Local Well-Posedness and Blow-Up for the Solutions to the Axisymmetric Inviscid Hall-MHD Equations

In this paper, we consider the regularity problem of the solutions to the axisymmetric, inviscid, and incompressible Hallmagnetohydrodynamics (Hall-MHD) equations. First, we obtain the local-in-time existence of sufficiently regular solutions to the axisymmetric inviscid Hall-MHD equations without resistivity. Second, we consider the inviscid axisymmetric Hall equations without fluids and prove that there exists a finite time blow-up of a classical solution due to the Hall term. Finally, we obtain some blow-up criteria for the axisymmetric resistive and inviscid Hall-MHD equations.


Introduction
Magnetohydrodynamics is the study of the dynamics of the electrically conducting fluids. The dynamics of the fluids can be described by the Navier-Stokes equations and the dynamics of the magnetic field can be described by the Maxwell equations for a perfect conductor. The Hall-magnetohydrodynamics (Hall-MHD) equations differ from the standard incompressible MHD equations by the Hall term ∇ × ((∇ × ) × ), which plays an important role in the study of the magnetic reconnection in the case of the large magnetic shear (see [1,2]). In [3], Hall-MHD equations have been formally derived from using the generalized Ohm's law instead of the usual simplified Ohm's law. The Cauchy problem for threedimensional incompressible Hall-MHD equations reads as follows: where , , and represent three-dimensional velocity vector field, the magnetic field, and scalar pressure, respectively. The initial data 0 and 0 satisfy Note that if ∇ ⋅ 0 = 0, then the divergence free condition is propagated by (1) 3 . We only consider R 3 for a spatial domain with vanishing at infinity condition for simplicity. The Hall magnetohydrodynamics were studied systematically by Lighthill [2]. The Hall-MHD is important, describing many physical phenomena, e.g., space plasmas, star formation, neutron stars, and geo-dynamo (see [1,[4][5][6][7][8] and references therein).
The Hall-MHD equations have been mathematically investigated in several works. In [9], Acheritogaray, Degond, Frouvelle, and Liu derived the Hall-MHD equations from either two fluids' model or kinetic models in a mathematically more rigorous way. In [10], the global existence of weak solutions to (1) and the local well-posedness of classical solution are established when ], > 0. Also, a blow-up criterion for smooth solution to (1) and the global existence of smooth solution for small initial data are obtained (see [10,Theorem 2.2 and 2.3]). Some of the results have been refined by many authors (see [11][12][13] and references therein). Recently, temporal decay for the weak solution and smooth 2 Advances in Mathematical Physics solution with small data to Hall-MHD are also established in [14]. Spatial and temporal decays of solutions to (1) have been investigated in [15].
It is well-known that the local-in-time classical solutions to axisymmetric Navier-Stokes equations without swirl persist to any time (see [16,17]). But the global well-posedness for the axisymmetric Navier-Stokes equations with swirl component is widely open and has been one of the most fundamental open problems in the Navier-Stokes equations.
The axisymmetric MHD equations can be written as follows: Lei [18] proved the global well-posedness of classical solutions to system (8) when ≥ 0.
Then axisymmetric Hall-MHD equations are reduced to the following: For axisymmetric Hall-MHD equations, the global wellposedness of the axisymmetric solutions to the viscous case (], > 0) was first established by Fan, Huang, and Nakamura [19]. Recently, Chae and Weng [20] showed that the incompressible Hall-MHD system without resistivity is not globally in time well-posed in any Sobolev space (R 3 ) with > 7/2. But local-in-time existence of smooth solution to (1) is totally open when ≡ 0. Compared with the work in [18], it seems very surprising that Hall term plays a dominant role for the occurrence of the singularity and even for the local well-posedness of the partially viscous Hall-MHD problems. In this paper, we intend to investigate the blow-up problem for the solutions to the partially viscous axisymmetric Hall-MHD equations and local-in-time existence of solutions to Advances in Mathematical Physics 3 such solution with the axisymmetry. Setting = − , Ω = / , and Π = / , (9) are equivalent to the following equations: First, we consider the local well-posedness of the axisymmetric Hall MHD equations with ] ≡ 0 and ≡ 0, and (10) can be rewritten as the equations with integer > 9/2 be axisymmetric initial data. There exist 0 > 0 and classical and axisymmetric solution (Π, Ω) to (11)- (12) such that Remark 2. Since the local-in-time regularity of solution to (1) is necessary to preserve the axisymmetry of the Hall-MHD equations locally in time, Theorem 1 cannot resolve the open question raised from [20]. We remark that the relation between (11)- (12) and (1) cannot be justified without local well-posedness of solution to (1) ( = ] ≡ 0).
Next, we consider the local well-posedness/blow-up problem for the axisymmetric Hall equations with zero fluid velocity and = 0. We rewrite the Hall equation for Π = / : The above equation has similar features to the inviscid Burgers equation.
Theorem 3. Assume Π 0 ∈ (R 3 ) for any integer > 5/2. Then there exist 0 > 0 and a classical solution to (14) such that Furthermore, for any Π 0 ̸ = 0, there exists * > 0 such that the above local solution Π( ) has singularity at a finite time = * . Remark 4. In [20], the authors showed that if the initial data Π 0 satisfies Π 0 (0, 0) ≥ 10 4 2 * for some constant * and Π 0 (0, 0) > 0, then the singularity of Π and Ω to axisymmetric inviscid Hall-MHD equations happens in a finite time. Theorem 3 implies that the singularity of Π which is a solution to (14) happens in a finite time without any restriction of the initial data.
Finally, we consider the incompressible Hall-MHD equations with zero fluid viscosity, for simplicity, assuming that ] ≡ 0 and ≡ 1.
For the solutions to (10), global a priori bounds can be obtained; that is, ‖Π‖ ∞ (0, ; ∞ ) + ‖Ω‖ ∞ (0, ; 2 ) < ∞ for all > 0. (16) We assume that our initial data ( 0 , 0 ) is axisymmetric and satisfies with > 5 2 , The local-in-time existence of a smooth solution to (1) was already obtained by Chae, Wan, and Wu [21]. We obtain the following blow-up criterion for the local-in-time solutions to the Hall-MHD equations with ] ≡ 0 and ≡ 1. Then, for the first blow-up time * < ∞ of the classical solution to (9), it holds that if and only if one of the following conditions holds: (i) 4
Remark 6. For the usual MHD equations, Lei [18] proved the global well-posedness for the axisymmetric MHD equations even for the case that ] ≡ 1 and ≡ 0. For Hall-MHD equations, even local well-posedness is widely open for this zero resisitivity case due to the Hall term (see [20]). Theorem 5 indicates that if there exists a finite time singularity to the axisymmetric equations with ] ≡ 0 and ≡ 1, then some norms of velocity and vorticity should approach infinity even for the outside of any infinite cylinder.
For simplicity, we denote for the harmless constant which changes from line to line, and ‖ ⋅ ‖ for -norm.

Proof of Theorem 1: Local-in-Time Existence
In this section, we consider the local-in-time existence of regular solution to (11)- (12). Even if this problem does not seem complicated, we have a few technical difficulties raised from the axisymmetry; e.g., mollifying equations do not preserve the axial symmetry. We briefly explain some steps to prove Theorem 1: First, we consider system (21) without giving any symmetry. We can obtain the regularized system (25) by using standard mollifier. Then we can obtain various estimates and local-in-time existence of a solution for (21). Finally, we consider the initial data which is axial symmetry and axisymmetry is also preserved by (21) and this argument gives a proof of local-in-time existence of solution to (11)- (12). We consider the equations where , , Π, and Ω are assumed to be independent scalar valued functions without assuming symmetry for a while, and the divergence free velocity field u = ( , , ) + ( , , ) is assumed to be obtained from the equation Thus, we have , then the divergence theorem and trace theorem induce the following estimates: We define a regularized system of (21) as follows: where J is a standard mollifier as in [22]. Next, we obtain apriori estimates to derive a time 0 which does not depend on > 0. Then we prove that (25) have a local-intime solution

Proposition 7. Let
with an integer > 5/2. Then, for some positive constant 0 and Advances in Mathematical Physics
Proof. We set First, we show that 1 is Lipschitz continuous on space. We estimate for < 1, ≥ 3 By the similar estimates as in (34), we obtain 6

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By the virtue of properties of mollifier, Lipschitz continuity of the remaining functions , = 2, 3, 4, can be obtained with constant / . Thus, we can deduce the following for ‖ ⋅ ‖ , ‖⋅‖ ≤ , with ≥ 3 and = ( 1 , 2 , 3 , 4 ). Now we use the Picard theorem with domain . By picking any initial data (0) ∈ and choosing 0 = 3/4 0 0 , we have, where ‖ ‖ ,‖̃‖ ≤ 4 0 . Therefore, the Picard theorem implies that, for each 0 < < 1, there exists a unique solution ( ) ∈ 1 (0, ; ) for a fixed time > 0. For simplicity, let be the maximal existence time of such solution. Suppose that, for some 0 < < 1, we have < 0 . Then by Proposition 7, for arbitrarily small > 0, we have If we apply the standard continuation argument, then we can have local-in-time solution at least until 0 . This contradicts the assumption that < 0 . Hence we prove that, for any 0 < < 1, there is a unique solution ( ) with a uniform time 0 , such that ( ) ∈ 1 (0, 0 ; ). This completes the proof.

Proposition 9.
For an integer > 7/2, the solutions obtained in Proposition 8 form the Cauchy sequences in the following spaces: Proof. Taking operator ( = 1, 2, 3) on both sides of (25) 1 and multiplying , we deduce that Advances in Mathematical Physics (48) 1 , . . . , 6 can be estimated as follows: Similarly, we can obtain the estimates for Π.
The other terms and Π + Ω can be estimated similarly, so we omit the details. Then we have for > 7/2. Gronwall's inequality gives us which implies that { } ∈ ([0, 0 ], 1 ) and this information completes the proof.
Proof of Theorem 1. With the bounds in Proposition 7, if we use the Sobolev inequality, then we can obtain the higher order convergence, i.e., ∈ (0, 0 ; ) for all < by the following inequality Now, to show ∈ (0, 0 ; ) ∩ Lip(0, 0 ; −1 ) where satisfies our equations in classical sense almost every time, we begin the process of obtaining the right continuity at = 0 first. Because is a reflexive Banach space, by Proposition 7, there exist a subsequence and limit functions If we use the above result, ( ) ∈ (0, 0 ; ) for any < , then ∈ (0, 0 ; ) is obtained by the following estimate.

Proof of Theorem 3: Blow-Up of Axisymmetric Hall Equations
The proof of Theorem 3 is split into two propositions: localin-time existence of a regular solution to (14)  Proof. First, we find the global solution to the following regularized equation of (14) without assuming the axisymmetry, Before proceeding further, we note that the divergence theorem can be applicable due to the mollifier. Let Hence, the image of the function defined on is included in for > 3/2. To use the Picard theorem on space ( > 5/2), we first obtain that is Lipschitz continuous on , i.e., is a Lipschitz continous function on a bounded open set in . Now we can apply Picard theorem. For each > 0, there exist a unique solution Π and a finite time , such that Π ∈ 1 (0, : ). Following the standard process of constructing local-in-time solution, we obtain an implicit form of the solution Since (0) = 0, we have Since the above regularized equation satisfies an energy estimate, we deduce that and hence For the higher order norm, Gronwall's inequality implies Advances in Mathematical Physics

11
The above inequality justifies that each solution Π is a global solution to regularized equation, and Π ∈ 1 (0, ∞ : ) , for all Π 0 ∈ (90) Second, we show that, for some finite time , the sequence {Π } >0 is a Cauchy sequence in (0, ; 2 ). We note that, for By the standard energy estimates, we have, for 0 ≤ ≤ , If ̸ = 0 or ̸ = , then we obtain easily that If = 0 or = , then we obtain Combining the above inequalities (93) and (94), we have The above inequality gives us By applying −1 at the regularized equation, we deduce that Now we are ready to show that {Π } ⊂ (0, ; 2 ) is a Cauchy sequence (as a sequence for → 0), where is chosen as above.
By the properties of regularizer J , for > 5/2, we have In summary, we have By Gronwall's inequality, we can conclude that {Π } is a Cauchy sequence in the (0, ; 2 ) space. And by boundness, if we apply the interpolation inequality, then we can see that {Π } is a cauchy sequence in (0, ; ), ∀ < . So we have the limit function Π ∈ (0, ; ). And { Π } is also a cauchy sequence in the (0, ; 2 ) space by the following estimates: For 5/2 < < , we have the limit function Π ∈ (0, ; −1 ).

Proposition 11. Let Π be an axisymmetric global classical solution to
Then Π ≡ 0.

Proof of Theorem 5: Blow-Up Criterion
In this section, we provide the proof of Theorem 5 which is the blow-up criterion for the axisymmetric Hall-MHD equations with ] = 0 and = 1: where Ω = / and Π = / . Known blow-up criterion for the partial viscous Hall-MHD equations (1) without symmetry (] = 0 and = 1) is as follows (see [13]).
With the axial symmetry, we can derive the following apriori estimates.
Proof. We first consider = 2 with ∈ N. Taking scalar product of (113) with Π −1 , we deduce that From the divergence free condition and the decay conditions like lim (116) can be reduced to It implies that For any > 0, we have As → ∞, we have ‖Π‖ ∞ (0, ; ∞ ) ≤ ‖Π 0 ‖ ∞ . If ∈ (2 , 2( + 1)), then we have Taking 2 scalar product of (112) with Ω, we have Then we have This completes the proof.
From the energy estimates of the velocity and magnetic fields, we have 14
By the Gagliardo-Nirenberg inequality (139) can be reduced to Since the last term in the above can be absorbed in the left hand side, we have ΔB ∈ 4 (0, ; 4 ) , ∇B ∈ 4 (0, ; ∞ ) .
The estimate of Ω, / , B, and∇Π can be obtained similarly to the proof of the condition of (19). This completes the proof.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.