Global existence of 2D nonhomogeneous incompressible magnetohydrodynamics with vacuum
© Su et al.; licensee Springer. 2014
Received: 4 January 2014
Accepted: 31 March 2014
Published: 6 May 2014
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 is permitted to contain vacuum states, and the initial velocity and magnetic fields can be arbitrarily large.
where is some fixed positive constant.
The global well-posedness 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, [3–20]). Here, we only mention some of them. Kawashima  obtained the global existence of smooth solutions in the two-dimensional case when the initial data are a small perturbation of some given constant state. Li-Xu-Zhang showed in  the global well-posedness and large-time 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. Umeda-Kawashima-Shizuta  studied the global existence and time decay rate of smooth solutions to the linearized two-dimensional compressible MHD equations. The optimal decay estimates of classical solutions to the compressible MHD system were obtained by Zhang-Zhao  when the initial data are close to a nonvacuum equilibrium. Hu-Wang [10, 11] and Fan-Yu  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  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. Craig-Huang-Wang  obtained the global existence and uniqueness of strong solutions for initial data with small -norm in the bounded or unbounded domain in .
In , Huang-Wang 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 . Here, we want to answer the question. Our main result is stated as follows.
The proof of Theorem 1.1 is mainly based on a critical Sobolev inequality of logarithmic type which was recently proved by Huang-Wang  and is originally due to Brezis-Wainger . The main difficulty compared with  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 and .
The paper is organized as follows. In Section 2, we state some well-known inequalities and basic facts which will be used frequently later. The proof of Theorem 1.1 will be cast in Section 3.
Lemma 2.1 Assume that the conditions of Theorem 1.1 hold. Then there exists a positive time such that the Cauchy problem (1)-(6) admits a unique strong solution on .
Next is the well-known Gagliardo-Nirenberg inequality (see ).
Next, we list the Poincaré type inequality, which yields even when the vacuum states appear.
where C depends only on , , and .
so the details are omitted here. □
To improve the regularity of the magnetic fields, we need the following result on the elliptic system.
with some constant C depending only on q.
To bound the -norm of the gradient of the velocity, we will apply a critical Sobolev inequality of logarithmic type which was proved by Huang-Wang . This is the key tool for the proof of Theorem 1.1.
3 Proof of Theorem 1.1
where . Then due to the local existence theorem (Lemma 2.1), it can easily be shown that the strong solution can be extended beyond . This conclusion contradicts the assumption on . Thus, the strong solution exists globally on for any . Hence the proof of Theorem 1.1 is therefore completed.
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 and may change from line to line.
First, the -norm of the density can be obtained easily by using the method of characteristics, we list the following lemma without proof.
Next, the basic energy inequalities are used.
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).
where is the material derivative of f.
from which we complete the proof of this lemma. □
where we have used (9), (16), (17), and (18).
The following lemma is devoted to improving the time regularity of u and B.
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.
Then, combining all the above estimates (36)-(40) together we show that (35). This completes the proof of Lemma 3.5. □
which implies . Similarly, we can obtain . Thus, we obtain (41), and thus complete the proof of Lemma 3.6. □
The proof of Theorem 1.1 is based on all the estimates that we deduced in Lemmas 3.1-3.6. From all the estimates obtained, we arrive at (15), and, finally, the proof of Theorem 1.1 is therefore completed.
This work was supported by NSFC-Union Science Foundation of Henan (No. U1304103) and Natural Science Foundation of Henan Province (No. 122300410261).
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