Abstract
In this paper, we focus on the generalized 3D magnetohydrodynamic equations. Two logarithmically blowup criteria of smooth solutions are established.
MSC: 76D03, 76W05.
Keywords:
generalized MHD equations; blowup criteria1 Introduction
We study blow up criteria of smooth solutions to the incompressible generalized magnetohydrodynamics (GMHD) equations in
with the initial condition
Here , and are nondimensional quantities corresponding to the flow velocity, the magnetic field and the total kinetic pressure at the point , while and are the given initial velocity and initial magnetic field with and , respectively.
The GMHD equations is a generalized model of MHD equations. It has important physical background. Therefore, the GMHD equations are also mathematically significant. For 3D NavierStokes equations, whether there exists a global smooth solution to 3D impressible GMHD equations is still an open problem. In the absence of global wellposedness, the development of blowup/ non blowup theory is of major importance for both theoretical and practical purposes. Fundamental mathematical issues such as the global regularity of their solutions have generated extensive research and many interesting results have been established (see [15]).
When , (1.1) reduces to MHD equations. There are numerous important progresses on the fundamental issue of the regularity for the weak solution to (1.1), (1.2) (see [618]). A criterion for the breakdown of classical solutions to (1.1) with zero viscosity and positive resistivity in was derived in [9]. Some sufficient integrability conditions on two components or the gradient of two components of and in MorreyCampanato spaces were obtained in [10]. A logarithmal improved blowup criterion of smooth solutions in an appropriate homogeneous Besov space was obtained by Wang et al.[11]. Zhou and Fan [15] established various logarithmically improved regularity criteria for the 3D MHD equations in terms of the velocity field and pressure, respectively. These regularity criteria can be regarded as log in time improvements of the standard Serrin criteria established before. Two new regularity criteria for the 3D incompressible MHD equations involving partial components of the velocity and magnetic fields were obtained by Jia and Zhou [17].
When , , (1.1) reduces to NavierStokes equations. Leray [19] and Hopf [20] constructed weak solutions to the NavierStokes equations, respectively. The solution is called the LerayHopf weak solution. Later on, much effort has been devoted to establish the global existence and uniqueness of smooth solutions to the NavierStokes equations. Different criteria for regularity of the weak solutions have been proposed and many interesting results have been obtained [2125].
In the paper, we obtain two logarithmically blowup criteria of smooth solutions to (1.1), (1.2) in MorreyCampanato spaces. We hope that the study of equations (1.1) can improve the understanding of the problem of NavierStokes equations and MHD equations.
Now we state our results as follows.
Theorem 1.1Let, , with, and. Assume thatis a smooth solution to (1.1), (1.2) on. Ifusatisfies
then the solutioncan be extended beyond.
We have the following corollary immediately.
Corollary 1.1Let, , with, and. Assume thatis a smooth solution to (1.1), (1.2) on. Suppose thatTis the maximal existence time, then
Theorem 1.2Let, , with, and. Assume thatis a smooth solution to (1.1), (1.2) on. If
then the solutioncan be extended beyond.
We have the following corollary immediately.
Corollary 1.2Let, , with, and. Assume thatis a smooth solution to (1.1), (1.2) on. Suppose thatTis the maximal existence time, then
The paper is organized as follows. We first state some preliminaries on function spaces and some important inequalities in Section 2. Then we prove main results in Section 3 and Section 4, respectively.
2 Preliminaries
Before stating our main results, we recall the definition and some properties of the homogeneous MorreyCampanato space.
Definition 2.1 For , the MorreyCampanato space is defined by
where denotes the ball of center x with radius R.
Let , we define the homogeneous space by
where is the space of all functions in with compact support. is a Banach space when it is equipped with the norm
where the infimum is taken over all possible decompositions.
Lemma 2.1Letandp, qsatisfy. Thenis the dual space of.
Then there exists a constantsuch that for anyand,
The following lemma comes from [21].
Lemma 2.3Assume that. For, and, , we have
The following inequality is the wellknown GagliardoNirenberg inequality.
Lemma 2.4Letj, mbe any integers satisfying, and let, and, be such that
Then, for all, there is a positive constantCdepending only onn, m, j, q, r, θsuch that the following inequality holds:
with the following exception: ifandis a nonnegative integer, then (2.2) holds only for a satisfying.
3 Proof of Theorem 1.1
Proof Let . We multiply the first equation of (1.1) by and use integration by parts. This yields
Similarly, we obtain
Summing up (3.1) and (3.2), we deduce that
By using Lemmas 2.1, 2.2 and (2.2), we have
We apply integration by parts, , Lemmas 2.1, 2.2 and (2.2). This gives
Similarly, we obtain
Substituting (3.4)(3.6) into (3.3) yields
Owing to (1.3), we know that for any small constant , there exists such that
By (3.7), we obtain
It follows from (3.8) and Gronwall’s inequality that
Applying to the first equation of (1.1), then taking inner product of the resulting equation with and using integration by parts, we get
Similarly, we have
Combining (3.10)(3.11), using , and integration by parts yields
In what follows, for simplicity, we set .
Using Hölder’s inequality and (2.1), (2.2), we have
Similarly, we have
and
Inserting (3.13)(3.16) into (3.12) yields
Gronwall’s inequality implies the boundedness of norm of u and B provided that , which can be achieved by the absolute continuous property of integral (1.3). We have completed the proof of Theorem 1.1. □
4 Proof of Theorem 1.2
Proof Let . Multiplying the first equation of (1.1) by and using integration by parts, we obtain
Similarly, we get
Summing up (4.1) and (4.2), we deduce that
Using Lemmas 2.1, 2.2 and (2.2), we obtain
Similarly, we obtain
and
Combining (4.3)(4.7) yields
Thanks to (1.5), we know that for any small constant , there exists such that
By (4.8) and (4.9), we obtain
Equation (4.10) and Gronwall’s inequality give the estimate
From (4.11), estimate for this case is the same as that for Theorem 1.1. Thus, Theorem 1.2 is proved. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
LH and YW carried out the proof of the main part of this article. All authors have read and approved the final manuscript.
References

Luo, Y: On the regularity of the generalized MHD equations. J. Math. Anal. Appl.. 365, 806–808 (2010). Publisher Full Text

Wu, G: Regularity criteria for the 3D generalized MHD equations in terms of vorticity. Nonlinear Anal.. 71, 4251–4258 (2009). Publisher Full Text

Yuan, J: Existence theorem and regularity criteria for the generalized MHD equations. Nonlinear Anal., Real World Appl.. 11, 1640–1649 (2010). Publisher Full Text

Zhang, Z: Remarks on the regularity criteria for generalized MHD equations. J. Math. Anal. Appl.. 375, 799–802 (2011). Publisher Full Text

Zhou, Y: Regularity criteria for the generalized viscous MHD equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 24, 491–505 (2007). Publisher Full Text

Gala, S: Extension criterion on regularity for weak solutions to the 3D MHD equations. Math. Methods Appl. Sci.. 33, 1496–1503 (2010)

He, C, Xin, Z: On the regularity of solutions to the magnetohydrodynamic equations. J. Differ. Equ.. 213, 235–254 (2005). Publisher Full Text

He, C, Wang, Y: On the regularity for weak solutions to the magnetohydrodynamic equations. J. Differ. Equ.. 238, 1–17 (2007). Publisher Full Text

Lei, Z, Zhou, Y: BKM’s criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete Contin. Dyn. Syst.. 25, 575–583 (2009)

Wang, Y, Wang, S, Wang, Y: Regularity criteria for weak solution to the 3D magnetohydrodynamic equations. Acta Math. Sci.. 32, 1063–1072 (2012)

Wang, Y, Zhao, H, Wang, Y: A logarithmically improved blow up criterion of smooth solutions for the threedimensional MHD equations. Int. J. Math.. 23, (2012) Article ID 1250027

Zhou, Y: Remarks on regularities for the 3D MHD equations. Discrete Contin. Dyn. Syst.. 12, 881–886 (2005)

Zhou, Y, Gala, S: Regularity criteria for the solutions to the 3D MHD equations in the multiplier space. Z. Angew. Math. Phys.. 61, 193–199 (2010). Publisher Full Text

Zhou, Y, Gala, S: A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field. Nonlinear Anal.. 72, 3643–3648 (2010). Publisher Full Text

Zhou, Y, Fan, J: Logarithmically improved regularity criteria for the 3D viscous MHD equations. Forum Math.. 24, 691–708 (2012)

Jia, X, Zhou, Y: Regularity criteria for the 3D MHD equations via partial derivatives. Kinet. Relat. Models. 5, 505–516 (2012)

Jia, X, Zhou, Y: Regularity criteria for the 3D MHD equations involving partial components. Nonlinear Anal., Real World Appl.. 13, 410–418 (2012). Publisher Full Text

Ni, L, Guo, Z, Zhou, Y: Some new regularity criteria for the 3D MHD equations. J. Math. Anal. Appl.. 396, 108–118 (2012). Publisher Full Text

Leray, J: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math.. 63, 183–248 (1934)

Hopf, E: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr.. 4, 213–231 (1951)

Kato, T, Ponce, G: Commutator estimates and the Euler and NavierStokes equations. Commun. Pure Appl. Math.. 41, 891–907 (1988). Publisher Full Text

LemariéRieusset, PG: Recent Developments in the NavierStokes Problem, CRC Press, Boca Raton (2004)

Mazya, VG: On the theory of the ndimensional Schrödinger operator. Izv. Akad. Nauk SSSR, Ser. Mat.. 28, 1145–1172 (1964)

Zhou, Y, Gala, S: Logarithmically improved regularity criteria for the NavierStokes equations in multiplier spaces. J. Math. Anal. Appl.. 356, 498–501 (2009). PubMed Abstract  Publisher Full Text

Fan, J, Jiang, S, Nakamura, G, Zhou, Y: Logarithmically improved regularity criteria for the NavierStokes and MHD equations. J. Math. Fluid Mech.. 13, 557–571 (2011). Publisher Full Text