Advances in Mathematical Physics
Volume 2018 (2018), Article ID 5343824, 16 pages
https://doi.org/10.1155/2018/5343824

Research Article

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

Eunji Jeong, Junha Kim, and Jihoon Lee

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

Copyright © 2018 Eunji Jeong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we consider the regularity problem of the solutions to the axisymmetric, inviscid, and incompressible Hall-magnetohydrodynamics (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.

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 , 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:

where , , and represent three-dimensional velocity vector field, the magnetic field, and scalar pressure, respectively. The initial data and satisfy

Note that if , then the divergence free condition is propagated by (1)₃. We only consider 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 . 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:

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

for almost every .

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

and standard basis vectors for the cylindrical coordinates

we set

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 .

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

For axisymmetric Hall-MHD equations, the global well-posedness of the axisymmetric solutions to the viscous case () 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 with . But local-in-time existence of smooth solution to (1) is totally open when . 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 , , and , (9) are equivalent to the following equations:

First, we consider the local well-posedness of the axisymmetric Hall MHD equations with and , and (10) can be rewritten as the equations

Theorem 1. Let with integer be axisymmetric initial data. There exist and classical and axisymmetric solution 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) .

Next, we consider the local well-posedness/blow-up problem for the axisymmetric Hall equations with zero fluid velocity and . We rewrite the Hall equation for :

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

Theorem 3. Assume for any integer . Then there exist and a classical solution to (14) such that

Furthermore, for any , there exists such that the above local solution has singularity at a finite time .

Remark 4. In [20], the authors showed that if the initial data satisfies for some constant and , then the singularity of and to axisymmetric inviscid Hall-MHD equations happens in a finite time. Theorem 3 implies that the singularity of 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 and .

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

We assume that our initial data is axisymmetric and satisfies

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 and .

Theorem 5. Let be a local-in-time classical solution to the axisymmetric Hall-MHD equations (9) with . 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)

(ii)

In the above, denotes the inside of infinite cylinder such that for any and is defined by .

Remark 6. For the usual MHD equations, Lei [18] proved the global well-posedness for the axisymmetric MHD equations even for the case that and . 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 and , 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.

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

where , , , and are assumed to be independent scalar valued functions without assuming symmetry for a while, and the divergence free velocity field is assumed to be obtained from the equation

Thus, we have

If , then the divergence theorem and trace theorem induce the following estimates:

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

where is a standard mollifier as in [22]. Next, we obtain apriori estimates to derive a time which does not depend on . Then we prove that (25) have a local-in-time solution space for each .

Proposition 7. Let

where with an integer . Then, for some positive constant and with , we have

Proof. For and , we have the following inequality by the calculus inequality and Hölder’s inequality:

Similarly, for and , we can obtain

and ,

For an integer and all integers , we have

Hence, we conclude that, for ,

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

Proposition 8. Assume and are the same as in Proposition 7. Let . Then for given initial data , and with an integer , there exists a unique solution to regularized system (25) such that , .

Proof. We set

First, we show that is Lipschitz continuous on space.
We estimate for ,

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

By the virtue of properties of mollifier, Lipschitz continuity of the remaining functions , , can be obtained with constant . Thus, we can deduce the following for ,

with and . Now we use the Picard theorem with domain . By picking any initial data and choosing , we have, for ,

where ,. Therefore, the Picard theorem implies that, for each , there exists a unique solution for a fixed time . For simplicity, let be the maximal existence time of such solution. Suppose that, for some , we have . Then by Proposition 7, for arbitrarily small , we have

If we apply the standard continuation argument, then we can have local-in-time solution at least until . This contradicts the assumption that . Hence we prove that, for any , there is a unique solution with a uniform time , such that . This completes the proof.

Proposition 9. For an integer , the solutions obtained in Proposition 8 form the Cauchy sequences in the following spaces:

Proof. Taking operator () on both sides of (25)₁ and multiplying , we deduce that

where

can be estimated as follows:

Similarly, we can obtain the estimates for .

where

These can be estimated similarly.

The other terms and can be estimated similarly, so we omit the details. Then we have

for . Gronwall’s inequality gives us

which implies that 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., for all by the following inequality

Now, to show where satisfies our equations in classical sense almost every time, we begin the process of obtaining the right continuity at first. Because is a reflexive Banach space, by Proposition 7, there exist a subsequence and limit functions which satisfies for any , in and in . This implies that , . Thus we have

If we use the above result, for any , then is obtained by the following estimate. For arbitrary and , there exists such that ,

where is a dual pairing on . If we choose , then by the weak continuity,

Similarly, we have

By inequalities (71) and (75), we have the right continuity of at . If we apply the standard time translation invariant property and the time reversal techique, we also have without any difficulty. Lipschitz continuity also can be obtained by

which means . Hence it is a classical solution to (21) almost every time.
Next we assume that the initial data is axisymmetric. Then the axisymmetry of a classical solution to (21) is preserved and is axisymmetric solution to (21). Now we go back to (9) with , and set

Then for

we know that there exists a unique solution , . But if we replace with and with , then also satisfy (21) with the initial data. So by the uniqueness, in and in . Next, we can show that . By the Poincaré lemma, satisfies the equation of (9). Then we can deduce that and satisfy (9) almost every time by finding the axisymmetric scalar pressure . Then the energy inequality (4) implies that and, almost every time,

which implies . The uniqueness of can be obtained from the standard techniques and we omit the details. Finally we can show that . Almost every time, we can rewrite the equation with

Then we can obtain the conclusion through the standard estimate with Gronwall’s 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

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,

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 .
To use the Picard theorem on space , 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 , there exist a unique solution and a finite time , such that . Following the standard process of constructing local-in-time solution, we obtain an implicit form of the solution

Since , 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

The above inequality justifies that each solution is a global solution to regularized equation, and

Second, we show that, for some finite time , the sequence is a Cauchy sequence in . We note that, for ,

By the standard energy estimates, we have, for ,

If or , then we obtain easily that

If or , then we obtain

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

The above inequality gives us

By applying at the regularized equation, we deduce that

Now we are ready to show that is a Cauchy sequence (as a sequence for ), where is chosen as above.

By the properties of regularizer , for , we have

In summary, we have

By Gronwall’s inequality, we can conclude that is a Cauchy sequence in the space. And by boundness, if we apply the interpolation inequality, then we can see that is a cauchy sequence in , . So we have the limit function . And is also a cauchy sequence in the space by the following estimates:

For , we have the limit function .
Finally, we can show that . By the Banach Alaoglu theorem, we have and , because we know that, for any , . It implies by the following estimate:

for any given , for some , . Now we show that

By the weak continuity, for any , there exists such that if , then , for all . Choose with . Then it gives us

Also we have

which implies that

By the boundness with weak convergence, it is deduced that

which implies

Thus we have the time continuity of at 0. For any and initial value , we can obtain a right continuity at by the time translation invariant property. By the fact that is also a solution to the euqation for , we have a left continuity at . Of course by the above process, the left continuity at also can be obtained. We have proved that .

Proposition 11. Let be an axisymmetric global classical solution to

Then

Proof. Define which satisfies the equation . By our assumption, for , almost every time. So we can find the explicit form of it by

Now we choose initial values and such that . Then it satisfies

Because if we suppose that , then by the explicit form of , for some , we have which makes a contradiction. Hence, is a nondecreasing funcion with respect to . Since this process is independent of the choice of , we can find that by the continuity and 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 and :

where and .

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

Proposition 12. Assume that , with . Let be a smooth solution to (1) ( and ) for . If satisfies

then the solution can be extended beyond .

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

Proposition 13. If is a solution to (112)–(113) satisfying , with then it holds that

Proof. We first consider with . Taking scalar product of (113) with , we deduce that

From the divergence free condition and the decay conditions like

(116) can be reduced to

It implies that

For any , we have

As , we have If , then we have

Taking scalar product of (112) with , we have

Then we have

This completes the proof.

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

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

Taking scalar product of (125) with , we have

Using Gronwall’s inequality, we have

Hence we have if we assume (19).
If we consider the equations for ()), then we obtain

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

Taking scalar product of (128) with , we deduce that

In the above, we used the fact that when . Gronwall’s inequality again gives us

Multiplying both sides of (112) with and integrating over , we have

Then it is immediate that .
Following the ideas in [18], we introduce the angular stream function such that

For all , we have

By the interpolation inequality , we have

Therefore, we have . Also we can have for all .
If we multiply on the both sides of (9)₄ and integrate over , then we have

Hence we deduce that

Similarly, we have .
Setting , we have

By the maximal inequality, we have

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

Then, from (125), we have, for all ,

By Gronwall’s inequality, we have

If we let , we obtain . Hence, for any , we obtain that 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

Using an inequality

we have

By Gronwall’s inequality, we conclude that

The estimate of , , , and 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.

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.

References

T. G. Forbes, “Magnetic Reconnection in Solar Flares,” Geophysical & Astrophysical Fluid Dynamics, vol. 62, no. 1-4, pp. 15–36, 1991.

M. J. Lighthill, “Studies on magneto-hydrodynamic waves and other anisotropic wave motions,” Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, vol. 252, pp. 397–430, 1960.

P. A. Sturrock, Plasma Physics, Cambridge University Press, 1994.

S. A. Balbus and C. Terquem, “Linear analysis of the hall effect in protostellar disks,” The Astrophysical Journal , vol. 552, no. 1, pp. 235–247, 2001.

H. Homann and R. Grauer, “Bifurcation analysis of magnetic reconnection in Hall-{MHD}-systems,” Physica D: Nonlinear Phenomena, vol. 208, no. 1-2, pp. 59–72, 2005.

P. D. Mininni, D. O. Gómez, and S. M. Mahajan, “Dynamo action in magnetohydrodynamics and Hall-magnetohydrodynamics,” The Astrophysical Journal , vol. 587, no. 1 I, pp. 472–481, 2003.

D. A. Shalybkov and V. A. Urpin, “The Hall effect and the decay of magnetic fields,” Astron. Astrophys, vol. 32, pp. 685–690, 1997.

M. Wardle, “Star formation and the hall effect,” Astrophysics and Space Science, vol. 292, no. 1-4, pp. 317–323, 2004.

M. Acheritogaray, P. Degond, A. Frouvelle, and J. Liu, “Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system,” Kinetic and Related Models, vol. 4, no. 4, pp. 901–918, 2011.

D. Chae, P. Degond, and J. Liu, “Well-posedness for Hall-magnetohydrodynamics,” Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire, vol. 31, no. 3, pp. 555–565, 2014.

D. Chae and J. Lee, “On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics,” Journal of Differential Equations, vol. 256, no. 11, pp. 3835–3858, 2014.

M. Dai, “Regularity criterion for the 3D Hall-magneto-hydrodynamics,” Journal of Differential Equations, vol. 261, no. 1, pp. 573–591, 2016.

Y.-Z. Wang and W. Zuo, “On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity,” Communications on Pure and Applied Analysis, vol. 13, no. 3, pp. 1327–1336, 2014.

D. Chae and M. Schonbek, “On the temporal decay for the Hall-magnetohydrodynamic equations,” Journal of Differential Equations, vol. 255, no. 11, pp. 3971–3982, 2013.

S. Weng, “On analyticity and temporal decay rates of solutions to the viscous resistive Hall-MHD system,” Journal of Differential Equations, vol. 260, no. 8, pp. 6504–6524, 2016.

O. A. Ladyzhenskaya, “Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry,” Zap. Nauchn. Sem. Leningrad. Otdel. Mati. Inst. Steklov. (LOMI), vol. 7, pp. 155–177, 1968.

M. R. Ukhovskii and V. I. Iudovich, “Axially symmetric flows of ideal and viscous fluids filling the whole space,” Journal of Applied Mathematics and Mechanics, vol. 32, pp. 52–61, 1968.

Z. Lei, “On axially symmetric incompressible magnetohydrodynamics in three dimensions,” Journal of Differential Equations, vol. 259, no. 7, pp. 3202–3215, 2015.

J. Fan, S. Huang, and G. Nakamura, “Well-posedness for the axisymmetric incompressible viscous Hall-magnetohydrodynamic equations,” Applied Mathematics Letters, vol. 26, no. 9, pp. 963–967, 2013.

D. Chae and S. Weng, “Singularity formation for the incompressible Hall-MHD equations without resistivity,” Ann. Inst. Henri Poincare-Analyse Nonlineaire, vol. 33, no. 4, pp. 1009–1022, 2016.

D. Chae, R. Wan, and J. Wu, “Local well-posedness for the Hall-{MHD} equations with fractional magnetic diffusion,” Journal of Mathematical Fluid Mechanics, vol. 17, no. 4, pp. 627–638, 2015.

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, UK, 2002.