On three-dimensional Hall-magnetohydrodynamic equations with partial dissipation

In this paper, we address the Hall-MHD equations with partial dissipation. Applying some important inequalities (such as the logarithmic Sobolev inequality using BMO space, bilinear estimates in BMO space, Young’s inequality, cancellation property, interpolation inequality) and delicate energy estimates, we establish an improved blow-up criterion for the strong solution. Moreover, we also obtain the existence of the strong solution for small initial data, the smallness conditions of which are given by the suitable Sobolev norms.


Introduction
The incompressible Hall-magnetohydrodynamic equations with full dissipation in three dimensions read as: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u t + (u · ∇)u + ∇(p + π) = κ 1 u x 1 x 1 + κ 2 u x 2 x 2 + κ 3 u x 3 x 3 + (B · ∇)B, div u = 0, Here u(t, x), B(t, x) denote velocity field and magnetic field, respectively; κ 1 , κ 2 , κ 3 are the kinematic viscosity, (t, x) ∈ R + × R 3 . Compared to usual MHD system and the Boussinesq equations, Hall-MHD equations involve ∇ × ((∇ × B) × B), it is Hall term and plays a crucial position in magnetic reconnection due to Ohm's law. Magnetic reconnection corresponds to changes in the topology of magnetic field lines, which are ubiquitously observed in space. The Hall term becomes important when large magnetic shear appears because it has second-order derivatives, and it restores the influence of the electric current in the Lorentz force occurring in Ohm's law, which was neglected in usual MHD. Therefore, Hall-MHD is very important for such problems as magnetic reconnection in neutron stars, geo-dynamo, space plasmas, and star formation. The paper [1] introduces the physical background to Hallmagnetohydrodynamics, and papers [7,8,13,[15][16][17][18]24] present the recent progress of the Hall-MHD system.
The nonlinear Jordan-Moore-Gibson-Thompson equation with memory read as where u = u(x, t) denotes the scalar acoustic velocity. The Jordan-Moore-Gibson-Thompson equation is one of the nonlinear sound equations that describe the propagation of sound waves in gases and liquids. Recent works on the Jordan-Moore-Gibson-Thompson equation can be found in [4,12]. The Hall-MHD Eqs. (1) describe the magnetic properties for a conductive fluid moving in a magnetic field, in which magnetic reconnection happens in the case of large magnetic shear. In the Hall-MHD Eqs. (1), u = u(x, t), B = B(x, t) are non-dimensional quantities corresponding to the fluid velocity field, the magnetic field. Many results on usual MHD system have been obtained in [10, 11, 14, 17, 21-23, 26-28, 30-33]. However, the Hall-MHD system had few results until recently. The paper [7] got the local existence and global small solutions for the Hall-magnetohydrodynamics. Some results on the Boussinesq and MHD equations with partial viscosity were obtained in [5,6,15,24]. Two new blow-up criteria for the system (1) with κ 1 = κ 2 = κ 3 = 1 were obtained by Chae and Lee in [8]. Fei and Xiang [19] got a blow-up criterion and small existence to (1) with κ 1 = κ 2 = 1, κ 3 = 0.
The paper [20] established regularity criterion for the Hall-MHD equations without viscosity and full dissipation, the papers [2,3] obtained regularity criterion for the Hall-MHD equations with full viscosity and full dissipation in different spaces. In this paper, we investigate the Hall-magnetohydrodynamic system with full viscosity and partial dissipation. Inspired by [8,13,19,33], we find a new blow-up criterion for strong solution, which imposes the condition is (u, ∇B) ∈ L 2 (0, T; BMO). Additionally, we also get the existence of the strong solution for small initial data.
The first aim of this paper is to get blow-up criterion for the strong solution to (1) with κ 1 , κ 2 > 0, κ 3 = 0.
, and div u 0 = div B 0 = 0, let T 0 < ∞ be the first blow-up time to the problem (1), then Remark 1.1 Compared to previous results, the blow-up condition [19].

Notations and preliminaries
Through the paper, ∂ k and u k represent the k th components of ∇ and u, and the following simplified notation will be adopted throughout the paper: Next, some lemmas are given.

Proof of Theorem 1.1
We adopt the following notations: Operating ∇ on (1) 1 and (1) 3 , and multiplying them by ∇ u and ∇ B, respectively, and then integrating by parts, one gets Using Lemma 2.2, we have By the above inequality, cancellation property and Young's inequality, one obtains We apply cancellation property and Lemma 2.3 to deduce that For H 3 , when | | = 0, the H 3 have cancelled. When | | = 1, by div u = 0, H 3 can be rewritten as follows Thus using the Höledr inequality and Lemma 2.3, one obtains When | | = 2, one can write H 3 as H 321 , H 322 can be further decomposed into three parts, respectively.

Proof of Theorem 1.2
Operating ∇ to (1) 1 , (1) 3 , taking the scalar product of them with ∇u, ∇B, one gets Firstly, applying the Hölder inequality, commutator estimate and interpolation, one gets here we use the fact that ∇B 3 ≤ C B 2 , ∇ 2 B 6 ≤ C ∇ 3 B 2 due to the Gagliardo-Nirenberg-Sobolev inequality. By the Hölder inequality, one obtains K 4 + K 5 can be written into two parts: By the Höledr inequality, we obtain Combining (22)- (26), we get Similarly to derivation of (22), one gets We apply cancellation property, the Hölder inequality, commutator estimate to estimate E 1 as follows E 2 can be split into two terms: Noticing the fact that ∇B 3 ≤ C B and Collecting (30) and (31), we have Similarly to the above calculation, one gets In similar manner as H 3213 and H 3212 , one gets Combining (33)-(37), we have One can split E 4 + E 5 into four terms: where κ 0 = min{κ, 1}. Choose ε * sufficiently small such that C ∇u 0 2 + ∇B 0 2 + u 0 2 + B 0 2 ≤ κ 0 2 .
By Theorem 1.1, applying continuation argument, we obtain the result of Theorem 1.2.