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 .where