On the Nonlinear Stability and Instability of the Boussinesq System for Magnetohydrodynamics Convection

: This paper is concerned with the nonlinear stability and instability of the two-dimensional (2D) Boussinesq-MHD equations around the equilibrium state ( u = 0, B = 0, θ = θ 0 ( y )) with the temperature-dependent fluid viscosity, thermal diffusivity and electrical conductivity in a channel. We prove that if a + ≥ a − , and d 2 dy 2 κ ( θ 0 ( y )) ≤ 0 or 0 < d 2 dy 2 κ ( θ 0 ( y )) ≤ β 0 , with β 0 > 0 small enough constant, and then this equilibrium state is nonlinearly asymptotically stable, and if a + < a − , this equilibrium state is nonlinearly unstable. Here, a + and a − are the values of the equilibrium temperature θ 0 ( y ) on the upper and lower boundary.

The situation for the case d dy θ 0 (y) < 0 is closely related to the Rayleigh-Taylor instability according to the well-known Boussinesq approximation, where the temperature difference is directly proportional to the density difference between the bottom and top of the layer of fluid. The Rayleigh-Taylor instability appears when a heavy fluid is on top of a light one. The linear instability for the incompressible fluid was first established by Rayleigh in 1883 [41] and Chandrasekhar in 1981 [42]. Grenier [43] gave some examples of nonlinearly unstable solutions of Euler equations and proved an instability result for Prandtl equations. Recently, Hwang and Guo [44] obtained the nonlinear Rayleigh-Taylor instability for the inviscid incompressible fluid. Guo and Tice [45,46] proved the linear Rayleigh-Taylor instability for inviscid and viscous compressible fluids by introducing a new variational method. Later on, using the new variational method, many authors considered the effects of magnetic field in the fluid equations, see Jiang-Jiang [47][48][49].
This paper is concerned with the nonlinear stability and instability for the full BMHD system with the temperature-dependent fluid viscosity, thermal diffusivity, and electrical conductivity. Our results are as follows.

Remark 2.
Our result also holds for the incompressible 3-D BMHD system.

Notations:
The space H 1 div is defined as H 1 div = {u ∈ H 1 : div u = 0} and this rule of definition is applied to the sapce H m div with m ≥ 0. We define The remainder of the paper is organized, as follows. In Section 2, we construct the growing solutions to the linearized Boussinesq-MHD system for the case a + < a − . With these precise growth rate λ, we construct an approximation solution with higher order growing modes in Section 3. In Section 4, after obtaining the crucial estimates of the linearized system, we present nonlinear energy estimates of the original perturbed equations for the case a + < a − . In Section 5, we will prove Theorem 1, which concludes the nonlinear instability and stability. Finally, in Section 6, we give the conlusions of this paper.

Variational Method for the Case a + < a −
In this section, we prove that, if a + < a − , then there exists a smooth linear growing mode of the forms (8) with the eigenvalue Λ > 0.
We first linearize (4) around σ = 0, u = b = 0, p = 0 as We want to find a dominate eigenvalue of the linearized equations (7), with its corresponding growing normal mode, which takes the form: , and satisfies the boundary condition (3) in the sense of the trace.
When a + < a − , that is, d dy θ 0 (y) = −1 κ(θ 0 ) , plugging (8) into (7) we can easily get the following equivalent system. Lemma 1. When a + < a − . Assume (8), then (7) takes the form of We define It is easy to check that I 1 (σ,ũ,b) and J 1 (σ,ũ,b) are well define on the space We know that I 1 (σ,ũ,b) has a upper bound on the set A. We are now in a position to prove that there exists a growing mode of (8) with the eigenvalue Λ > 0.

The Exponential Growth Rate Λ
The goal of this section is to prove that the eigenvalue Λ in Section 2 is the sharp exponential growth rate for the linearized BMHD equations (7). The results are as follows.
Subsequently, there exists a global unique solution (σ, u, b), satisfying to the linearized BMHD system (7). Moreover, for any m 1 ∈ N with m 1 ≤ m, and t > 0, it holds that Proof. Both the existence and uniqueness of a solution to (7) essentially follow from some a priori estimates. We now establish the estimates. For the L 2 estimate, multiplying the first equation of (7) by κ(θ 0 )σ, and then integrating the result equation with respect to (x, y) ∈ Ω, one obtains Thanks to integration by parts again, one has Multiplying the second and third equation in (7) with u and b, respectively, and using the fact that integrating by parts, we have This, together with (16), implies From the definition of Λ in Lemma 2, we get from (17) that which implies that, for any t ≥ 0 Notice that the quantities κ(θ 0 ), µ(θ 0 ) and γ(θ 0 ) have positive lower bounds, and all of their derivatives are bounded in R. It follows from (17) that which together with (18) implies that, for all t > 0 Therefore, combining (19) with (18), we can show that, for all t ≥ 0 In order to get the H 1 estimate, applying the operator ∂ i (with i = 1, 2 and ∂ 1 = ∂ x , ∂ 2 = ∂ y ) to the equations (7), we obtain which is equivalent to where Taking the L 2 inner product of the first equation in (22) with κ(θ 0 )∂ i σ, and then using integration by parts, one gets Similarly, taking the L 2 inner product of the second and third equations in (21) with ∂ i u and ∂ i b, respectively, and then using integration by parts, we obtain which, along with (23), gives This, together with (20), implies that for all t > 0 By an induction argument, we can get (14) and (15), which completes the proof.

Nonlinear Energy Estimates for the Case a + < a −
In this section, we prove the nonlinear estimates for the nonlinear perturbation (4) for the case a + < a − .
(Ω) and assume the boundary conditions Subsequently, there exists a unique global solution (σ, u, b) satisfying to the perturbed BMHD (4) with initial data (σ 0 , u 0 , b 0 ) and the corresponding boundary conditions. Moreover, there are two positive constants δ ∈ (0, 1] and T > 0, such that, for any t ∈ [0, T] and (σ, where the constant C only depends on θ and Ω. Proof. Similar to the proof of Lemma 3, we just need to present some necessary a priori estimates for sufficiently smooth solutions to (4).
Multiplying the three equations in (4) by σ, u, and b, respectively, and then integrating by parts, we obtain Therefore, applying Young's inequality, we can show Integrating the above inequality gives that for all t < T In order to get the H 1 estimate, applying the operator ∂ i (with i = 1, 2 and ∂ 1 = ∂ x , ∂ 2 = ∂ y ) to the equations (4), we have Multiplying the three equations in (26) by ∂ i σ, ∂ i u and ∂ i b, respectively, and then using integration by parts, we have Thanks to the boundness of κ(θ 0 ) and its derivatives, it follows from Holder's inequality that Similarly, by Poincare inequality, we can estimate F 3 and F 4 , as follows Therefore, it follows from (28) that, for some positive constant c 0 and any t > 0, which, together with (25), leads to Then we have if η > 0 is sufficiently small and (σ, u, b)(t) H 1 ≤ δ with δ small enough and t ∈ [0, T] for some positive time T. Similarly, one can obtain that which implies (24).

Proof of Theorem 1
The goal of this section is to prove Theorem 1.
is also a solution to (7) with the initial data (σ δ 0 , u δ 0 , b δ 0 ), so (σ d , u d , b d ) solves the following problem: Similar to the proof of (17), one can get from (32) that Hence, by the definition of Λ, one has For the remainder terms on the right-hand-side of (33), one first can deduce that Similarly, it holds that and By integration by parts, one can control the term Similarly, one can show that Noticing that Substituting (34)- (38) into (33), yields which, along with the Gronwall's inequality, gives rise to It follows from (31) and (39) that, for all t ≤ min (T δ , T * , T * * ), it holds that which yields that for some positive constant C 3 independent of δ. Now, we claim that if ε 0 is taken to be so small that In fact, if T * = min (T δ , T * , T * * ) < T δ , then (31) implies that which contradicts with the definition of T * in (29).
On the other hand, if T * * = min (T δ , T * , T * * ) < T δ , then it follows from (31), that which contradicts with the definition of T * * in (30), and, thus, (41) holds. Therefore, thanks to the fact that we get from (40) and (42) that which completes the proof of the case (i).