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

1. 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 [nabla] x (([nabla] x B) x B), 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 three-dimensional incompressible Hall-MHD equations reads as follows:

[mathematical expression not reproducible], (1)

where u, B, and p represent three-dimensional velocity vector field, the magnetic field, and scalar pressure, respectively. The initial data [u.sub.0] and [B.sub.0] satisfy

V x [u.sub.0] = [nabla] x [B.sub.0] = 0. (2)

Note that if [nabla] x [B.sub.0] = 0, then the divergence free condition is propagated by [(1).sub.3]. We only consider [R.sup.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, 4-8] 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 v, [kappa] > 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 [11-13] and references therein). Recently, temporal decay for the weak solution and smooth 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].

Using vector identity, we can rewrite (1) as follows:

[mathematical expression not reproducible]. (3)

Note that a weak solution (u, B) to (1) satisfies the following energy inequality (see [10]):

[mathematical expression not reproducible], (4)

for almost every t [member of] [0, [infinity]).

Next we consider the mathematical setting for the axisymmetric Hall-MHD equations. Introducing the cylindrical coordinates

r = [square root of [x.sup.2.sub.1] + [x.sup.2.sub.2]],

[theta] = arctan [x.sub.2]/[x.sub.1], (5)

z = [x.sub.3], (5)

and standard basis vectors for the cylindrical coordinates

[e.sub.r] = (cos [theta], sin [theta], 0),

[e.sub.[theta]] = (- sin [theta], cos [theta], 0),

[e.sub.z] = (0,0,1), (6)

we set

u = [u.sup.r] (r, z, t) [e.sub.r] + [u.sup.z] (r, z, t) [e.sub.z],

B = [B.sup.[theta]] (r, z, t) [e.sub.[theta]]. (7)

It is well-known that the local-in-time classical solutions to axisymmetric Navier-Stokes equations without swirl persist to anytime (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:

[mathematical expression not reproducible]. (8)

Lei [18] proved the global well-posedness of classical solutions to system (8) when [kappa] [greater than or equal to] 0.

Then axisymmetric Hall-MHD equations are reduced to the following:

[mathematical expression not reproducible]. (9)

For axisymmetric Hall-MHD equations, the global well-posedness of the axisymmetric solutions to the viscous case (v, [kappa] > 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 [H.sup.m]([R.sup.3]) with m > 7/2. But local-in-time existence of smooth solution to (1) is totally open when k = 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 such solution with the axisymmetry. Setting [[omega].sup.[theta]] = [[partial derivative].sub.z][u.sup.r] - [[partial derivative].sub.r][u.sup.z], [OMEGA] = [[omega].sup.[theta]]/r, and [PI] = [B.sup.[theta]]/r, (9) are equivalent to the following equations:

[mathematical expression not reproducible]. (10)

First, we consider the local well-posedness of the axisymmetric Hall MHD equations with v [equivalent to] 0 and [kappa] [equivalent to] 0, and (10) can be rewritten as the equations

[[partial derivative].sub.t][PI] + ([u.sup.r][[partial derivative].sub.r] + [u.sup.z][[partial derivative].sub.z])[PI] = 2[PI][[partial derivative].sub.z][PI] (11)

[[partial derivative].sub.t][OMEGA] + ([u.sup.r][[partial derivative].sub.r] + [u.sup.z][[partial derivative].sub.z])[OMEGA] = 2[PI][[partial derivative].sub.z][PI] (11). (12)

Theorem 1. Let ([u.sub.0],[B.sub.0],[B.sub.0]/r, curl [u.sub.0]/r) [member of] [H.sup.m]([R.sup.3]) x [H.sup.m]([R.sup.3]) x [H.sup.m]([R.sup.3]) x [H.sup.m-1]([R.sup.3]) with integer m > 9/2 be axisymmetric initial data. There exist [T.sub.0] > 0 and classical and axisymmetric solution ([PI],[OMEGA]) to (11)-(12) such that

[mathematical expression not reproducible]. (13)

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) ([kappa] = v [equivalent to] 0).

Next, we consider the local well-posedness/blow-up problem for the axisymmetric Hall equations with zero fluid velocity and [kappa] = 0. We rewrite the Hall equation for [PI] = [B.sup.[theta]]/r:

[[partial derivative].sub.t][PI] = -2[PI][[partial derivative].sub.z][PI], [PI](x,0) = [[PI].sub.0](x). (14)

The above equation has similar features to the inviscid Burgers equation.

Theorem 3. Assume [[PI].sub.0] [member of] [H.sup.m]([R.sup.3]) for any integer m > 5/2. Then there exist [T.sub.0] > 0 and a classical solution to (14) such that

[mathematical expression not reproducible]. (15)

Furthermore, for any [[PI].sub.0] [not equal to] 0, there exists [T.sup.*] >0 such that the above local solution n(i) has singularity at a finite time t = [T.sup.*].

Remark 4. In [20], the authors showed that if the initial data [[PI].sub.0] satisfies [[partial derivative].sub.z][[PI].sub.0](0,0) [greater than or equal to] [10.sup.4][C.sup.2.sub.*] for some constant [C.sub.*] and [[PI].sub.0](0,0) > 0, then the singularity of [PI] and [OMEGA] to axisymmetric inviscid Hall-MHD equations happens in a finite time. Theorem 3 implies that the singularity of [PI] 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 v [equivalent to] 0 and [kappa] [equivalent to] 1.

For the solutions to (10), global a priori bounds can be obtained; that is,

[mathematical expression not reproducible]. (16)

We assume that our initial data ([u.sub.0], [B.sub.0]) is axisymmetric and satisfies

([u.sub.0],[B.sub.0],[[OMEGA].sub.0],[[PI].sub.0]) [member of] (H.sup.m][([R.sup.3]).sup.4] with m > 5/2, [nabla] x [u.sub.0] = [nabla] x [B.sub.0] = 0. (17)

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 v [equivalent to] 0 and [kappa] [equivalent to] 1.

Theorem 5. Let (u,B,p) be a local-in-time classical solution to the axisymmetric Hall-MHD equations (9) with v = 0, [kappa] = 1. Then, for the first blow-up time [T.sup.*] < [infinity] of the classical solution to (9), it holds that

[mathematical expression not reproducible], (18)

if and only if one of the following conditions holds:

(i)

[mathematical expression not reproducible]. (19)

(ii)

[mathematical expression not reproducible]. (20)

In the above, [C.sub.R] denotes the inside of infinite cylinder such that [C.sub.R] = {(x,y,z) | [x.sup.2] + [y.sup.2] < [R.sup.2]} for any R > 0 and [f.sub.+](x) is defined by max{f(x), 0}.

Remark 6. For the usual MHD equations, Lei [18] proved the global well-posedness for the axisymmetric MHD equations even for the case that v [equivalent to] 1 and [kappa] [equivalent to] 0. For HallMHD 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 v [equivalent to] 0 and [kappa] [equivalent to] 1, then some norms of velocity and vorticity should approach infinity even for the outside of any infinite cylinder.

For simplicity, we denote C for the harmless constant which changes from line to line, and [[parallel]*[parallel].sub.m] for [H.sup.m]-norm.

2. 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

[mathematical expression not reproducible], (21)

where [omega], B, [PI], and [OMEGA] are assumed to be independent scalar valued functions without assuming symmetry for a while, and the divergence free velocity field u = [u.sup.r](r, [theta], z)[e.sub.r] + [u.sup.z](r, [theta], z)[e.sub.z] is assumed to be obtained from the equation

-[DELTA]u = [nabla] x ([omega][e.sub.[theta]]). (22)

Thus, we have

u (x) = [PHI] * ([nabla] x ([omega][e.sub.[theta])), where [PHI] (x) = 1/4[pi] [absolute value of (x)]. (23)

If [omega] [member of] [H.sup.1]([R.sup.3]) [intersection] ([R.sup.3]), then the divergence theorem andtrace theorem induce the following estimates:

[mathematical expression not reproducible]. (24)

We define a regularized system of (21) as follows:

[mathematical expression not reproducible], (25)

where [J.sub.[epsilon]] is a standard mollifier as in [22]. Next, we obtain apriori estimates to derive a time [T.sub.0] which does not depend on [epsilon] > 0. Then we prove that (25) have a local-in-time solution Se = ([[omega].sub.[epsilon]],[B.sub.[epsilon]],[[PI].sub.[epsilon]],[[OMEGA].sub.[epsilon]]) [member of] [C.sup.1](0,[T.sub.0];[H.sup.m]) x [C.sup.1](0,[T.sub.0];[H.sup.m]) x [C.sup.1](0,[T.sub.0]; [H.sup.m+1]) x [C.sup.1](0,[T.sub.0]; [H.sup.m]) space for each [epsilon] > 0.

Proposition 7. Let

[mathematical expression not reproducible] (26)

where [X.sup.m] = [H.sup.m]([R.sup.3]) x [H.sup.m]([R.sup.3]) x [H.sup.m+1]([R.sup.3]) x [H.sup.m]([R.sup.3]) with an integer m > 5/2. Then, for some positive constant [C.sub.0] and T < 1/[C.sub.0][S.sub.0] with [mathematical expression not reproducible], we have

[mathematical expression not reproducible]. (27)

Proof. For m > 5/2 and 1 [less than or equal to] s [less than or equal to] m, we have the following inequality by the calculus inequality and Holder's inequality:

[mathematical expression not reproducible] (28)

Similarly, for m > 5/2 and 1 [less than or equal to] s [less than or equal to] m, we can obtain

[mathematical expression not reproducible], (29)

and 1 [less than or equal to] s [less than or equal to] m + 1,

[mathematical expression not reproducible]. (30)

For an integer m > 5/2 and all integers 1 [less than or equal to] s [less than or equal to] m, we have

[mathematical expression not reproducible]. (31)

Hence, we conclude that, for m > 5/2,

[mathematical expression not reproducible]. (32)

Inequality (32) implies that (27) holds true.

Proposition 8. Assume [C.sub.0] and [S.sub.0] are the same as in Proposition 7. Let [T.sub.0] = 3/4[C.sub.0][S.sub.0]. Then for given initial data [[omega].sub.0],[B.sub.0],[[OMEGA].sub.0] [member of] [H.sup.m]([R.sup.3]), and [[PI].sub.0] [member of] [H.sup.m+1]([R.sup.3]) with an integer m > 5/2, there exists a unique solution ([omega].sub.[epsilon]],[B.sub.[epsilon]],[[PI].sub.[epsilon]],[[OMEGA].sub.[epsilon]]) to regularized system (25) such that [[omega].sub.[epsilon]],[B.sub.[epsilon]],[[OMEGA].sub.[epsilon]] [member of] [C.sup.1](0,[T.sub.0];[H.sup.m]), [[PI].sub.[epsilon]] [member of] [C.sup.1](0,[T.sub.0];[H.sup.m+1]).

Proof. We set

[mathematical expression not reproducible]. (33)

First, we show that [F.sup.1.sub.[epsilon]] is Lipschitz continuous on [H.sup.m] space.

We estimate for [epsilon] < 1, m [greater than or equal to] 3

[mathematical expression not reproducible]. (34)

By the similar estimates as in (34), we obtain

[mathematical expression not reproducible], (35)

[mathematical expression not reproducible], (36)

[mathematical expression not reproducible]. (37)

By the virtue of properties of mollifier, Lipschitz continuity of the remaining functions [F.sup.j.sub.[epsilon]], j = 2, 3,4, can be obtained with constant C/[epsilon]. Thus, we can deduce the following for [mathematical expression not reproducible],

[mathematical expression not reproducible] (38)

with m [greater than or equal to] 3 and [F.sup.1.sub.[epsilon]] = ([F.sup.2.sub.[epsilon]], [F.sup.3.sub.[epsilon]], [F.sup.4.sub.[epsilon]], [F.sub.[epsilon]]). Now we use the Picard theorem with domain [X.sup.m]. By picking any initial data [S.sub.[epsilon]](0) [member of] [X.sup.m] and choosing [T.sub.0] = 3/4[C.sub.0][S.sub.0], we have, for R = [S.sub.0]/(1 - [C.sub.0][T.sub.0][S.sub.0],

[mathematical expression not reproducible] (39)

where [mathematical expression not reproducible]. Therefore, the Picard theorem implies that, for each 0 < [epsilon] < 1, there exists a unique solution [S.sub.[epsilon]](t) [member of] [C.sup.1] (0, [T.sub.[epsilon]];[X.sup.m]) for a fixed time [T.sub.[epsilon]] > 0. For simplicity, let [T.sub.[epsilon]] be the maximal existence time of such solution. Suppose that, for some 0 < [epsilon] < 1, we have [T.sub.[epsilon]] < [T.sub.0]. Then by Proposition 7, for arbitrarily small [delta] > 0, we have

[mathematical expression not reproducible]. (40)

If we apply the standard continuation argument, then we can have local-in-time solution [S.sub.[epsilon]] at least until [T.sub.0]. This contradicts the assumption that [T.sub.[epsilon]] < To. Hence we prove that, for any 0 < [epsilon] < 1, there is a unique solution Se (t) with a uniform time T0> such that [S.sub.[epsilon]] (t) [member of] [C.sup.1] (0, [T.sub.0]; [X.sup.m]). This completes the proof.

Proposition 9. For an integer m > 7/2, the solutions obtained in Proposition 8 form the Cauchy sequences in the following spaces:

{[[T.sub.[epsilon]]} [member of] C (0,[T.sub.0];[X.sup.1]) {[[partial derivative].sub.t] [S.sub.[epsilon]]} [member of] C(0,[T.sub.0];[X.sup.0]). (41)

Proof. Taking [[partial derivative].sub.i] operator (i = 1,2,3) on both sides of (25)1 and multiplying [[partial derivative].sub.i][[omega].sub.[epsilon]], we deduce that

[mathematical expression not reproducible], (42)

where

[mathematical expression not reproducible], (43)

[mathematical expression not reproducible], (44)

[mathematical expression not reproducible], (45)

[mathematical expression not reproducible], (46)

[mathematical expression not reproducible], (47)

[mathematical expression not reproducible]. (48)

[A.sub.1], ..., [A.sub.6] can be estimated as follows:

[mathematical expression not reproducible], (49)

[mathematical expression not reproducible], (50)

[mathematical expression not reproducible], (51)

[mathematical expression not reproducible], (52)

[mathematical expression not reproducible], (53)

[mathematical expression not reproducible]. (54)

Similarly, we can obtain the estimates for [PI].

[mathematical expression not reproducible] (55)

where

[mathematical expression not reproducible], (56)

[mathematical expression not reproducible], (57)

[mathematical expression not reproducible], (58)

[mathematical expression not reproducible], (59)

[mathematical expression not reproducible], (60)

[mathematical expression not reproducible]. (61)

These [B.sub.1], ..., [B.sub.6] can be estimated similarly.

[mathematical expression not reproducible], (62)

[mathematical expression not reproducible], (63)

[mathematical expression not reproducible], (64)

[mathematical expression not reproducible], (65)

[mathematical expression not reproducible], (66)

[mathematical expression not reproducible]. (67)

The other terms B and [PI] + [OMEGA] can be estimated similarly, so we omit the details. Then we have

[mathematical expression not reproducible], (68)

for m > 7/2. Gronwall's inequality gives us

[mathematical expression not reproducible], (69)

which implies that {[S.sub.[epsilon]]} [member of] C([0, [T.sub.0]], [X.sup.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., S [member of] C(0, [T.sub.0];[X.sup.s]) for all s < m by the following inequality

[mathematical expression not reproducible]. (70)

Now, to show S [member of] C(0,[T.sub.0];[X.sup.m]) [intersection] Lip(0, [T.sub.0]; [X.sup.m-1]) where S satisfies our equations in classical sense almost every time, we begin the process of obtaining the right continuity at t = 0 first. Because [X.sup.m] is a reflexive Banach space, by Proposition 7, there exist a subsequence and limit functions S(t) [member of] [L.sup.[infinity]](0,[T.sub.0];[H.sup.m]) which satisfies [[partial derivative].sub.t]S(t) [member of] [L.sup.[infinity]](0,[T.sub.0];[H.sup.m-1]) for any [mathematical expression not reproducible]. This implies that [mathematical expression not reproducible]. Thus we have

[mathematical expression not reproducible]. (71)

If we use the above result, S(t) [member of] C(0, [T.sub.0];[X.sup.s]) for any s < m, then S [member of] [C.sub.w](0, [T.sub.0];[X.sup.m]) is obtained by the following estimate.

For arbitrary [delta] > 0 and [phi] [member of] [X.sup.-m], there exists [??] [member of] [X.sup.-s] such that [mathematical expression not reproducible],

[mathematical expression not reproducible], (72)

where [mathematical expression not reproducible] is a dual pairing on [X.sup.-k] x [X.sup.k]. If we choose [phi] = [[nabla].sup.2m][[omega].sub.0] [member of] [H.sup.-m], then by the weak continuity,

[mathematical expression not reproducible], (73)

[mathematical expression not reproducible], (74)

Similarly, we have

[mathematical expression not reproducible]. (75)

By inequalities (71) and (75), we have the right continuity of S(t) at t = 0. If we apply the standard time translation invariant property and the time reversal techique, we also have S(t) [member of] C(0, [T.sub.0]; [H.sup.m]) without any difficulty. Lipschitz continuity also can be obtained by

[mathematical expression not reproducible], (76)

which means S [member of] C(0, [T.sub.0]; [X.sup.m]) [intersection] Lip(0, [T.sub.0]; [X.sup.m-1]). Hence it is a classical solution to (21) almost every time.

Next we assume that the initial data ([u.sub.0], [B.sub.0],[[PI].sub.0],[[OMEGA].sub.0]) is axisymmetric. Then the axisymmetry of a classical solution to (21) is preserved and ([omega],B,[PI],[OMEGA]) is axisymmetric solution to (21). Now we go back to (9) with v = [kappa] = 0, and set

[mathematical expression not reproducible]. (77)

Then for

[mathematical expression not reproducible], (78)

we know that there exists a unique solution (w,B,Q) e C(0,[T.sub.0];[H.sup.m-1]) [intersection] Lip C(0,[T.sub.0];[H.sup.m-2]), [PI] [member of] C(0,[T.sub.0];[H.sup.m]) [intersection] Lip C(0,[T.sub.0];[H.sup.m-1]). But if we replace [OMEGA] with [omega]/r and [PI] with B/r, then [omega], B, [omega]/r, B/r also satisfy (21) with the initial data. So by the uniqueness, [OMEGA] = w/r in C(0,[T.sub.0];[H.sup.m-1]) [intersection] Lip C(0,[T.sub.0];[H.sup.m-2]) and [PI] = B/r in C(0,[T.sub.0];[H.sup.m]) [intersection] Lip C(0,[T.sub.0];[H.sup.m-1]). Next, we can show that u = [PHI] * curl[omega] [member of] C(0,[T.sub.0];[H.sup.m]) [intersection] Lip C(0,[T.sub.0];[H.sup.m-1]). By the Poincare lemma, curlu = [omega] satisfies the [omega] equation of (9). Then we can deduce that u and B satisfy (9) almost every time by finding the axisymmetric scalar pressure p. Then the energy inequality (4) implies that u [member of] [L.sup.[infinity]](0, [T.sub.0]; [L.sup.2]) and, almost every time,

[mathematical expression not reproducible], (79)

which implies u [member of] C(0,[T.sub.0];[H.sup.m]) [intersection] Lip(0,[T.sub.0];[H.sup.m-1]). The uniqueness of u can be obtained from the standard techniques and we omit the details. Finally we can show that B [member of] C(0,[T.sub.0];[H.sup.m]) [intersection] Lip(0,[T.sub.0];[H.sup.m-1]). Almost every time, we can rewrite the B equation with

[[partial derivative].sub.t]B + ([u.sup.r][[partial derivative].sub.t] + [u.sup.z][[partial derivative].sub.z] B = [u.sup.r][PI] + 2[PI][[partial derivative].sub.z]B, (B/r = [PI]). (80)

Then we can obtain the conclusion through the standard [H.sup.m] estimate with Gronwalls inequality.

3. Proof of Theorem 3: Blow-Up of Axisymmetric Hall Equations

The proof of Theorem 3 is split into two propositions: local-in-time existence of a regular solution to (14) and the finite time blow-up of the local-in-time solution.

Proposition 10. The equation

[[partial derivative].sub.t][PI] = -2[PI][[partial derivative].sub.z][PI] [PI](x,0) = [[PI].sub.0](x) [member of] [H.sup.m], m > 5/2 (81)

has a unique local-in-time solution.

Proof. First, we find the global solution to the following regularized equation of (14) without assuming the axisymmetry,

[mathematical expression not reproducible] (82)

Before proceeding further, we note that the divergence theorem can be applicable due to the mollifier. Let

F([[PI].sub.[epsilon]]) = -2[J.sub.[epsilon]]([J.sub.[epsilon]][[PI].sub.[epsilon]] x [[partial derivative].sub.z][J.sub.[epsilon]][[PI].sub.[epsilon]]. (83)

Hence, the image of the function F defined on [H.sup.m] is included in [H.sup.m] for m >3/2.

To use the Picard theorem on [H.sup.m] space (m > 5/2), we first obtain that F is Lipschitz continuous on [H.sup.m], i.e.,

[mathematical expression not reproducible]. (84)

F is a Lipschitz continous function on a bounded open set O in [H.sup.m]. Now we can apply Picard theorem. For each [epsilon] > 0, there exist a unique solution n and a finite time [T.sub.[epsilon]], such that [[PI].sub.[epsilon]] [member of] [C.sup.1](0,[T.sub.[epsilon]] : [H.sup.m]). Following the standard process of constructing local-in-time solution, we obtain an implicit form of the solution

[[PI].sub.[epsilon]] (t) = [[PI].sub.0] + [[integral].sup.t.sub.0] F([[PI].sub.[epsilon]](s)) ds, 0 [less than or equal to] t < [T.sub.[epsilon]]. (85)

Since F(0) = 0, we have

[mathematical expression not reproducible]. (86)

Since the above regularized equation satisfies an energy estimate, we deduce that

[mathematical expression not reproducible], (87)

and hence

[mathematical expression not reproducible]. (88)

For the higher order norm, Gronwall's inequality implies

[mathematical expression not reproducible]. (89)

The above inequality justifies that each solution [[PI].sub.[epsilon]] is a global solution to regularized equation, and

[[PI].sub.[epsilon]] [member of] [C.sup.1](0,[infinity]:[H.sup.m]), for all [[PI].sub.0] [member of] [H.sup.m] (90)

Second, we show that, for some finite time T, the sequence [{[[PI].sub.[epsilon]]}.sub.[epsilon]>0] is a Cauchy sequence in C(0, T; [L.sup.2]). We note that, for T < 1/[C.sub.m] [[parallel][[PI].sub.0][parallel].sub.m],

[mathematical expression not reproducible] (91)

By the standard energy estimates, we have, for 0 [less than or equal to] s [less than or equal to] m,

[mathematical expression not reproducible] (92)

If j [not equal to] 0 or j [not equal to] s, then we obtain easily that

[mathematical expression not reproducible] (93)

If j = 0 or j = s, then we obtain

[mathematical expression not reproducible] (94)

Combining the above inequalities (93) and (94), we have

d/dt [[parallel][[PI].sub.[epsilon][parallel].sub.m] [less than or equal to] [C.sub.m] [[parallel][[PI].sub.[epsilon][parallel].sup.2.sub.m]. (95)

The above inequality gives us

[mathematical expression not reproducible] (96)

By applying [H.sup.m-1] norm at the regularized equation, we deduce that

[mathematical expression not reproducible] (97)

Now we are ready to show that {[[PI].sub.[epsilon]]} [subset] C(0, T; [L.sup.2]) is a Cauchy sequence (as a sequence for [[epsilon].sub.n] [right arrow] 0), where T is chosen as above.

[mathematical expression not reproducible] (98)

By the properties of regularizer [J.sub.[epsilon]], for m > 5/2, we have

[mathematical expression not reproducible] (99)

In summary, we have

[mathematical expression not reproducible] (100)

By Gronwall's inequality, we can conclude that {[[PI].sub.[epsilon]]} is a Cauchy sequence in the C(0, T; [L.sup.2]) space. And by [H.sup.m] boundness, if we apply the interpolation inequality, then we can see that {[[PI].sub.[epsilon]} is a cauchy sequence in C(0, T; [H.sup.S]), [for all]s < m. So we have the limit function [PI] [member of] C(0, T; [H.sup.S]). And {[[partial derivative].sub.t][[PI].sub.[epsilon]]} is also a cauchy sequence in the C(0, T; [L.sup.2]) space by the following estimates:

[mathematical expression not reproducible] (101)

For 5/2 < s < m, we have the limit function [[partial derivative].sub.t][PI] [member of] C(0, T; [H.sup.s-1]).

Finally, we can show that [PI] [member of] C(0, T; [H.sup.m]) [intersection] Lip(0, T; [H.sup.m-1]). By the Banach Alaoglu theorem, we have [PI] [member of] [L.sup.[infinity]](0, [T.sub.0]; [H.sup.m]) and [[partial derivative].sub.t][PI] [member of] [L.sup.[infinity]](0, [T.sub.0]; [H.sup.m-1]), because we know that, for any s < m, [PI] [member of] C(0, T; [H.sup.S]). It implies [PI] [member of] [C.sub.w](0, T; [H.sup.m]) by the following estimate:

[mathematical expression not reproducible] (102)

for any given [phi] [member of] [H.sup.-m], for some [??] [member of] [H.sup.-s], s < m. Now we show that

[mathematical expression not reproducible] (103)

By the weak continuity, for any [delta] > 0, there exists r > 0 such that if 0 < t < r, then -[delta] < [[PI](r)-[[PI].sub.0], [phi]] < [delta], for all [phi] [member of] [H.sup.-m]. Choose [phi] = [[nabla].sup.2s][[PI].sub.0] with s [less than or equal to] m. Then it gives us

[mathematical expression not reproducible] (104)

Also we have

[[parallel][[PI].sub.0][parallel].sup.2.sub.m] [less than or equal to] [[parallel][PI] (t)[parallel].sup.2.sub.m] + [2.sup.m+1][delta], (105)

which implies that

[mathematical expression not reproducible] (106)

By the [H.sup.m] boundness with weak convergence, it is deduced that

[[parallel][PI] (t)[parallel].sub.m] [less than or equal to] [[parallel][[PI].sub.0][parallel].sub.m]/1 - t[C.sub.m] [[parallel][[PI].sub.0][parallel].sub.m], 0 [less than or equal to] t [less than or equal to] T (107)

which implies

[mathematical expression not reproducible] (108)

Thus we have the time continuity of n at 0. For any [t.sub.0] [less than or equal to] T and initial value [PI]([t.sub.0]), we can obtain a right continuity at [t.sub.0] by the time translation invariant property. By the fact that [PI](-x, [t.sub.0] - t) is also a solution to the equation for 0 [less than or equal to] t [less than or equal to] [t.sub.0], we have a left continuity at [t.sub.0]. Of course by the above process, the left continuity at T also can be obtained. We have proved that [PI] [member of] C(0, T; [H.sup.m]).

Proposition 11. Let [PI] be an axisymmetric global classical solution to

[[partial derivative].sub.t][PI] + 2[PI][[partial derivative].sub.z][PI] = 0. (109)

Then [PI] [equivalent to] 0.

Proof. Define [mathematical expression not reproducible] which satisfies the equation [mathematical expression not reproducible]. By our assumption, [PI] [member of] C([0, [infinity]); [H.sup.m]([R.sup.3])) [intersection] Lip([0, [infinity]); [H.sup.m-1]([R.sup.3])) for m > 5/2, [mathematical expression not reproducible] almost every time. So we can find the explicit form of it by

[mathematical expression not reproducible] (110)

Now we choose initial values [mathematical expression not reproducible] and [mathematical expression not reproducible] such that [mathematical expression not reproducible]. Then it satisfies

[mathematical expression not reproducible] (111)

Because if we suppose that [mathematical expression not reproducible], then by the explicit form of [phi], for some [t.sub.0] > 0, we have [mathematical expression not reproducible] which makes a contradiction. Hence, [[PI].sub.0]([r.sub.0], z) is a nondecreasing funcion with respect to z. Since this process is independent of the choice of [r.sub.0], we can find that [[PI].sub.0] [equivalent to] 0 by the continuity and [L.sup.2] boundness.

4. 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 v =0 and [kappa] = 1:

[partial derivative][OMEGA]/[partial derivative]t + ([u.sup.r][[partial derivative].sub.r] + [u.sup.z][[partial derivative].sub.z]) [OMEGA] + 2[PI][[partial derivative].sub.z][PI] = 0, (112)

[partial derivative][PI]/[partial derivative]t + ([u.sup.r][[partial derivative].sub.r] + [u.sup.z][[partial derivative].sub.z]) [PI] - 2[PI][[partial derivative].sub.z][PI] = ([[partial derivative].sup.2.sub.r] + 3/r [[partial derivative].sub.r] + [[partial derivative].sup.2.sub.z]) [PI], (113)

where [OMEGA] = [w.sup.[theta]]/r and [PI] = [B.sup.[theta]]/r.

Known blow-up criterion for the partial viscous Hall-MHD equations (1) without symmetry (v =0 and [kappa] = 1) is as follows (see [13]).

Proposition 12. Assume that ([u.sub.0], [B.sub.0]) [member of] [H.sup.m]([R.sup.3]), m [greater than or equal to] 3 with [nabla] x [u.sub.0] = [nabla] x [B.sub.0] = 0. Let (u, B) be a smooth solution to (1) (v = 0 and [kappa] = 1)for 0 [less than or equal to] t < T. If (u, B) satisfies

[mathematical expression not reproducible] (114)

then the solution (u, B) can be extended beyond t = T.

With the axial symmetry, we can derive the following apriori estimates.

Proposition 13. If ([OMEGA], [PI]) is a solution to (112)-(113) satisfying ([OMEGA], [PI]) [member of] C([0, T); [H.sup.m]), with m [greater than or equal to] 3 then it holds that

[mathematical expression not reproducible] (115)

Proof. We first consider p = 2n with n [member of] N. Taking scalar product of (113) with [[PI].sup.p-1], we deduce that

[mathematical expression not reproducible] (116)

From the divergence free condition and the decay conditions like

[mathematical expression not reproducible] (117)

(116) can be reduced to

[mathematical expression not reproducible] (118)

It implies that

[mathematical expression not reproducible] (119)

For any R > 0, we have

[mathematical expression not reproducible] (120)

As n [right arrow] [infinity], we have [mathematical expression not reproducible]. If p [member of] (2n, 2(n + 1)), then we have

[mathematical expression not reproducible] (121)

Taking [L.sup.2] scalar product of (112) with [OMEGA], we have

[mathematical expression not reproducible] (122)

Then we have

[mathematical expression not reproducible] (123)

This completes the proof.

From the energy estimates of the velocity and magnetic fields, we have

[mathematical expression not reproducible] (124)

Proof of Theorem 5. First, we assume that assumption (19) holds. If we consider the equation of the vorticity [[omega].sup.[theta]], then we have

[mathematical expression not reproducible] (125)

Taking [L.sup.2] scalar product of (125) with [[omega].sup.[theta]], we have

[mathematical expression not reproducible] (126)

Using Gronwall's inequality, we have

[mathematical expression not reproducible] (127)

Hence we have [[omega].sup.[theta]] [member of] [L.sup.[infinity]] (0, T; [L.sup.2]) if we assume (19).

If we consider the equations for [??][PI] ([??] = ([[partial derivative].sub.r], [[partial derivative].sub.z])), then we obtain

[mathematical expression not reproducible] (128)

Via an interpolation inequality and Young's inequality, we have

[mathematical expression not reproducible] (129)

Taking scalar product of (128) with [??][PI], we deduce that

[mathematical expression not reproducible] (130)

In the above, we used the fact that (1/r)[[partial derivative].sub.r]([[absolute value of ([[partial derivative].sub.r][PI])].sup.2] + [[partial derivative].sub.z][PI])].sup.2]) [member of] [L.sup.1]([R.sup.3]) when t < [T.sup.*]. Gronwall's inequality again gives us

[??][PI] [member of] [L.sup.[infinity]] (0, T; [L.sup.2]),

[[??].sup.2][PI] [member of] [L.sup.2] (0, T; [L.sup.2]). (131)

Multiplying both sides of (112) with [[absolute value of ([OMEGA])].sup.4][OMEGA] and integrating over [R.sup.3], we have

[mathematical expression not reproducible] (132)

Then it is immediate that [OMEGA] [member of] [L.sup.[infinity]](0, T; [L.sup.6]).

Following the ideas in [18], we introduce the angular stream function [[psi].sup.[theta]] such that

-([[partial derivative].sup.2.sub.r] + 1/r [[partial derivative].sub.r] - 1/[r.sup.2] + [[partial derivative].sup.2.sub.z]) [[psi].sup.[theta]] = [[omega].sup.[theta]]. (133)

For all 1 < p < [infinity], we have

[mathematical expression not reproducible] (134)

By the interpolation inequality [mathematical expression not reproducible], we have

[mathematical expression not reproducible] (135)

Therefore, we have [u.sup.r]/r [member of] [L.sup.1](0, T; [L.sup.[infinity]]). Also we can have [u.sup.r]/r [member of] [L.sup.p](0, T; [L.sup.[infinity]]) for all p < [infinity].

If we multiply [([B.sup.[theta]).sup.2n-1] on the both sides of [(9).sub.4] and integrate over [R.sup.3], then we have

[mathematical expression not reproducible] (136)

Hence we deduce that

[mathematical expression not reproducible] (137)

Similarly, we have [mathematical expression not reproducible]

Setting B = [B.sup.[theta]][e.sub.[theta]], we have

[[partial derivative].sub.t]B + (u x [nabla]) B - [DELTA]B = [u.sup.r]/r B + 2[[partial derivative].sub.z]B[PI]. (138)

By the maximal inequality, we have

[mathematical expression not reproducible] (139)

By the Gagliardo-Nirenberg inequality

[mathematical expression not reproducible] (140)

(139) can be reduced to

[mathematical expression not reproducible] (141)

Since the last term in the above can be absorbed in the left hand side, we have

[DELTA]B [member of] [L.sup.4] (0, T; [L.sup.4]),

[nabla]B [member of] [L.sup.4] (0, T; [L.sup.[infinty]]). (142)

Then, from (125), we have, for all 1 < p < [infinity],

[mathematical expression not reproducible] (143)

By Gronwall's inequality, we have

[mathematical expression not reproducible] (144)

If we let p [right arrow] [infinity], we obtain [[omega].sup.[theta]] [member of] [L.sup.[infinity]](0, T; [L.sup.[infinity]]). Hence, for any T < [infinity], we obtain that [[omega].sup.[theta]], [nabla]B [member of] [L.sup.2](0, T; [L.sup.[infinity]]) and conclude that there does not exist a finite time blow-up if we assume (19).

Next, we assume that condition (20) holds. If we apply (129) to (128), we obtain

[mathematical expression not reproducible] (145)

Using an inequality

[mathematical expression not reproducible] (146)

we have

[mathematical expression not reproducible] (147)

By Gronwall's inequality, we conclude that

[??][PI] [member of] [L.sup.[infinity]] (0, T; [L.sup.2]),

[[??}.sup.2][PI] [member of] [L.sup.2] (0, T; [L.sup.2]). (148)

The estimate of [OMEGA], [u.sup.r]/r, B, and [nabla][PI] can be obtained similarly to the proof of the condition of (19). This completes the proof.

https://doi.org/10.1155/2018/5343824

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

E. Jeong was partially supported by Chung-Ang University Excellent Student Scholarship and J. Kim and J. Lee were partially supported by NRF Grant No. 2016R1A2B3011647. The authors thank Professor Dongho Chae for helpful remarks.

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Eunji Jeong, Junha Kim, and Jihoon Lee (iD)

Department of Mathematics, Chung-Ang University, Seoul 06974, Republic of Korea

Correspondence should be addressed to Jihoon Lee; jhleepde@cau.ac.kr

Received 16 July 2018; Accepted 26 September 2018; Published 1 November 2018

Academic Editor: Carlo Bianca
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