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

Jeong, Eunji; Kim, Junha; Lee, Jihoon

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

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Research Article | Open Access

Volume 2018 | Article ID 5343824 | https://doi.org/10.1155/2018/5343824

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

Eunji Jeong,¹Junha Kim,¹and Jihoon Lee¹

Academic Editor: Carlo Bianca

Received16 Jul 2018

Accepted26 Sept 2018

Published01 Nov 2018

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 satisfyNote 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 coordinatesand standard basis vectors for the cylindrical coordinateswe setIt 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 thatFurthermore, 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 satisfiesThe 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 thatif 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 equationswhere , , , 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 equationThus, we haveIf , 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. Letwhere 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 obtainand ,For an integer and all integers , we haveHence, 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 setFirst, we show that is Lipschitz continuous on space.
We estimate for , By the similar estimates as in (34), we obtainBy 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 haveIf 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 thatwhere can be estimated as follows:Similarly, we can obtain the estimates for .whereThese can be estimated similarly.The other terms and can be estimated similarly, so we omit the details. Then we havefor . Gronwall’s inequality gives uswhich 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 inequalityNow, 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 haveIf 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 haveBy 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 bywhich 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 setThen forwe 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 withThen 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 equationhas 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. LetHence, 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 solutionSince , we haveSince the above regularized equation satisfies an energy estimate, we deduce thatand henceFor the higher order norm, Gronwall’s inequality impliesThe above inequality justifies that each solution is a global solution to regularized equation, andSecond, 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 thatIf or , then we obtainCombining the above inequalities (93) and (94), we haveThe above inequality gives usBy applying at the regularized equation, we deduce thatNow 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 haveIn summary, we haveBy 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 thatBy the weak continuity, for any , there exists such that if , then , for all . Choose with . Then it gives us Also we havewhich implies thatBy the boundness with weak convergence, it is deduced thatwhich impliesThus 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 toThen

Proof. Define which satisfies the equation . By our assumption, for , almost every time. So we can find the explicit form of it byNow 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 satisfiesthen 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 thatFrom the divergence free condition and the decay conditions like (116) can be reduced toIt implies thatFor any , we haveAs , we have If , then we haveTaking scalar product of (112) with , we haveThen we haveThis 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 haveTaking scalar product of (125) with , we haveUsing Gronwall’s inequality, we haveHence we have if we assume (19).
If we consider the equations for ()), then we obtainVia an interpolation inequality and Young’s inequality, we haveTaking scalar product of (128) with , we deduce thatIn the above, we used the fact that when . Gronwall’s inequality again gives usMultiplying both sides of (112) with and integrating over , we haveThen it is immediate that .
Following the ideas in [18], we introduce the angular stream function such thatFor all , we haveBy the interpolation inequality , we haveTherefore, we have . Also we can have for all .
If we multiply on the both sides of (9)₄ and integrate over , then we haveHence we deduce thatSimilarly, we have .
Setting , we haveBy the maximal inequality, we haveBy the Gagliardo-Nirenberg inequality (139) can be reduced toSince the last term in the above can be absorbed in the left hand side, we haveThen, from (125), we have, for all ,By Gronwall’s inequality, we haveIf 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 obtainUsing an inequalitywe haveBy Gronwall’s inequality, we conclude thatThe 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.

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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.

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