Global Well-Posedness for the 3D Axisymmetric Hall-MHD System with Horizontal Dissipation

Studied in this paper is the Cauchy problem for the 3D incompressible Hall-MHD system with horizontal dissipation. It is shown that if the initial data is axisymmetric and the swirl component of the velocity and the magnetic vorticity are trivial, such a system is globally well-posed for the large initial data. The key is to take full advantage of the structure of the Hall-MHD system in axisymmetric case to overcome the main difficulty due to the absence of vertical dissipation.


Introduction
In this paper, we consider the following 3D incompressible Hall-MHD system with horizontal dissipation for velocity (1.1) ⎧ ⎪ ⎨ ⎪ ⎩ Definition 1. 1 We say a vector field f(x, t) is axisymmrtric, if it can be written as Here (r, , z) is the usual cylindrical coordinates in ℝ 3 , that is, for any x = (x 1 , x 2 , x 3 ), and f (t, x) = f r (t, r, z)e r + f (t, r, z)e + f z (t, r, z)e z . Moreover, we call an axisymmetric vector field f is without swirl if f = 0.
The well-known work by Lei in [17] established the global well-posedness of the usual MHD system without any smallness assumptions for a class of specific axisymmetric initial data. More precisely, under the assumption that u 0 = B r 0 = B z 0 = 0 , he proved that there exists a unique global solution if the initial data satisfies Ai and the first author in [1] weakened the initial condition (1.2) to (u 0 , B 0 ) ∈ H 1 × H 2 (ℝ 3 ) and ∇×u 0 r ∈ L 2 (ℝ 3 ) . In addition, Fan-Huang-Nakamura in [9] extended the result of [17] to the usual incompressible Hall-MHD system. For the axisymmetric MHD system with horizontal dissipation and vertical magnetic diffusion, Jiu-Liu in [12] proved that there exists a unique global solution under the assumption that the initial data satisfies u 0 = B r 0 = 0 and smooth enough. See [21][22][23], the authors also investigated the global well-posedness for 3D Boussinesq system with anisotropic dissipation corresponding to large axisymmetric data without swirl.
Inspired by the ideas in [12,21], our main result in this paper is concerning the global existence and the uniqueness of axisymmetric smooth solution to the system (1.1) which does not have the swirl component for velocity field and magnetic vorticity. This means solution of the form: By direct computations, we find that and Then the system (1.1) can be equivalently written in the cylindrical coordinates where the Laplacian operator Δ = rr + 1 r r + zz and the horizontal Laplacian operator Δ h = rr + 1 r r . On the other hand, we know that the vorticity ∶= ∇ × u of the vector field u takes the form and in the cylindrical coordinates. It follows from (1.3) that the quantity obeys to the equation Now, let us state our main result as follows. There are two main difficulties in the proof of Theorem 1.2. The first one is there is no smoothing effect on the vertical derivative due to the absence of vertical viscosity. The other is how to deal with the Hall term ∇ × ((∇ × B) × B) , which is quadratic in the magnetic field and contains the second-order derivatives. To overcome these two difficulties, we need to fully use the structure of (1.1) in axisymmtric case with u 0 = B r 0 = B z 0 = 0 . Now we briefly sketch the proof of Theorem 1.2. In order to absorb the magnetic stretching term B ⋅ ∇u and the vortex stretching term ⋅ ∇u into the convection term, we define which strongly relies on the geometric structure of axisymmetric flows. Then by virtue of (1.3) and (1.4), the quantity (Ω, Π) verifies .

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We see that in Ω equation leaving only one term involving the Π as a forcing one. Thus, we can obtain the desired control on Ω by studying the properties of Π . Moreover, using Lemma 3.3, we can obtain the estimates of ‖u‖ L ∞ and ‖ u r r ‖ L ∞ , which allows us to get the H 1 bound of the u and B (Proposition 3.6). To make up for the shortage of vertical diffusion, we deeply use anisotropic inequality to get the H 2 estimate of u (Proposition 3.8). Finally, we need to get the Lipschitz estimates of u and B, which play the key role in our proof. To do this, by using the axisymmetric structure and the incompressible condition, we have the following estimate Taking advantage of the smooth effect of the velocity in the horizontal direction and Sobolev's embedding, we obtain the estimate every term in the right side. In addition, by using the regularity theory of the parabolic equation, we get the bound of ‖∇B‖ L 2 ([0,t];L ∞ ) (Proposition 3.9). With the Lipschitz estimate of B in hand, we can show the H 2 estimate of B (Proposition 3.10).
The present paper is built up as follows. Section 2 is devoted to collecting some useful inequalities, which will be used later. With these inequalities in hand, we will show some a priori estimate in Sect. 3. Finally, we give the proof of Theorem 1.2 in Sect. 4.
We end up this section with some notations we are going to use in this context. Notations: For simplicity, we denote With the denote in hand, we will frequently use the following facts In addition, the letter C stands for some generic constant, which may vary from line to line. For A ≲ B , we mean that there is a uniform constant C such that A ≤ CB . We always denote ∫ ⋅dx ∶= ∫ ℝ 3 ⋅dx and L p ([0, t];L q ) ∶= L p ([0, t];L q (ℝ 3 )).

Preliminaries
In this section, we list some useful lemmas, which will be used in the next section.

A priori estimates
The main goal of this section is to establish some a priori estimates needed for the proof of Theorem 1.2. Let us begin with the basic L 2 estimate of (u, B). It should be noted that in this estimate we do not need the axisymmetric assumption. Notice that divu = divB = 0 and the operator ∇× is symmetric, we have and Inserting the above fact into (3.1) and integrating on t, it gives which implies the desired result. ◻ Next, our task is to obtain the H 1 estimate of (u, B). Let us first show some estimates for Π and Ω.  Letting p → ∞ in above inequality, we obtain In particular, taking p = 2 in (3.2) and using Gronwall's inequality, it infers On the other hand, by taking the L 2 -inner product of the Ω equation in (1.5) with Ω , we deduce from the incompressible condition divu = 0 that Using the decay condition lim r→∞ Π(r, z, t) = 0 , one has

3
The Hölder inequality and Young inequality yield Hence, putting all the above estimates into (3.5), we have An application of Gronwall's inequality yields where we used the estimates (3.3) and (3.4). This ends the proof of Proposition 3.

◻
Using the L p boundedness of Riesz operator and the Biot-Savart law, we have the following important lemma, which links the velocity to the vorticity.
which gives the desired result. ◻ To achieve the H 1 estimate of (u, B), we also need the following estimate of Proof Firstly, from the incompressibility divu = 0 , we get by taking the L 2 -inner product of the (3.10)

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which implies the second desired estimate. We finish the proof of Proposition 3.6. ◻ In order to get the H 2 estimate of (u, B), we first give the following proposition. (3.11) Here we used in the cylindrical coordinates the Laplacian operator Δ = rr + 1 r r + zz and the incompressible condition divu = r u r + u r r + z u z = 0. In the following, we estimate I i term by term. Applying Lemmas 2.1 and 2.4, we get and Using Hölder's inequality and Young's inequality, we have and

3
Summing up all the estimates I 1 -I 6 leads to The Gronwall inequality implies where we used Propositions 3.2, 3.6, 3.7 and Corollary 3.4. This ends the proof of Proposition 3.8. ◻ The following proposition is devoted to studying the Lipschitz estimate for (u, B), which plays a key role in what follows.

Then
Proof From the structure of axisymmetric flows and the incompressible condition, we know that divu = r u r + u r r + z u z = 0 and = z u r − r u z . Hence We first deduce from Lemma 3.3 and Proposition 3.8 that Next, we turn to bound the quantity ‖ r u r ‖ L 1 ([0,t];L ∞ ) and ‖ r u z ‖ L 1 ([0,t];L ∞ ) . Notice that for a divergence free vector function f, we have d dt Hence, acting the operator ∇× to the first equation of (1. By making use of the L p ([0, t];L q ) (1 < p, q < +∞) estimates for the parabolic equation of singular integral and potentials, see [18,25] (3.14) For the first term in the right side of (3.15), using Hölder's inequality and Lemma 2.3, we obtain As for the second term in the right side of (3.15), we get from Lemma 2.3 again that From these estimates and (3.15), we see Therefore, thanks to the Gronwall inequality, we deduce from Propositions 3.8 and 3.9 that The proof is completed. ◻

Proof of Theorem 1.2
Now, we are in a position to complete the proof of Theorem 1.2.
Proof The existence part can be obtained by the classical Friedrichs method (see [3] for more details): For n ≥ 1 , the cut-off operator J n is defined as We consider the following truncated system in the space Here (u 0 , B 0 ) is a divergence free axisymmetric vector field. Then (u n (x, 0), B n (x, 0)) is also axisymmetric due to the radial property of the function . Since the operators J n and PJ n are the orthogonal projectors for the L 2 -inner product, the above formal calculations remain unchanged. Based on Propositions 3.1, 3.8, 3.9 and 3.10, we get by using standard arguments that the system (4.1) has a unique global solution (u n , B n ) n∈ℕ such that and The control is uniform with respect to the parameter n. Furthermore, by using a standard compactness argument, we obtain that the approximate solutions (u n , B n ) n∈ℕ converges to some (u, B) which satisfies our initial data. This completes the proof of the existence part. We omit here the details, see for example [13,22,24].
We next prove the uniqueness part. Let (u 1 , B 1 ) and (u 2 , B 2 ) be two solutions of the system (1.1) with the same initial data such that and We denote u ∶= u 1 − u 2 , B ∶= B 1 − B 2 and P ∶= P 1 − P 2 . Then ( u, B) satisfieŝ Journal of Nonlinear Mathematical Physics (2022) 29:794-817 Taking the L 2 -inner product of the u and B equations of (4.2) with u and B , respectively, we get which we used the facts that and due to divu i = divB i = 0 , we have Using Hölder's inequality and Young's inequality, one has and Therefore, substituting the above estimates into (4.3), we obtain

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which along with Gronwall's inequality applied implies that u = 0 and B = 0 . This ends the proof of Theorem 1.2. ◻

Conclusion
We consider global axisymmetric smooth solutions for the 3D incompressible Hall-MHD system with horizontal disspation. We prove that if the initial data is axisymmetric and the swirl component of the velocity and the magnetic vorticity are trivial, such a system is globally well-posed for the large initial data.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflict of interest
The authors declare that there is no conflict of interests regarding the publication of this paper.
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