T ], gives Setting t = (k + 1)T yields (3.14) From iteration we have Integrating (3.10) with respect to time from kT to t ∈ (kT, (k + 1)T ] and using (3.15) imply (3.8). This concludes the proof. and Then where t ∈ (kT, (k + 1)T ], k ∈ N 0 , c * p = min{1, c p } and c 0 appears below (3.21). Proof. Multiplying (3.1) by ω 1 , integrating over Ω, integrating by parts and using the boundary conditions (3.5) 1 yield Applying the Hölder and the Young inequalities to the first term on the r.h.s. of (3.18) implies We apply the Poincaré inequality to the second term on the l.h.s. and set c * p = 2 cp+1 . Then (3.19) takes the form Estimating the last term on the r.h.s. gives (3.23) Integrating (3.23) with respect to time from t = kT to t ∈ (kT, (k + 1)T ) yields (3.24) Simplifying we get (3.25) Setting t = (k + 1)T we get (3.26) Using (3.15) yields (3.27) Hence, iteration implies Integrating (3.29) with respect to time from t = kT to t ∈ (kT, (k + 1)T ], k ∈ N 0 , and using (3.8) we obtain (3.17). This concludes the proof.
Let H 1,r + 2µH 1,rφ where the dot denotes derivative with respect to r. Differentiating (3.35) with respect to r gives 1,r | ∂(Ω∩supp ϕ (1) ) = 0. Then we get ,r H 1,r dx. (3.37) Now, we examine the particular terms in (3.37). The third term on the l.h.s. of (3.37) can be expressed in the form where v r = −(rψ 1 ) ,z , v z = rψ 1,r +2ψ 1 , so v r,r = −rψ 1,rz −ψ 1,z , v z,r = rψ 1,rr +3ψ 1,r . Applying the Hölder inequality and interpolation yields 1,r r) ,r drdz which is estimated by where it is used that suppφ (1) is located in a positive distance from the axis of symmetry. The last term in I 2 is bounded by L2(Ω) . The fifth term on the l.h.s. of (3.37) equals 1,zz dx.
We can integrate by parts with respect to z in the fifth term on the l.h.s. of (3.39) using that v · ∇H 1 | S2 = 0 and v z | S2 = 0. Therefore, the term takes the form where the first integral vanishes and the second is bounded by . The sixth term on the l.h.s. of (3.39) is bounded by   To examine (3.2) in a neighborhood located in a positive distance from the axis of symmetry we multiply (3.2) by ϕ (2) and obtain the equation 1 , integrating over Ω and using the boundary condition H (3.43) The first term on the r.h.s. is bounded by Using that the problem 1 , H 1 | ∂(Ω∩supp ϕ (2) ) = 0 implies the estimate we obtain from (3.43) for sufficiently small ε the inequality Integrating (3.45) with respect to time from t = kT to t ∈ (kT, (k + 1)T ] we derive (3.33). This concludes the proof.
In Lemma 3.5 quantity H 1 (kT ) H 1 (Ω) is not estimated in terms of data. To make it we need. Lemma 3.6. Let k ∈ N 0 . Assume that where H 1 is estimated in (3.8) and ω 1 in (3.17). Assume that H 1 (0) ∈ H 1 (Ω). Then the following estimate holds Proof. Expresing (3.45) in the form Integrating (3.47) with respect to time from t = kT to t = (k + 1)T and using definition of B 1 we get and (3.50) For given F 2 and F 3 we prove existence of solutions to problems (3.49) and (3.50) in the norms of ω 1 and H 1 introduced in Lemmas 3.2 and 3.5 by the Faedo-Galerkin method. Then by the method of successive approximations and [16] we can prove existence of solutions to problem (3.1)-(3.6) in the normed appeared in Lemmas 3.1-3.6. Now we describe the statement more precisely.
To prove existence of solutions to problem (3.1)-(3.6) we use the following method of successive approximations. Let ψ 1n be given. Then v n = v rnēr + v znēz = −(rψ 1n ) ,zēr + 1 r (r 2 ψ 1n ) ,rēz . From this construction we have that div v n = 0. Then we calculate and finally the second step of iteration is described by (3.53) Consider problem (3.51).
Next we show convergence. For this we introduce the differences Applying the Hölder and Young inequalities to the term on the r.h.s. yields   The first term on the r.h.s. of (3.83) is treated in the following way Integrating the inequality with respect to time implies where γ < 1 for h sufficiently small. Inequality (3.88) implies convergence. This concludes the proof.

Stability of the special solutions.
To prove stability of the special solutions (3.1)-(3.6) we first derive a differential inequality for solutions to problem (1.40)-(1.48). Hence we introduce the quantities (4.1) Proposition 1. Assume that all quantites in (4.1) are finite. Then there exists a constant c 0 such that d dt where ν * = min{ν, µ}.
Lemma 4.1. Assume that the following quantities are finite . Then the following inequality holds Proof. Multiplying (1.40) by v 1 |v 1 | 2 , integrating the result over Ω and using the boundary conditions (1.44) 1 , we obtain Integrating by parts and using the boundary conditions (1.44) 1 the first term on the r.h.s. of (4.3) equals Hence, In view of boundary conditions (1.44) 1 and integration by parts the second term on the r.h.s. of (4.3) takes the form Hence where the second integral is bounded by The third term on the r.h.s. of (4.3) is bounded by Applying the Hölder inequality the last term on the r.h.s. of (4.3) is bounded by

WOJCIECH M. ZAJA CZKOWSKI
Employing the above estimates in (4.3) and using that ε = ν we get   Now we obtain an analogous estimate for H 1 . Multiplying (1.42) by H 1 |H 1 | 2 , integrating over Ω and by parts applying the boundary conditions (1.44) 2 , we derive The first term on the r.h.s. of (4.10) is bounded by (Ω) , where (4.8) for v is used. Integrating by parts in the second term on the r.h.s. of (4.10) yields where the second integral is bounded by where (4.8) was used. Using the above estimates in (4.10) and assuming that ε is sufficiently small we get (4.11) Adding (4.9) and (4.11), using that ε is sufficiently small and employing notation from the assumptions of the lemma we have This proves Lemma 4.1.
Next, we shall obtain an inequality with X 2 2 under the time derivative. Lemma 4.2. Assume that the following quantities are finite . Then the following differential inequality holds (4.14) Integrating by parts and using the boundary conditions (1.44) 3 in the first term on the r.h.s. of (4.14), it equals Integrating by parts in the second term on the r.h.s. of (4.14) we have Similarly, integrating by parts in the third term on the r.h.s. of (4.14) yields − Ω H · ∇ω 1 j 1 dx which is bounded by The last but one term on the r.h.s. of (4.14) equals H1ω1,zdx which is estimated by Finally, the last term on the r.h.s. of (4.14) is estimated by Employing the above estimates in (4.14) and using that ε is sufficiently small yields (Ω) , where (1.46) and estimate (4.8) 2 for solutions to (1.47) are used. The last but one term on the r.h.s. of (4.15) is bounded by and the third term on the r.h.s. of (4.15) by Employing the above estimates in (4.15) we derive the inequality  Since on the r.h.s. of (4.16) the norm H 1 2

L2
(Ω) appears we need additional differential inequality. For this purpose we multiply (1.42) by H 1 and integrate the result over Ω. Then we get Applying the Hölder and Young inequalities to the r.h.s. of the above inequality gives Multiplying (1.40) by v 1 and integrating over Ω yields H1v1dx Now, we estimate the particular terms from the r.h.s. of (4.18). The first two terms are bounded by The third and fourth terms are bounded by The fifth and sixth terms are estimated by Finally, the last term is bounded by Employing the above estimates in (4.18) and using that ε is sufficiently small we get  (4.20) In view of assumptions of this lemma inequality (4.20) takes the form of (4.13). This concludes the proof.
From (4.16) and the above inequality we obtain (4.13). This concludes the proof of Lemma 4.2.
Lemma 4.3. Assume that the following quantities are finite Now, we estimate the particular terms from the r.h.s. of (4.22). The first term on the r.h.s. of (4.22) can be expressed in the form Next we estimate the fourth term on the r.h.s. of (4.22). For simplicity, we first consider the following part of this term (v r,r − v z,z )(H r,z + H z,r )j 1 dx.
We estimate the first term in K 1 in a similar way as the first term in J 1 was estimated. Denoting it by K 3 , we have Finally, the last term in K 1 can be expressed in the form