Abstract
In this paper, we prove the existence of global strong solutions to the Cauchy problem
of 2D incompressible magnetohydrodynamics (MHD) flows. Here, we emphasize that the
initial density
Keywords:
global strong solutions; 2D incompressible magnetohydrodynamics flows; vacuum states1 Introduction
The mathematical model of magnetohydrodynamics (MHD) is used to simulate the motion of a conducting fluid under the effect of the electromagnetic field and has a very wide range of applications in astrophysics, plasma, and so on. The governing equations of nonhomogeneous MHD can be stated as follows [1,2]:
with
and farfield behavior
where
The global wellposedness and dynamical behaviors of MHD system are rather difficult
to investigate because of the strong coupling and interplay interaction between the
fluid motion and the magnetic fields. Recently, there is much more important progress
on the mathematical analysis of these topics for the (nonhomogeneous or homogeneous)
MHD system (see, for example, [320]). Here, we only mention some of them. Kawashima [14] obtained the global existence of smooth solutions in the twodimensional case when
the initial data are a small perturbation of some given constant state. LiXuZhang
showed in [15] the global wellposedness and largetime behavior of classical solutions to the Cauchy
problem of compressible MHD for regular initial data with small energy but possibly
large oscillations. In [9,18], Hoff and Tsyganov obtained the global existence and uniqueness of weak solutions
with small initial energy. UmedaKawashimaShizuta [17] studied the global existence and time decay rate of smooth solutions to the linearized
twodimensional compressible MHD equations. The optimal decay estimates of classical
solutions to the compressible MHD system were obtained by ZhangZhao [20] when the initial data are close to a nonvacuum equilibrium. HuWang [10,11] and FanYu [8] proved the global existence of renormalized solutions to the compressible MHD equations
for general large initial data. When the viscosity and resistivity go to zero, Zhang
[19] showed that the solution of the Cauchy problem for the nonhomogeneous incompressible
MHD system converges to the solution of the ideal MHD system and the convergence rate
was also obtained. CraigHuangWang [7] obtained the global existence and uniqueness of strong solutions for initial data
with small
In [12], HuangWang considered the global strong solutions to (1)(4) in the bounded domain
with suitable boundary conditions on u and B. Their arguments actually depend on the size of the domain, and so they cannot be
applied to the Cauchy problem directly. Then one natural question may be raised: whether
the global strong solutions exist in the whole space
Theorem 1.1Assume that the initial data
where
The proof of Theorem 1.1 is mainly based on a critical Sobolev inequality of logarithmic type which was recently proved by HuangWang [12] and is originally due to BrezisWainger [21]. The main difficulty compared with [12] is that we should bound all the desired estimates without the restriction on the size of the domain, especially that the Poincaré inequality is not the same from the bounded domain to the whole spaces.
For convenience, we explain the notions used throughout this paper. Set
The standard homogeneous and inhomogeneous Sobolev spaces are defined as follows:
for
The paper is organized as follows. In Section 2, we state some wellknown inequalities and basic facts which will be used frequently later. The proof of Theorem 1.1 will be cast in Section 3.
2 Preliminaries
In this section, we list some useful lemmas which will be frequently used in the next sections. We start from the local existence of strong solutions, which is similar to [4] or [14].
Lemma 2.1Assume that the conditions of Theorem 1.1 hold. Then there exists a positive time
Next is the wellknown GagliardoNirenberg inequality (see [22]).
Lemma 2.2For
where
Next, we list the Poincaré type inequality, which yields
Lemma 2.3Assume that
whereCdepends only on
Proof The proof of this lemma can easily be deduced by (9), Hölder’s inequality and the following equality:
so the details are omitted here. □
In the following, in order to improve the regularity of the velocity, we need to use the estimates of the Stokes equations. We refer the reader to [23,24] for details.
Lemma 2.4Consider the following stationary Stokes equations:
Then for any
To improve the regularity of the magnetic fields, we need the following result on the elliptic system.
Lemma 2.5Assume that
where
with some constantCdepending only onq.
To bound the
Lemma 2.6For
3 Proof of Theorem 1.1
This section is devoted to obtaining the proof of Theorem 1.1. According to Lemma 2.1,
a local strong solution of the Cauchy problem (1)(6) exists. Suppose
where
The proof of (15) is based on a series of lemmas. For simplicity, throughout the remainder
of this paper, we denote by C a generic constant which depends only on the initial data and
First, the
Lemma 3.1For every
Next, the basic energy inequalities are used.
Lemma 3.2For every
The following estimates are the key estimates in the proof of Theorem 1.1, which depends on the critical Sobolev inequality of logarithmic type (see Lemma 2.6).
Lemma 3.3For every
where
Proof First, multiplying (2) by
For the first term on the righthand side of (19), using Young’s inequality and (16), one shows that
Next, the second term can be deduced as follows:
Then, substituting the above two estimates into (19), one obtains
Multiplying (3) by
The term
Then multiplying (21) by
To proceed, we have to estimate
For convenience, we denote
Then, combining (17), (22), and (23), we conclude that
To proceed, we have to get the appropriate bound on
where we have used (11) and (24). Similarly, we conclude from (12), (11), and (24) that
Hence, combining (25) and (26), we obtain
Thus, keeping the definition of
Substituting the above estimate into (25), we conclude that
It follows from the basic energy estimate that one can choose the interval
Substituting the above estimate into (25), we conclude that
which implies that
from which we complete the proof of this lemma. □
Remark 3.1 Due to (18) and the definition of the material derivative
by the following simple fact, i.e.:
where we have used (9), (16), (17), and (18).
The following lemma is devoted to improving the time regularity of u and B.
Lemma 3.4For every
Proof Differentiating (2) with respect to t, we obtain
Multiplying the above equation by
Now, we estimate each term on the righthand side of (31). First, due to (1), we have
Next, it follows from (1), (18), Hölder’s inequality, and Young’s inequality that
Then one obtains
where we have used (9), (11), and (24). Finally, as for
Hence, substituting all the above estimates into (31), we conclude that
From now on, we focus on the estimate for B. Differentiating equation (3) with respect to t, multiplying the resulting equation by
We estimate each term on the righthand side of (33). First, for
Similarly, for
For
Then, substituting the above estimates on
Thus, combining (32) and (34), together with Gronwall’s inequality, one easily completes the proof of (30). This completes the proof of Lemma 3.4. □
Next, we will apply (12) and (13) to improve the higher regularity on the velocity u and magnetic fields B, respectively.
Lemma 3.5For every
Proof Let us rewrite (2) in the following form:
Then, using Lemma 2.4, we conclude that
Similarly, due to Lemma 2.5, we obtain
Thus, combining the above two inequalities and Young’s inequality, we arrive at
Then, by Lemmas 2.4 and 2.5, we have
and
Then, combining all the above estimates (36)(40) together we show that (35). This completes the proof of Lemma 3.5. □
Lemma 3.6For every
Proof Differentiating (1) with respect to
which, combined with (35) and Gronwall’s inequality, yields
Similarly, we can also obtain from (1) that
which combined with (42), together with (35) and Gronwall’s inequality, yields
It follows from (12) that
which implies
The proof of Theorem 1.1 is based on all the estimates that we deduced in Lemmas 3.13.6. From all the estimates obtained, we arrive at (15), and, finally, the proof of Theorem 1.1 is therefore completed.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
MS carried out the main work and drafted the manuscript. XQ participated in completing the proof of Lemma 3.5. JW participated in completing the proof of Lemma 3.6. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by NSFCUnion Science Foundation of Henan (No. U1304103) and Natural Science Foundation of Henan Province (No. 122300410261).
References

Cabannes, H: Theoretical Magnetofluiddynamics, Academic Press, New York (1970)

Kulikovskiy, AF, Lyubimov, GA: Magnetohydrodynamics, AddisonWesley, Reading (1965)

Abidi, H, Paicu, M: Global existence for the magnetohydrodynamic system in critical spaces. Proc. R. Soc. Edinb., Sect. A. 138, 447–476 (2008)

Chen, Q, Tan, Z, Wang, Y: Strong solutions to the incompressible magnetohydrodynamic equations. Math. Methods Appl. Sci.. 34, 94–107 (2011). Publisher Full Text

Chen, GQ, Wang, D: Global solution of nonlinear magnetohydrodynamics with large initial data. J. Differ. Equ.. 182, 344–376 (2002). Publisher Full Text

Chen, GQ, Wang, D: Existence and continuous dependence of large solutions for the magnetohydrodynamic equations. Z. Angew. Math. Phys.. 54, 608–632 (2003). Publisher Full Text

Fan, J, Yu, W: Strong solution to the compressible MHD equations with vacuum. Nonlinear Anal., Real World Appl.. 10, 392–409 (2009). Publisher Full Text

Hoff, D, Tsyganov, E: Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics. Z. Angew. Math. Phys.. 56, 791–804 (2005). Publisher Full Text

Hu, X, Wang, D: Global solutions to the threedimensional full compressible magnetohydrodynamic flows. Commun. Math. Phys.. 283, 255–284 (2008). Publisher Full Text

Hu, X, Wang, D: Global existence and largetime behavior of solutions to the threedimensional equations of compressible magnetohydrodynamic flows. Arch. Ration. Mech. Anal.. 197, 203–238 (2010). Publisher Full Text

Huang, X, Wang, Y: Global strong solution to the 2D nonhomogeneous incompressible MHD system. J. Differ. Equ.. 254, 511–527 (2013). Publisher Full Text

Kawashima, S: Smooth global solutions for twodimensional equations of electromagnetofluid dynamics. Jpn. J. Appl. Math.. 1, 207–222 (1984). Publisher Full Text

Li, H, Xu, X, Zhang, J: Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum. SIAM J. Math. Anal.. 45, 1356–1387 (2013). Publisher Full Text

Wang, D: Large solutions to the initialboundary value problem for planar magnetohydrodynamics. SIAM J. Appl. Math.. 63, 1424–1441 (2003). Publisher Full Text

Umeda, T, Kawashima, S, Shizuta, Y: On the decay of solutions to the linearized equations of electromagnetofluid dynamics. Jpn. J. Appl. Math.. 1, 435–457 (1984). Publisher Full Text

Tsyganov, E, Hoff, D: Systems of partial differential equations of mixed hyperbolicparabolic type. J. Differ. Equ.. 204, 163–201 (2004). Publisher Full Text

Zhang, J: The inviscid and nonresistive limit in the Cauchy problem for 3D nonhomogeneous incompressible magnetohydrodynamics. Acta Math. Sci.. 31, 882–896 (2011). Publisher Full Text

Zhang, J, Zhao, J: Some decay estimates of solutions for the 3D compressible isentropic magnetohydrodynamics. Commun. Math. Sci.. 8, 835–850 (2010). Publisher Full Text

Brezis, H, Wainger, S: A note on the limiting cases of Sobolev embedding and convolution inequalities. Commun. Partial Differ. Equ.. 5, 773–789 (1980). Publisher Full Text

Galdi, GP: An Introduction to the Mathematical Theory of NavierStokes Equations. Volume I: Linearized Steady Problems, Springer, Berlin (1994)

Ladyzhenskaya, OA: The Mathematical Theory of Viscous Incompressible Fluids, Gordon & Breach, New York (1969)