On the blow-up criterion of strong solutions for the MHD equations with the Hall and ion-slip effects in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^{3}}$$\end{document}

In this paper, we establish a blow-up criterion of strong solutions to the 3D incompressible magnetohydrodynamics equations including two nonlinear extra terms: the Hall term (quadratic with respect to the magnetic field) and the ion-slip term (cubic with respect to the magnetic field). This is an improvement of the recent results given by Fan et al. (Z Angew Math Phys, 2015).


Introduction
We are interested in the regularity criterion of the following incompressible magnetohydrodynamics equations with the Hall and ion-slip effects in R 3 : ∇ · u = ∇ · B = 0, (u, B)(x, 0) = (u 0 (x), B 0 (x)), (1.1) where x ∈ R 3 and t > 0. Here u = u(x, t) ∈ R 3 , B = B(x, t) ∈ R 3 and π = π(x, t) are nondimensional quantities corresponding to the flow velocity, the magnetic field and the pressure at the point (x, t), while u 0 (x) and B 0 (x) are the given initial velocity and initial magnetic field with ∇ · u 0 = 0 and ∇ · B 0 = 0, respectively.
Magnetohydrodynamics (MHD) is a fluid theory that describes plasma physics by treating the plasma as a fluid of charged particles. Hence, the equations that describe the plasma form a nonlinear system that couples Navier-Stokes equations with Maxwell's equations. This model describes some important physical phenomena. Comparing with the usual viscous incompressible MHD equations, the system (1.1) contains the extra term ∇ × ((∇ × B) × B) which is the so-called Hall term and ∇ × [((∇ × B) × B) × B] the ion-slip effect.
The above system has been studied a lot by physicists and mathematicians because of its physical importance, complexity, rich phenomena and mathematical challenges; see [1][2][3][4]6] and the references cited therein.
The mathematical study of the above system was initiated by Mulone and Solonnikov [19], who proved the small data global existence of strong solutions in a bounded domain. Refer to [19][20][21]. Recently, Chae and Lee [2] established an existence result on strong solutions and proved the following regularity criteria or u ∈ L 2 0, T ; BMO R 3 and ∇B ∈ L 2 0, T ; BMO R 3 . Here BMO is the space of functions of bounded mean oscillations.
Very recently, the local well posedness of the strong solution to the incompressible magnetohydrodynamic equations with the Hall and ion-slip effects was established for the whole space R 3 by Fan et al. [12] and they proved that if In this paper, we extend the results of [12] to the critical Besov space. We prove a blow-up criterion for local strong solutions in terms of the critical Besov space ∞;∞ and multiplier spaces.

Multiplier and Besov spaces
Before stating our results precisely, we first recall the definition of the homogeneous Besov space with negative indicesḂ −α ∞,∞ in R 3 and the homogeneous Sobolev spaceḢ α q of exponent α > 0. It is known [24] We introduce now the homogeneous Sobolev spaceḢ s q R 3 , which is defined as the set of functions Here H s q R 3 is the standard inhomogeneous Sobolev space. We introduce the following two lemmas.
Lemma 2.2. Let 1 < p < q < ∞ and let s = α q p − 1 > 0. Then there exists a constant depending only on α, p and q such that the estimate . This result is due to Meyer-Gerard-Oru [18], and the proof is based on the Littlewood-Paley decomposition, please refer to [18] for the details.
Next, let us define the function spaces in which the blow-up criterion of strong solution (u, B, π) is going to be established.

ZAMP
On the blow-up criterion of strong solutions for the MHD Page 3 of 10 18 Definition 2.3. [16,26] For 0 ≤ r < 3 2 , the space where we denote by We have the homogeneity properties: Proof. Let f ∈ L 3 r R 3 . By using the following well-known Sobolev embedding Then, it follows that Remark 2.1. We note from Proposition 2.5 in [16] that By Banach fixed point theorem and energy estimates, it is easy to prove the following well posedness of local strong solutions to the problem (1.1), and hence we omit the details here (see for e.g., [12]).
First our main result reads as follows.
then the solution (u, B) can be extended beyond T . Next, we consider an improvement of Theorem 2.6 for the case r = 1 by using critical Besov space

2)
and then the solution (u, B) can be extended beyond T .

Remark 2.2.
Since the multiplier spaceẊ r (R 3 ) with 0 ≤ r ≤ 1 is wider than the Lebesgue space L 3 r (R 3 ), hence our result is an improvement of the recent result obtained by Chae-Lee [2] and Fan et al. [12].

Remark 2.4.
In the sequel, we will use the following inequality for all 1 < q < ∞

Proof of Theorem 2.6
We are now in a position to the proof of our first result.
Proof. We want to establish an a priori estimate for the smooth solution.
First, multiplying (1.1) 1 by u, after integration by parts and taking the divergence-free property into account, we have Similarly, multiplying (1.1) 2 by B, and integrating over R 3 , we have Combining (3.1) and (3.2), we have the well-known energy equality Integrating the above equality over time interval [0, t], we obtain the energy equality and this proves that Take operator Δ on Eq. (1.1) 1 and take scalar product of them with Δu, we obtain ZAMP On the blow-up criterion of strong solutions for the MHD Page 5 of 10 18 Now let us remind the following well-known interpolation inequality: if 0 ≤ r ≤ 1, then Using this inequality together with Young's inequality, we obtain In the following calculations we will use the following Gagliardo-Nirenberg inequality To estimate J 2 , we apply interpolation inequality and Young's inequality, Applying Δ to (1.1) 2 , then multiplying it by ΔB, after integration by parts, we find that, 1 2 d dt From the calculus inequality, interpolation inequality and Young's inequality, we bound I 1 as follows: Similarly, from the calculus inequality, interpolation inequality and Young's inequality, we bound I 2 as follows: By integration by parts, I 3 can be rewritten as: For I 3,1 , using Young's inequality and the interpolation inequality, we obtain For the second term on the right of I 3 , by the simple vector identity and Hölder inequality, I 3,2 is bounded by Inserting the estimates above in (3.3)-(3.4), we obtain 1 2 d dt

ZAMP
On the blow-up criterion of strong solutions for the MHD Page 7 of 10 18 Consequently, using the Gronwall's inequality, one has This completes the proof of Theorem 2.6.

Proof of Theorem 2.7
In this section, we shall give the proof of theorem 2.7.
Proof. Testing (1.1) 2 by |u| 2 u, taking the divergence-free property into account and using the Hölder inequality, we get 1 4 d dt Testing (1.1) 2 by |B| 2 B, taking the divergence-free property into account and using the Hölder inequality, we get 1 4 d dt where we used the fact ∇w Due to the following ones (0 ≤ r ≤ 1)

.
By applying the standard Gronwall inequality, we obtain, for every t ∈ [0, T ], From (4.1) and the standard Serrin regularity criterion (see e.g., [12]), we have (u, B) smooth on (0, T ) × R 3 . Then, by using the standard arguments of the continuation of local solutions, it is easy to conclude that the above estimate (4.1) implies that the solution (u(x, t), B(x, t)) can be extended beyond t = T . This completes the proof by Theorem 2.7.