Abstract
In this paper we consider an initial value problem of the 2D MHD equations with velocity dissipation and without magnetic diffusion. We establish a new magnetic regularity criterion in terms of the magnetic field. In contrast to the magnetic regularity criterion \(\nabla b\in L^{1}(0,T; BMO(\mathbb {R}^{2}))\), our regularity criterion \(\int_{0}^{T} (\Vert b\otimes b(s)\Vert _{B_{\infty,1}^{0}(\mathbb {R}^{2})} +\Vert b\otimes b(s)\Vert _{L^{2}(\mathbb {R}^{2})} )\,ds<\infty\) is different; for example, our simplified regularity criterion \(\int_{0}^{T}\Vert b(s)\Vert ^{2}_{B_{\infty,1}^{\varepsilon}(\mathbb {R}^{2})}\,ds<\infty\) requires higher time integrability and lower regularity of space.
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1 Introduction
In this paper we consider the global regularity on the 2D incompressible magnetohydrodynamic (MHD) equations with velocity dissipation and without magnetic diffusion,
where \(u=u(t,x)\) stands for the velocity of the fluid, \(b=b(t,x)\) for the magnetic field, and \(p=p(t,x)\) for the scalar pressure. Due to the lack of magnetic diffusion, the global well-posedness is extremely difficult and remains open.
The study of basic equations in fluid kinematics is one of the interesting fields. For example, we have the MHD equations [1, 2], the Benjamin-Bona-Mahony equations [3, 4], and the quasi-geostrophic equations [5–7]. Since the MHD equations have strong physical backgrounds [1, 2], the study of the MHD equations has attracted considerable interest and much progress has been made in the last few years. One of the fundamental problems regarding the MHD equations is that they develop singularities. This is due to the nonlinear coupling between the Navier-Stokes equations with a forcing induced by the magnetic field and the induction equation (see [8–10]).
We first recall some of the recent progress in this direction. The local well-posedness for the system (1.1) has been proved in [11–13], respectively. Furthermore, Jiu and Niu [13] proved that the solution keeps smoothness up to time T if
Jiu and Liu [14] discussed the global regularity for the 3D axisymmetric MHD equations with horizontal dissipation and vertical magnetic diffusion. Fan and Ozawa [15], and Zhou and Fan [16] obtained a regularity criterion on the velocity field \(\nabla u\in L^{1}(0,T; L^{\infty}(\mathbb {R}^{2}))\) and the magnetic field \(\nabla b\in L^{1}(0,T; BMO(\mathbb {R}^{2}))\), respectively. To the best of our knowledge, for the MHD equations (1.1), when the indicator of dissipation is larger than 1 and without zero magnetic diffusion the global well-posedness is also open. Due to lack of the magnetic diffusion, it is very difficult to get global estimates of the local solution in any Sobolev spaces. Very recently, Jiu, Niu, and Wu et al. [17] gave a new regularity criterion by the Besov space technique. Motivated by the ideas described in [17] and [18], the main goal of this paper is to establish another regularity criterion in terms of the condition on the magnetic field. Our main result is the following.
Theorem 1.1
Assume that \((u_{0}(x), b_{0}(x))\in H^{s}(\mathbb {R}^{2}) \) (\(s>2\)) with \(\nabla\cdot u_{0}(x)=\nabla\cdot b_{0}(x)=0\). Let \((u(t,x), b(t,x))\) be a local smooth solution of system (1.1). Then \((u(t,x), b(t,x))\) can be extended beyond time T if
Remark 1.1
\(\forall\varepsilon>0\), \(B^{\varepsilon }_{\infty,1}\) is a Banach algebra and the embedding \(B_{\infty ,1}^{\varepsilon}\hookrightarrow B_{\infty,1}^{0}\hookrightarrow L^{\infty}\) hold, the condition (1.2) can be replaced by
Based on the above observation, the condition (1.3) demands higher time integrability and lower regularity of space than the regularity condition imposed by Zhou and Fan [16].
The plan of this paper is as follows. In the next section, we state some notations and preliminary results in the standard theory of Besov spaces. In the third section, we first establish all tools needed to get a magnetic regularity, then we divide the proof into three steps to get the magnetic regularity criterion.
2 Notations and preliminaries
We first introduce the following notations. C denotes a positive constant which may vary from line to line. \(X\lesssim Y\) means that there exists a positive harmless constant C such that \(X\leq CY\). We use the sub-indices (like \(C_{s}\) or \(\lesssim_{s}\)) to indicate the parameter dependence of the constant C. Let \(\mathcal{S}(\mathbb {R}^{d})\) be the Schwartz class of rapidly decreasing functions and \(\mathcal{S'}(\mathbb{R}^{d})\) the space of tempered distributions. For all \(u\in\mathcal{S}'\), the Fourier transform \(\mathcal{F}u\), also denoted by û, is defined by
The inverse Fourier transform allows us to recover u from û:
\([A,B]\) stands for the commutator operator \(AB-BA\), where A and B are any pair of operators on some Banach space.
We now recall some standard theories of Besov space (more details see [19]).
Let \(\mathcal{C}\) denote the annulus \(\{\xi\in \mathbb {R}^{d}:3/4\leq \vert \xi \vert \leq 8/3\}\). There exist two radial functions \(\chi\in C_{c}^{\infty}(\mathcal {B}(0,4/3))\) and \(\varphi\in C_{c}^{\infty}(\mathcal{C})\) both taking values in \([0,1]\) such that
For every \(u\in\mathcal{S'}(\mathbb{R}^{d})\), the inhomogeneous dyadic blocks \(\triangle_{j}\) are defined as follows: \(\triangle_{-1}u=\chi(D)u\) and \(\triangle_{j}u=\varphi(2^{-j}D)u\), \(\forall j\geq0\). The inhomogeneous low-frequency cut-off operator \(S_{j}\) is defined by
In the inhomogeneous case, the following Littlewood-Paley decomposition makes sense:
Let \(s \in \mathbb {R}\) and \(p, q \in[1,\infty]\), the inhomogeneous Besov space \(B_{p,q}^{s}(\mathbb{R}^{d})\) is defined by
In this paper, two kinds of the coupled space-time Besov spaces \(L_{T}^{r}B_{p,q}^{s}\) and \(\widetilde{L}_{T}^{r}B_{p,q}^{s}\) (\(r\geq1\)) are defined, respectively, as follows:
The following links between these spaces are direct results due to the Minkowski inequality:
In particular,
Bernstein’s inequalities are fundamental in the analysis involving Besov spaces and these inequalities trade integrability for derivatives.
Lemma 2.1
[19]
Let \(\mathcal{C}\) be an annulus and \(\mathcal{B}\) a ball. Then there is a constant such that for all \(k\in\mathbb{N}\cup\{0\}\), \(1\leq p \leq q \leq\infty\), and \(f\in L^{p}\), we have
The Biot-Savart law will be used often to get the control between the gradient of velocity and the vorticity.
Lemma 2.2
[19]
For any divergence-free vector field u, there exists a universally positive constant C such that, for any \(1< p<\infty\), we have
here \(w=\operatorname{curl} u=\nabla\times u\) is the vorticity.
Next, we state a commutator estimate involving the Riesz operator \(\mathcal{R}=(-\Delta)^{-1}\operatorname{curldiv}\).
Lemma 2.3
[20]
The standard Riesz operator \(\mathcal{R}=(-\Delta)^{-1}\operatorname{curldiv}\) is continuous and linear, it maps \(L^{p}(\mathbb {R}^{d})\) into \(L^{p}(\mathbb {R}^{d})\) for any \(1 < p < \infty\). In particular, for all \(f\in L^{p}(\mathbb {R}^{d})\) the following estimate holds true:
Lemma 2.4
[20]
If u is a smooth divergence-free vector field of \(\mathbb {R}^{2}\) with vorticity, and f is a smooth function, then for all \(p\in(1, \infty)\),
Proof
For the sake of convenience, we sketch the proof. Without loss of generality, we assume that the functions \(u\in C_{c}^{\infty}(\mathbb {R}^{2})\) and \(f\in C_{c}^{\infty}(\mathbb {R}^{2})\). It is easy to verify directly that
Due to Hölder’s inequality, Bernstein’s inequality, and the embedding \(B_{\infty,1}^{0}\hookrightarrow L^{\infty}\), we get
Moreover, it is easy to see that both inequalities (2.8) and (2.9) can be extended to all functions by a simple density argument. □
3 Proof of Theorem 1.1
In this section we prove our main result Theorem 1.1. The strategy of the proof is as follows. We first prove the global a priori bounds for \(\Vert w\Vert _{H^{1}}\) and \(\Vert j\Vert _{H^{1}}\). Then we divide the proof into three steps: (1) the \(L^{p}\) (\(2< p<\infty\)) estimate of the vorticity ω, (2) the gradient estimate of the velocity u, (3) the energy estimate of the vorticity ω and j.
Now we act with the operator curl on the velocity equation in (1.1) and obtain the following vorticity equation:
Multiplying the ith component of the magnetic equation (1.1) by \(b_{j}\), we obtain
similarly, multiplying the jth component of the magnetic equation (1.1) by \(b_{i}\), we have
Adding (3.2) and (3.3), we know that the \((i,j)\)th component of \(b\otimes b\) satisfies
equivalently,
where \(\nabla^{\top}u\) denotes the transposed matrix to ∇u.
Applying \(\mathcal{R}=(-\Delta)^{-1}\operatorname{curldiv}\) to (3.5) yields
Set \(G=w-\mathcal{R}(b\otimes b)\). Combining (3.1) and (3.6), we get
By the transport-diffusion equation (3.7), we can obtain the following desired bounded estimate.
Lemma 3.1
Assume that \((u_{0}(x), b_{0}(x))\) fulfills the conditions in Theorem 1.1. Let \((u(t,x), b(t,x))\) be the corresponding solution of the initial value problem (1.1). Then, for \(p\in(2,\infty)\) and for any \(T>0\), we have
where C is a positive constant depending only on T and the initial data.
Proof
Multiplying equation (3.7) by \(\vert G\vert ^{p-2}G\) and integrating over \(\mathbb {R}^{2}\), using the integration by parts and \(\operatorname{div} u=0\), we have
Due to the pointwise inequality \(\int_{\mathbb {R}^{2}}(-\Delta) G\vert G\vert ^{p-2}G\,dx\geq 0\) and the Hölder inequality, we have
Since the singular integral type operator \(\mathcal{R}\) is bounded on \(L^{p}(\mathbb {R}^{2})\) (\(1< p<\infty\)), we have
Due to Lemma 2.4, we have
Putting (3.12) into (3.11) and using the classical embedding \(B_{\infty,1}^{0}\hookrightarrow B_{\infty ,2}^{0}\hookrightarrow L^{\infty}\), we get
by the Biot-Savart law (Lemma 2.2), we have
as \(\omega=G+\mathcal{R}(b\otimes b)\), we have
where the \(L^{p}\) boundedness of Riesz operator has been used in the last inequality.
Multiplying equation (3.5) by \(\vert b\otimes b\vert ^{p-2}(b\otimes b)\) and integrating over \(\mathbb {R}^{2}\), using integration by parts and \(\operatorname{div}u=0\), we have
Hölder’s inequality and the Biot-Savart law (Lemma 2.2) yield
Combining the estimates (3.13) and (3.14), we get
Assuming
by Gronwall’s inequality we have
This implies that, for any \(2< p<\infty\),
This completes the proof of Lemma 3.1. □
Next, we give a key bounded estimate which is crucial in the proof of Theorem 1.1.
Lemma 3.2
Assume that \((u_{0}(x), b_{0}(x))\) fulfills the conditions in Theorem 1.1. Let \((u(t,x), b(t,x))\) be the corresponding solution of the initial value problem (1.1). Then, for \(p\in(2,\infty)\) and for any \(T>0\),
where C is a positive constant depending only on T and the initial data.
Proof
In view of (3.7), for \(j\geq-1\), acting with the Littlewood-Paley operator \(\triangle_{j}\) on (3.7), one has
For convenience, we take
Thus, equation (3.20) is written as
Multiplying equation (3.21) by \(\vert \triangle_{j}G \vert ^{q-2}\triangle_{j}G \) with \(q>2 \) and integrating over \(\mathbb {R}^{2}\), with the help of the Hölder inequality and \(\operatorname{div}u=0\), we derive that
For \(j\geq0\), the Fourier transform of \(\triangle_{j}G\) is supported away from the origin and the dissipative part possesses a lower bound,
where c is an absolute positive constant independent of q.
Therefore, we have
thus Gronwall’s inequality implies that
Taking the \(L^{1}[0,t]\) norm and using Young’s inequality, we obtain
For \(j=-1\), we have
Gathering the above high-low-frequency estimations, multiplying the corresponding inequality by \(2^{j\frac{2}{q}}\), and then summing over j from −1 to ∞, one has
due to the fact \(L_{t}^{1}B_{q,1}^{\frac{2}{q}}\approx\widetilde {L}_{t}^{1}B_{q,1}^{\frac{2}{q}}\), we have
thus
Next, we estimate the last term in the right hand side of the above inequality. Due to the commutator estimate in Lemma 2.4 and the boundedness of \(\mathcal{R}\) in \(L^{p} \) (\(1< p<\infty\)), we have
For the second term, using the Bernstein inequality, we have
where we have used the inequalities: \(\Vert u\Vert _{L^{2q}}\leq C(t)\) and \(\Vert G\Vert _{L^{2q}}\leq C(t)\) for \(q>2\). Putting the above estimates together, one has
Consequently, for any fixed \(t>0\), we get
With the aid of the standard embedding \(B_{q,1}^{\frac{2}{q}}(\mathbb {R}^{2})\hookrightarrow B_{\infty,1}^{0}(\mathbb {R}^{2})\), we have
Furthermore, we have the following estimate:
Consequently, we derive the following key bound:
This completes the proof of Lemma 3.2. □
With the aid of the boundedness of \(\int_{0}^{t}\Vert \nabla u(s)\Vert _{L^{\infty }}\,ds\), we obtain the global bounds of \(\Vert w\Vert _{H^{1}}\) and \(\Vert j\Vert _{H^{1}}\).
Lemma 3.3
If \((u(t,x), b(t,x))\) is a solution of system (1.1), then for any \(T>0\),
where C is a positive constant depending only on T and the initial data.
Proof
Taking the inner products of the first equation in (1.1) with u and the second equation in (1.1) with b, respectively, adding the resulting equations, and integrating by parts, we obtain
Now, w and j satisfy the equations
respectively, where
Taking the inner product of (3.36) with w and (3.37) with j, respectively, adding the resulting equations and integrating by parts, we obtain
With the help of the estimate (3.32) and Gronwall’s inequality, we obtain
Taking the inner product of (3.36) with \(-\Delta w\) yields
Similarly, taking the inner products of (3.37) with \(-\Delta j\) yields
Adding the above equations and integrating by parts, we have
where
Obviously, \(I_{3}=I_{4}\). We only need to estimate the other four terms.
For the terms \(I_{1}\) and \(I_{2}\), by the Hölder inequality, we have
For the term \(I_{3}\), by the Hölder inequality, we have
By the Gagliardo-Nirenberg inequality \(\Vert f\Vert _{L^{4}}\leq C\Vert f\Vert _{L^{2}}^{\frac{1}{2}}\Vert \nabla f\Vert _{L^{2}}^{\frac{1}{2}}\), one has
For the term \(I_{5}\), it is easy to obtain
Adding the estimates of \(I_{i}\) (\(i=1,2,3,4,5\)), we get
Due to Lemma 3.2, \(\int_{0}^{t}\Vert \nabla w\Vert _{L^{2}}^{2}\,ds\leq C\), Gronwall’s inequality immediately yields
This completes the proof of Lemma 3.3. □
Proof of Theorem 1.1
By the estimate (3.45), we know that \(\Vert \nabla w(t)\Vert _{L^{2}}^{2}+\Vert \nabla j(t)\Vert _{L^{2}}^{2}\leq C\), \(\forall t\in[0,T]\). Due to the classical embedding \(H^{1}(\mathbb {R}^{2})\hookrightarrow BMO(\mathbb {R}^{2})\), we obtain
By an argument which generalizes the classical BKM criterion [21] to the MHD system, we complete the proof of Theorem 1.1. □
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Acknowledgements
The authors are grateful to the anonymous referees for helpful comments and suggestions, which greatly improved the presentation of this paper. This work was supported by National Natural Science Foundation of China (Grant No. 11471129).
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YY carried out the theory of Besov spaces, XW carried out the Fourier localization technique and YT carried out the well-posedness. All authors read and approved the final manuscript.
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Yu, Y., Wu, X. & Tang, Y. A magnetic regularity criterion for the 2D MHD equations with velocity dissipation. Bound Value Probl 2016, 113 (2016). https://doi.org/10.1186/s13661-016-0623-6
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DOI: https://doi.org/10.1186/s13661-016-0623-6