Global well-posedness to the Cauchy problem of 2D inhomogeneous incompressible magnetic B ́enard equations with large initial data and vacuum

Abstract: In this paper, we are concerned with the Cauchy problem of inhomogeneous incompressible magnetic Bénard equations with vacuum as far-field density in R2. We prove that if the initial density and magnetic field decay not too slowly at infinity, the system admits a unique global strong solution. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states and even has compact support. Moreover, we extend the result of [16,17] to the global one.

(1.1) which is equipped the following initial conditions and far-field behavior: (ρ, ρu, ρθ, b)(x, 0) = (ρ 0 , ρu 0 , ρθ 0 , b 0 )(x) for x ∈ R 2 , (ρ, u, θ, b)(x, ·) → (0, 0, 0, 0), as |x| → ∞, (1.2) where ρ, u, θ, b and p denote the density, velocity, temperature, magnetic field, and pressure of the fluid, respectively. µ > 0 is the viscosity coefficient, κ > 0 is the heat conductivity coefficient, and ν > 0 is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field. e 2 = (0, 1) T , where T is the transpose. The magnetic Bénard equations (1.1) illuminates the heat convection phenomenon under the dynamics of the velocity and magnetic fields in electrically conducting fluids such as plasmas (see [10,11] for details). If we ignore the Rayleigh-Bénard convection term u · e 2 , system (1.1) recovers the inhomogeneous incompressible MHD equations (i.e., θ ≡ 0). Let us review some previous works about the standard incompressible MHD equations. In the absence of vacuum, Abidi-Paicu [1] established the local and global (with small initial data) existence of strong solutions in the framework of Besov spaces. Chen et al. [2] proved a global solution for the global well-posedness to the 3D Cauchy problem for the bounded density. In the presence of vacuum, imposing the following compatibility condition, for some (p 0 , g) ∈ H 1 × L 2 . Chen et al. [3] obtained the unique local strong solutions to the 3D Cauchy problem with general initial data. Song [13] studied the local well-posedness of strong solutions without additional compatibility condition (1.3), which extended the main result of [3]. Recently, Gao-Li [4] shown the global strong solutions with vacuum in bounded domain, provided that initial data is suitable small. Later on, Zhang-Yu [15] extended this result to the whole space. For the 2D case, Huang-Wang [5] investigated the global existence of strong solution with general large data in bounded domain provided that the compatibility condition (1.3) holds. Recently, Lv et al. [8] showed the global existence of strong solutions to the 2D Cauchy problem with the large data and vacuum. Let us go back to the system (1.1). Very recently, by weighted energy method, Zhong [16] showed the local existence of strong solutions to the Cauchy problem of (1.1) in R 2 . However, the global existence of strong solution to the 2D Cauchy problem of (1.1) with vacuum and general initial data is not addressed. In fact, this is the main aim of this paper.
Before stating the main results, we first explain the notations and conventions used throughout this paper. For R > 0. Set Moreover, for 1 ≤ r ≤ ∞ and k ≥ 1, the standard Sobolev spaces are defined as follows: Without loss of generality, we assume that initial density ρ 0 satisfies which implies that there exists a positive constant N 0 such that Throughout this paper, always denotē x := (e + |x| 2 ) 1/2 log 1+σ 0 (e + |x| 2 ), (1.6) with σ 0 > 0 fixed. The main result of this paper is stated as the following theorem: Theorem 1.1. In addition to (1.4) and (1.5), assume that the initial data (ρ 0 , u 0 , θ 0 , b 0 ) satisfies for any given numbers a > 1 and q > 2, (1.7) Then the problems (1.1) and (1.2) has a unique global strong solution (ρ ≥ 0, u, θ, b, p) satisfying that for any 0 < T < ∞, for some positive constant N 1 depending only on √ ρ 0 u 0 , N 0 , and T .
Remark 1.1. We remark that Theorem 1.1 is proved without any smallness on the initial data. Moreover, the initial density can contain vacuum states and even has compact support. We also point out that Theorem 1.1 extends the result of Zhong [16] to the global one. In particular, when b = 0, the incompressible magnetic Bénard equations (1.1) reduces to the incompressible Bénard equations, Theorem 1.1 also extends Zhong [17] to the global one.
We now make some comments on the key ingredients of the analysis in this paper. For the initial data satisfying (1.7), Zhong [16] recently established the local existence and uniqueness of strong solutions to the Cauchy problems (1.1) and (1.2) (see Lemma 2.1). Thus, to extend the local strong solution to be a global one, we need to obtain global a priori estimates on strong solutions to (1.1) and (1.2) in suitable higher norms. However, due to critically of Sobolev's inequality in R 2 , it seems difficult to bound ∥u∥ L p just in term of ∥ √ ρu∥ L 2 and ∥∇u∥ L 2 for any p ≥ 2. Moreover, compared with [9], for the systems (1.1) and (1.2) here, the strong coupling terms and Rayleigh-Bénard convection terms, such as u · ∇b, ρu · e 2 , and ρθe 2 , will bring out some new difficulties.
To overcome these difficulties mentioned above, some new ideas are needed. First, using the structure of the 2D magnetic equations, we multiply (1.1) 4 by 4|b| 2 b and thus obtain the useful a priori estimate on L 2 (R 2 ×(0, T ))-norm of |b||∇b| (see (3.5)), which is crucial in deriving the L ∞ (0, T ; L 2 (R 2 ))norm of ∇u, ∇θ and ∇b. Next, in order to derive the estimates on L ∞ (0, T ; L 2 (R 2 ))-norm of ∇u, ∇θ, motivated by [9], multiplying (1.1) 2 and (1.1) 3 byu := u t + u · ∇u andθ := θ t + u · ∇θ instead of usual u t and θ t respectively, we deduce that the key point to obtain the estimate on the L ∞ (0, T ; L 2 (R 2 ))-norm of the gradient of the velocity u and temperature θ is to bound the terms We find I 2 in fact can be bounded by ∥∇p∥ L 2 ∥∇u∥ 2 L 2 (see (3.8)), since ∂ j u i ∂ i u j ∈ H 1 due to the fact that divu = 0 and ∇ ⊥ · ∇u = 0 (see Lemma 2.4). Moreover, the usual L 2 (R 2 × (0, T ))-norm of b t cannot be directly estimated due to the strong coupled term u · ∇b. Thus, we multiplying (1.1) 4 by ∆b instead of usual b t , the coupled term can be controlled after integration by parts. Thirdly, to tackle the difficulty caused by the lack of the Sobolev inequality, motivated by [8,16,17], by introducing a weighted function to the density, as well as a Hardy-type inequality in [7] by Lions, the ∥ρ η v∥ σ (η > 0, σ > max{2, 2 η }) is controlled in term of ∥ √ ρv∥ L 2 and ∥∇v∥ L 2 (see (3.18)), which plays an important role in bounding the Rayleigh-Bénard convection terms ρu · e 2 and ρθe 2 , and deriving the estimates on the L ∞ (0, T ; L 2 (R 2 )) of √ t √ ρu t and √ t √ ρθ t . Finally, with these a priori estimates on the velocity, temperature and magnetic field at hand, some useful spatial weighted estimates on both b, ∇u and ∇θ are derived, which yields the bounded of L ∞ (0, T ; L 2 (R 2 ))-norm of √ t∇ 2 b (see Lemma 3.7). The rest of the paper is organized as follows. In Section 2, we collect some elementary facts and inequalities which will be needed in later analysis. Section 3 is devoted to the a priori estimates. Finally, we give the proof of Theorem 1.1 in Section 4.

Preliminaries
In this section, we will recall some known facts and elementary inequalities which will be used frequently later.
We start with the local existence of strong solutions whose proof can be found in [16].
Next, the following Gagliardo-Nirenberg inequalities will be stated, which see [12] for the detailed proof.
there exists a positive constant C depending only on j, m, n, p ′ , q, and r such that and m − j − n r is not a nonnegative integer. If 1 < r < ∞ and m − j − n r is a nonnegative integer, (2.1) holds with ϑ ∈ [ j m , 1).
As a key technical ingredient for our approach, we need the following weighted bounds for functions in the spaceD 1,2 (R 2 ) ≜ {v ∈ H 1 loc (R 2 ) : ∇v ∈ L 2 (R 2 )}, whose proof can be found in [6,Lemma 2.4]. Lemma 2.3. Letx be as in (1.6). Assume that ρ ∈ L 1 ∩ L ∞ be a non-negative function satisfying Moreover, for any η > 0 and σ > max{2, 2 Finally, let H 1 and BMO stand for the usual Hardy and BMO spaces (see [14,Section 4]). Then the following well-known facts play a key role in the proof of Lemma 3.2, whose proof can be found in [9].
(ii) There is a positive constant C such that for all v ∈ D 1,2 (R 2 ), it holds

A priori estimates
In this section, we will establish some necessary a priori bounds for strong solutions (ρ, u, θ, b, p) to the Cauchy problems (1.1) and (1.2) to extend the local strong solution. Thus, let T > 0 be a fixed time and (ρ, u, θ, b, p) be the strong solution to (1.1) and (1.
In what follows, we will use the convention that C denotes a generic positive constant depending on initial data and T .
We begin with the following standard energy estimate and the estimate on the L ∞ (0, T ; L 1 ∩ L ∞ )norm of the density.
Proof of Theorem 1.1. By Lemma 2.1, there exists a T * > 0 such that the problems 1.1 and 1.2 has a unique strong solution (ρ, u, θ, b, p) on R 2 × (0, T * ]. Now, we will extend the local solution to all time. Set T * = sup{T | (ρ, u, θ, b, p) is a strong solution on R 2 × (0, T]}. satisfies the initial condition (1.7) at t = T * . Thus, taking (ρ, u, θ, b)(x, T * ) as the initial data, Lemma 2.1 implies that one can extend the strong solutions beyond T * . This contradicts the assumption of T * in (4.1). The proof of Theorem 1.1 is completed. □

Conclusions
In this paper, we are concerned with the Cauchy problem of inhomogeneous incompressible magnetic Bénard equations with vacuum as far-field density in R 2 . Using the weighted function to the density, as well as the Hardy-type inequality, we have successfully established the time-uniform a priori estimates of solutions. Thus, we can extend the local strong solutions to the global one.