Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the non-collinearity condition

We consider a free boundary problem for the axially symmetric incompressible ideal magnetohydrodynamic equations that describes the motion of the plasma in vacuum. Both the plasma magnetic field and vacuum magnetic field are tangent along the plasma-vacuum interface. Moreover, the vacuum magnetic field is composed in a non-simply connected domain and hence is non-trivial. Under the non-collinearity condition on the free surface, we prove the local well-posedness of the problem in Sobolev spaces.

1. Introduction 1.1. Eulerian formulation. In this paper, we consider the free boundary problem of the axially symmetric incompressible ideal MHD equations: in Ω(t), in Ω(t), in Ω(t), ∂ t B z + (u r ∂ r + u z ∂ z )B z = (B r ∂ r + B z ∂ z )u z in Ω(t), ∂ r u r + u r r + ∂ z u z = 0 in Ω(t), in Ω(t).
(1.1) In the equation (1.1), u(t, x) = u r (t, r, z)e r + u θ (t, r, z)e θ + u z (t, r, z)e z is the Eulerian or spatial velocity field, B = B r (t, r, z)e r + B θ (t, r, z)e θ + B z (t, r, z)e z is the magnet field, and P denotes the pressure function of the fluid which occupies the moving vessel domain: Ω(t) : {(x 1 , x 2 , z)|0 ≤ r < r(z, t), z ∈ T}.
The system (1.1)-(1.3) can be used to describe the motion of the plasma confined inside a rigid wall and isolated from it by vacuum, which is one of laboratory plasma model problems (see [12,Chapter 4.6.1]). In the general setting, the plasma region Ω(t) is surrounded by the vacuum region Ω v (t), and the moving plasma-vacuum interface Γ(t) = ∂Ω(t) does not intersect with the outer wall ∂Ω w , where Ω w = Ω(t) ∪ Γ(t) ∪ Ω v (t) is a fixed domain. When the characteristic plasma velocity is very small compared to the speed of sound, the motion of the plasma is governed by the incompressible ideal MHD in Ω(t), i.e., (1.1). In the vacuum region Ω v (t), we neglect the displacement current in the Maxwell equations as usual in the non-relativistic MHD and assume the pre-Maxwell equations: (1.5) In the equations (1.5), B and E denotes the magnetic and electric fields in vacuum, respectively. The motion of the plasma is connected with the vacuum through the jump condition on the interface Γ(t): (1.6) Here E is the electric field of the plasma, i.e., (1.7) We also require the jump condition on ∂Ω w : HereB andÊ denotes the magnetic and electric fields outside the wall ∂Ω w and ν is the outward unit normal of ∂Ω w . The well-posedness of the general plasma-vacuum interface problem is still an open question. In this paper we restrict to an axially symmetric case and Ω w is a cylinder domain: {(x 1 , x 2 , z)|0 ≤ r < R S , z ∈ T}, ∂Ω w := {(x 1 , x 2 , z)|r = R S , z ∈ T} is a perfectly conducting wall and when the plasma is a perfect conductor. In this setting, we have the following boundary conditions B · ν = 0, E × ν = 0, on ∂Ω w (1.8) and B · n = B · n = 0 on Γ(t).
(1.9) Now with the first equation of (1.8) and the first equation of (1.9), we can derive a formula for the vacuum magnet filed B = B r (r, z, t)e r + B θ (r, z, t)e θ + B z (r, z, t)e z . In fact, we can transfer the first equation of (1.5) and the first equation of (1.8) into the two decoupled system: with no boundary condition and    ∂ r B r + 1 r B r + ∂ z B z = 0, with boundary condition B r + B z ∂ z r = 0, on Γ(t) : r = r(z, t), B r = 0, on r = R S .
Then we see B r = B z = 0, B θ = C(t) r (1.10) are solutions to the above two systems. In order to determine C(t), we consider the elliptic system of the vacuum electronic field: in Ω v (t), The third equation is obtained by using the first equation of (1.6) and the equation (1.7). Now by using integration by parts, we have where we used ∇ × B = 0 in Ω v (t) and ν × E = 0 on ∂Ω w . Thus, we have Hence the plasma-vacuum interface problem reduces to the free boundary problem (1.1)-(1.3). Our purpose of this paper is to establishing the local well-posedness for this problem.
Remark 1.1. If we set C(0) = 0 in the formula (1.4), then we have B = 0 for all time. With this trivial vacuum magnet field, the local well-posedness was proved in [11] without axially symmetric assumption. In fact, if the vacuum domain Ω v is simply connected, we only have trivial magnet field. It is interesting to study the non-trivial vacuum magnet field and its interaction with plasma magnet field, this is also the main reason for us to consider the problem with a nonsimply-connected vacuum domain in this paper.

Lagrangian reformulation.
We tranform the Eulerian problem (1.1)-(1.3) on the moving domain Ω(t) to be one on the fixed domain Ω by the use of Lagrangian coordinates. Let x ∈ Ω be the Lagrangian coordinate and η(x, t) be the Eulerian coordinate, which means η(x, t) ∈ Ω(t) denote the "position" of the fluid particle x at t. Thus, (1.11) and R(x, 0) = r, Θ(x, 0) = θ, Z(x, 0) = z.
(1.12) From the first and third equation of (1.11) and the initial data (1.12), we have R(x, t) = R(r, z, t), Z(x, t) = Z(r, z, t). (1.13) And for Θ, we have (1.14) (1.15) and denote the deformation tensor between (R, Z) and (r, z) as F ij = ∂ a j ζ i (r, z, t), where ζ = (R, Z), a = (r, z), (e.g. F 11 = ∂ r R). Then we have the following Lagrangian version of (1.1) in the fixed reference domain Ω: in Ω, Two dynamic boundary conditions become: where Γ := {(x 1 , x 2 , z)|(x 1 ) 2 + (x 2 ) 2 = R 0 , z ∈ T}, N = (1, 0) and Now we follow the idea used in [11] to transfer the system (1.16) to a free-surface incompressible Euler system with a forcing term induced by the flow map. In fact, it can be checked directly that With this equality, it can be checked that if ∂ r (rb r 0 ) + ∂ z (rb z 0 ) = 0 in Ω and b r 0 = 0 on Γ, then the condition ∂ A R (Rb r ) + ∂ A Z (Rb z ) = 0 and the second condition of (1.2) are satiesfied naturally. Then we plug (1.18) into the equation for b θ , we have and hence in Ω, in Ω, (1.20) with boundary condition: In the system (1.20), the initial magnet field b 0 can be regarded as a parameter vector that satisfies 1.3. Previous works. Free boundary problems in fluid mechanics have important physical background and have been studied intensively in the mathematical community. There are a huge amount of mathematical works, and we only mention briefly some of them below that are closely related to the present work, that is, those of the incompressible Euler equations and the related ideal MHD models. For the incompressible Euler equations, the early works were focused on the irrotational fluids, which began with the pioneering work of Nalimov [20] of the local well-posedness for the small initial data and was generalized to the general initial data by the breakthrough of Wu [30,31] (see also Lannes [16]). For the irrotational inviscid fluids, certain dispersive effects can be used to establish the global well-posedness for the small initial data; we refer to Wu [32,33], Germain, Masmoudi and Shatah [9,10], Ionescu and Pusateri [14,15] and Alazard and Delort [1]. For the general incompressible Euler equations, the first local well-posedness in 3D was obtained by Lindblad [17] for the case without surface tension (see Christodoulou and Lindblad [4] for the a priori estimates) and by Coutand and Shkoller [6] for the case with (and without) surface tension. We also refer to the results of Shatah and Zeng [23] and Zhang and Zhang [34]. Recently, the well-posedness in conormal Sobolev spaces can be found by the the inviscid limit of the free-surface incompressible Navier-Stokes equations, see Masmoudi and Rousset [18] and Wang and Xin [29].
The study of free boundary problems for the ideal MHD models seems far from being complete; it attracts many research interests, but up to now only few well-posedness theory for the nonlinear problem could be found. For the plasma-vacuum interface model that a surface current J is added as an outer force term to the vacuum pre-Maxwell system (1.5), with the non-collinearity condition holding for two magnet fields on the boundary: the well-posedness of the nonlinear compressible problem was proved in Secchi and Trakhinin [22] by the Nash-Moser iteration based on the previous results on the linearized problem [28,21]. The well-posedness of the linearized incompressible problem was proved by Morando, Trakhinin and Trebeschi [19], the nonlinear incompressible problem was solved by Sun, Wang and Zhang [25] very recently. On the other hand, Hao and Luo [13] established a priori estimates for the incompressible plasma-vacuum interface problem under the Taylors sign condition: (1.24) under the assumption that the strength of the magnetic field is constant on the free surface by adopting a geometrical point of view [4]. Recently, Gu and Wang proved the well-posedness of the incompressible plasma-vacuum problem under (1.24) with the vacuum magnet field is zero and the well-posedness of the axially symmertic ideal MHD equation (1.1) under (1.24) will be addressed in the forth coming paper. However, without axially symmetric assumption, the well-posedness of the plasma-vacuum interface problem under (1.24) is still unknown when the vacuum magnetic field B is non trivial. Finally, we also mention some works about the current-vortex sheet problem, which describes a velocity and magnet field discontinuity in two ideal MHD flows. The nonlinear stability of compressible current-vortex sheets was solved independently by Chen and Wang [3] and Trakinin [27] by using the Nash-Moser iteration. For incompressible current-vortex sheets, Coulombel, Morando, Secchi and Trebeschi [5] proved an a priori estimate for the nonlinear problem under a strong stability condition, and Sun, Wang and Zhang [24] solved the nonlinear stability.

Main Result
Before stating our results of this paper, we may refer the readers to our notations and conveniences in Section 3.1.
We define the higher order energy functional Then the main result in this paper is stated as follows.
holds initially. Then there exists a T 0 > 0 and a unique solution where P is a generic polynomial.

2.1.
Strategy of the proof. The strategy of proving the local well-posedness for the inviscid free boundary problems consists of three main parts: the a priori estimates in certain energy functional spaces, a suitable approximate problem which is asymptotically consistent with the a priori estimates, and the construction of solutions to the approximate problem. For the incompressible MHD equations (1.20), we derive our a priori estimates in the following way. First, we divide (1.20) into two sub-systems: one is for (v r , v z , q, R, Z), the other one is for (v θ , Θ) (see (4.1) and (4.2)). The a priori estimates for (v θ , Θ) can be obtained by standard energy method. This is because there is no pressure in this subsystem and no boundary integral needs to be considered. Here, one will meet the difficulty to deal with the singularity brought by the cylinder coordinates, i.e. the estimates of v θ r . Hence, we will apply the high order Hardy inequality to control these terms. On the other hand, the estimates for (v r , v z , q, R, Z) is more complicated. We shall use tangential energy estimates combining with divergence and curl estimates to close the a priori estimates of (ν, q, ζ), where we denote ν = (v r , v z ), ζ = (R, Z).
During this process, there are several difficulties to deal with. In the usual derivation of the a priori tangential energy estimates of (4.1) in the H 4 r,z setting, one deduces 1 2 The first difficulty one will meet is the loss of derivatives in estimating R Q (by recalling the energy functional E(t) defined by (2.1)). Our idea to overcome this difficulty is, motivated by [18,29,11], to use Alinhac's good unknowns which derives a crucial cancellation observed by Alinhac [2], i.e., when considering the equations for V and Q, the term R Q disappears. The second difficulty is to estimate the boundary integral I b . Recalling the boundary condition for q, one needs to control ∂ 4 z ζ 1/2 , which means a loss of derivatives again. To overcome this difficulty, we use the following important observation: with the non-collinearity condition (2.2) and boundary condition (1.22), one can have This means the non-collinearity condition (2.2) can actually improve one order boundary regularity, which plays a big role here. Hence, the boundary integral I b now can be estimated by using (H −1/2 , H 1/2 ) dual estimate and the tangential energy estimates can be finished. Doing the divergence and curl estimates is somehow standard and combining with the tangential energy estimates, we can close the a priori estimates.
After we obtaining the a priori estimates, we use linearization method to construct approximate system to (1.20). Again, we have two linearized sub-systems: (4.3) and (4.4). Thanks to the boundary smoothing effect of non-collinearity condition (2.2), we can avoid losing derivatives on the boundary estimates in the linearization iteration, which means the approximate system is asymptotically consistent with the a priori estimates for the original system. Then by a contraction argument, the solutions to (1.20) can be obtained based on the approximated solutions to the linearized system. What now remains in the proof of the local well-posedness of (1.20) is to constructing solutions to the linearized approximate problem (4.3) and (4.4). This solvability can be obtained by the viscosity vanishing method used in [11,Section 5.1]. Consequently, the construction of solutions to the incompressible MHD equations (1.20) is completed.

3.1.
Notation. Einstein's summation convention is used throughout the paper, and repeated Latin indices i, j, etc., are summed from 1 to 2. We use C to denote generic constants, which only depends on the domain Ω and the boundary Γ, and use f g to denote f ≤ Cg. We use P to denote a generic polynomial function of its arguments, and the polynomial coefficients are generic constants C. We use D to denote the spatial derives: ∂ r , ∂ z .
3.1.1. Sobolev spaces. For integers k ≥ 0, we define the axially symmetic Sobolev space H k r,z (Ω) to be the completion of the functions in C ∞ (Ω) in the norm for a multi-index α ∈ Z 2 + . For real numbers s ≥ 0, the Sobolev spaces H s r,z (Ω) are defined by interpolation.
On the boundary Γ, for functions w ∈ H k (Γ), k ≥ 0, we set 3.2. Product and commutator estimates. We recall the following product and commutator estimates.
Then we have Then we have Proof. The proof of these estimates is standard; we first use the Leibniz formula to expand these terms as sums of products and then control the L 2 r,z norm of each product with the lower order derivative term in L ∞ ⊂ H 2 r,z and the higher order derivative term in L 2 r,z . See for instance Lemma A.1 of [29].
We will also use the following lemma.

Lemma 3.2. It holds that
Proof. It is direct to check that |gh| s |g| W 1,∞ |h| s for s = 0, 1. Then the estimate (3.4) follows by the interpolation.
3.3. Hardy-type inequality. We recall the following Hardy inequality: Lemma 3.3 (A higher order Hardy-type inequality). Let s ≥ 1 be a given integer, and suppose that g ∈ H s r,z (Ω) and g(0, z) = 0, we have g r s−1 ≤ C g s . 3.4. Geometry Identities.
where ∂ can be ∂ r , ∂ z and ∂ t operators.
From the incompressible constraint, we have (3.8)

Linearized approximate system
In this section, we construct the approximate system by linearizing method and then derive a priori estimates for this system and also prove the solvability of the system. 4.1. Approximate system. First, we denote ζ = (R, Z), ν = (v r , v z ) and reformulate the the system (1.20) into two coupled sub-systems which are both defined in Ω: in Ω. (4.1) and , we introduce our approximate system as two-step linearized system: firstly, we solve After solving the system (4.3), then we solve the following system: We define We take the time T > 0 sufficiently small so that for t ∈ [0, T ], From the definition ofC(t): we can also have sup We define the high order energy functional: We will prove that E remains bounded on a time interval dependent of M , which is stated as the following theorem.

4.2.
A priori estimates for system (4.3). For system (4.3), we have the following a priori estimates: , it holds that: Proof. TakingJ divĀ on the second equation of (4.3) to get: Note that by (4.6) the matrixĒ is symmetric and positive. We denoteĥ(r, z, t) as the harmonic extension ofC 2 (t) and by the Trace theorem, we have (4.13) And thenq = q −ĥ satisfying the following elliptic equation with zero Dirichlet boundary condition: TimingR rq on the equation (4.14), integrating on Ω and using integration-by-parts, we have Thus, with a priori assumption (4.6) and Poincare's inequality, we arrive at and hence with (4.13), we have Next, applying ∂ k z , k = 1, 2, 3 to the equation (4.14) leads to Thus, similarly, we obtain Combining with (4.13) again, we have In order to obtain other high order derivatives of q, we denote g =Ē 1j ∂ a j q R and rewrite the first equation of (4.12) as 1 R ∂ r (R 2 g) =R∂ r g + 2∂ rR g = G (4.18) where Then we obtain With integration-by-parts and a priori assumption (4.5), (4.6), we have by taking T sufficiently small (only depend on M ). Thus, we arrive at and as a consequence, we have Then by using (4.17) and a priori assumption (4.6) again, we have (4.24) Next, by acting ∂ z , ∂ 2 z on the equation (4.18), we havē Then by a similar approach from (4.20) to (4.24), for k = 1, 2, we can obtain and hence Finally, we act ∂ r , ∂ 2 r on the equation (4.18) to obtain for k = 1, 2, and hence Combining with (4.17), we prove the proposition.

4.2.2.
Tangential estimates for ν = (v r , v z ). We start with the basic L 2 energy estimates.
Proof. Taking the L 2 (Ω) inner product of the second equation in (4.1) with ν yields Using the pressure estimates (4.11), we have By Hardy's inequality, we have   Integrating directly in time of the above yields (4.27).
In order to perform higher order tangential energy estimates, one needs to compute the equations satisfied by (∂ 4 z ν, ∂ 4 z q, ∂ 4 z ζ), which requires to commutate ∂ 4 z with each term of ∂Ā ζ i . It is thus useful to establish the following general expressions and estimates for commutators. we have By the identity (3.7), we have that It then holds that where the commutator C i (g) is given by It was first observed by Alinhac [2] that the highest order term of ζ will be cancelled when one uses the good unknown ∂ 4 z g − ∂ 4 z ζ · ∇Āg, which allows one to perform high order energy estimates.
The following lemma deals with the estimates of the commutator C i (g).
We now introduce the good unknowns With the condition (1.21), we have Applying ∂ 4 z to the second equation of (4.3), by (4.35), one gets and We shall now derive the ∂ 4 z -energy estimates and have the following proposition Proposition 4.6. For t ∈ [0, T ], it holds that Proof. Taking the L 2 (Ω) inner product of (4.43) with V yields 1 2 Firstly, the right hand side of (4.46) can be bounded by (4.47) Here we use (4.37) and we estimate by using Hardy's inequality (3.3) and ∂ 4 z (R(b 0 · ∇Θ) 2 ) 0 can also be bounded by C(M ). Next, with (1.15), we have By integration-by-parts and (4.44), we have (4.49) Here we used the proposition 4.3, the condition |b z 0 | ≥ δ on Γ and the trace theorem ∂ 3 z (b z 0 ∂ zζ ) 1/2 ≤ C b 0 · ∇ζ 4 to control the last term on the RHS of (4.49). Moreover, we also estimate g 3 0 as Then by the definition of V and the first equation of (4.3), we prove the proposition.

4.3.
Normal estimates for ν = (v r , v z ). In this subsection, we control the normal derivatives of ν = (v r , v z ) by using the equaton (4.44), and the curl equation and begin with the energy estimates for (4.50).
Proof. Apply D 3 to (4.50) to get Taking the L 2 inner product of (4.52) with D 3 (∂Ā Z v r − ∂Ā R v z ), by the integration by parts, we get 1 2 Recalling (1.15), we have (4.54) By the identity (3.7), we have Hence, we obtain We now turn to estimate the right hand side of (4.53). By the identity (3.7), we may have Similarly, (4.57) Consequently, plugging the estimates (4.55)-(4.57) into (4.53), we obtain Integrating (4.58) directly in time, and applying the fundamental theorem of calculous, we then conclude the proposition.
We now derive the divergence estimates. We denote div v : Proof. From the third equation of (4.3) andĀ| t=0 = I, we see that Hence, it is clear that by the identity (3.7), From the third equation of (4.3) again, we have This together with the equation ν = ∂ t ζ, we have This implies that, by doing the D 3 energy estimate and using the identity (3.7),   Proof. First, by using the Proposition 4.7 and the Proposition 4.6, we have (4.67) By direct calculation, we have and by integration-by-parts, the last term of the above equality can be calculated as Thus, we arrive at Next, we use the Proposition 4.7 and (4.70) to obtain and use the Proposition 4.8 and (4.67) to obtain By direct calculation, we have and by integration-by-parts, the last term of the above equality can be calculated as Thus, we arrive at Last, we can repeat the above process inductively to bound the ∂ z ∂ 3 The estimates for b 0 · ∇R, b 0 · ∇Z can be obtained by a similar way and the proposition is proved.
Combining the Proposition 4.4 and the Proposition 4.9, we arrive at the Proposition 4.2.
4.4. Solvability of the system (4.3). With the above a priori estimates, we can solve the system (4.3) by applying the artificial viscosity approach used in [11,Section 5.1]. Here we just give a sketch and the reader can refer to [11] for the details.
Firstly, we construct the linear ε-approximate problem by adding artificial viscosity: in Ω. (4.75) Secondly, we solve this artificial viscosity system (4.75) by using a fixed point argument which is based on the solvability of three linear problems, i.e., (4.76), (4.78) and (4.80).

4.4.1.
Three linear problems. The first linear problem is the following linear degenerate parabolic problem of ζ with given f 1 : (4.76) Proposition 4.10. Given f 1 ∈ L 2 (0, T ; H 4 r,z (Ω)), then the problem (4.76) admits a unique solution (R, Z) that achieves the initial data and satisfies (4.77) The second linear problem is the simple transport problem of v with given f 2 : (4.78) Proposition 4.11. Given f 2 ∈ L 1 (0, T ; H 4 (Ω)) and suppose that (v r 0 , v z 0 ) ∈ H 4 (Ω). Then the problem (4.78) admits a unique solution ν = (v r , v z ) that achieves the initial data (v r 0 , v z 0 ) and satisfies The last linear problem is the most substantial elliptic problem of q with given f 3 : Note that X(M, T ) is a Banach space. We then define a mapping M : X(M, T ) → X(M, T ) as M(w, π, ξ) = (ν, q, ζ), where ζ, ν and q are determined as follows. Given (w, π, ξ) ∈ X(M, T ), we first define ζ = (R, Z) as the solution to (4.76) with f 1 = w and the initial data η 0 = Id, and ν = (v r , v z ) as the solution to (4.78) with f 2 = −∇Āπ + (b 0 · ∇) 2 ξ and the initial data v 0 . Next we define q as the solution to (4.80) with where ν and ζ are the functions constructed in the above. Hence, if M is taken to be sufficiently large with respect to b 0 , v 0 ,R,Z and ε and then 0 < T < 1 is taken to be sufficiently small (depending on M and ε), then (ν, q, ζ) ∈ X(M, T ). This implies that the mapping M : X(M, T ) → X(M, T ) is well-defined. And then we can show that the mapping M has a fixed point in the space X(M, T ) by proving the contraction and can verify that the unique fixed point (ν, q, ζ) is a solution to (4.75). The details are omit here, the reader can refer to [11,Section 5.1].
Lastly, after we finding the solutions to the system (4.75), we derive an ε-independent estimates of the solutions, which allows us to pass to the limit as ε → 0 to produce the solution to the system (4.3). Recalling that since b r 0 = 0 on Γ, we do not need to impose boundary conditions for (R, Z) and thus there is no boundary layer appearing as ε → 0. The details are omit here. Finally, the existence of a unique solution to (4.3) is recorded in Theorem 4.13.

4.5.
A priori estimates for approximate system (4.4). We derive the high order energy estimates for (v θ , Θ) by standard energy method. Proof. Taking L 2 inner product with v θ and using integration-by-parts yields For ΩR b 0 · ∇Θb 0 · ∇v θ dx, we have ). (4.84) Acting D 4 on the second equation in (4.4), taking L 2 innner product with D 4 v θ and using integration-by-parts yields By using the commutator estimate (3.3), we have where we estimate and by using Hardy's inequality.
Using (1.11) and (1.15), we have Hence, we arrive at the conclusion of this proposition.
4.6. Solvability of the system (4.4). The solvability of the system (4.4) can be established by a quite similar method we used in Section 4.4. We first construct strong solutions to the ε-system: . and then by ε-independent a priori estimates, we can pass to the limit as ε → 0 to produce the solution to the system (4.4). The details are omit. Finally, the existence of a unique solution to (4.3) is recorded in Theorem 4.15.

4.7.
Proof of Theorem 4.1. We now collect the estimates derived previously to conclude our estimates and also verify the a priori assumptions (4.6). That is, we shall now present the This provides us with a time of existence T 1 dependent of M and an estimate on [0, T 1 ] of the type: sup Since by A κ (0) = I and J κ (0) = 1, the bound (4.9) and (4.12) verify in turn the a priori bounds (4.6) by the fundamental theorem of calculous with taking T 1 smaller if necessary. The proof of Theorem 4.1 is thus completed.

5.
Constructing solutions to the system (1.20) 5.1. Iteration scheme. In order to produce the solution to the nonlinear problem (1.20), we will pass to the limit as n → ∞ in a sequence of approximate solutions {(v r , v z , v θ , R, Z, Θ, q) (n) } ∞ n=0 , which are constructed via the linearization iteration by using the linearized approximate problem (4.3)-(4.4).
in Ω, We define a higher order energy functional We shall prove the claim (5.5) by the induction. First it holds for n = 0, 1. Now suppose that it holds for n ≤ m for m ≥ 1, then we prove that it holds for n = m + 1. First, it holds from the induction assumption that Then from the estimates (4.11) and (4.83), we obtain Taking T sufficiently small (only depends on M 0 ), we conclude the claim (5.5). Now we will prove that the sequence {(v r , v z , v θ , R, Z, Θ, q) (n) } ∞ n=0 converges in certain strong norm. Let n ≥ 3 and denote the differences: Also, we denoteÃ (n) = A (n) − A (n−1) ,C (n) = C (n) − C (n−1) .
in Ω, in Ω, in Ω, (Θ (n) ,ṽ θ(n) )| t=0 = 0, We will now estimate the differences in H 3 r,z norm. We will use a similar strategy which was used in Sections 4.2 and 4.5, and we divide our estimates into several steps.
Proof of Theorem 2.1. The strong convergence of {(v (n) , q (n) , η (n) )} is more than sufficient to pass to the limit as n → 0 in (5.2) and (5.3) to produce a solution to (1.20). The estimate (2.3) follows from (5.5), while the uniqueness follows by the exactly same arguments as that of showing the convergence.