Singular Limit of the Rotational Compressible Magnetohydrodynamic Flows

We consider the compressible models of magnetohydrodynamic flows giving rise to a variety of mathematical problems in many areas. We derive a rigorous quasi-geostrophic equation governed by magnetic field from the stratified flows of the rotational compressible magnetohydrodynamic flows with the well-prepared initial data and the tool of proof is based on the relative entropy. Furthermore, the convergence rates are obtained.


Introduction
Magnetohydrodynamic flows arise in science and engineering in a variety of practical applications such as in plasma confinement, liquid-metal cooling of nuclear reactors, and electromagnetic casting.The fundamental concept behind MHD flows is that magnetic fields can induce currents in a moving conductive fluid, which in turn polarizes the fluid and reciprocally changes the magnetic field itself.The set of equations that describe MHD flows are a combination of the Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism.These differential equations must be solved simultaneously, either analytically or numerically.Here we consider the viscous rotational compressible magnetohydrodynamic flows in the 2-dimensional whole space Ω fl R 2 : where u  is the vector field,   is the density, B  is the magnetic field,  > 3/2, and  ∈  3  (Ω), and we also assume that with ,  > 0, as  tends to 0. We first notice that the global-in-time existence solutions for systems ((1)-( 3)), supplemented with physically relevant constitutive relations, has been studied by Hu and Wang [1].
It should be pointed out that the incompressible inviscid limit problems to the compressible Navier-Stokes equations and related models are very interesting.For the case without rotational force, Masmoudi [2] proved the convergence of the weak solution of isentropic Navier-Stokes equations to the strong solution of the incompressible Euler equations in the 2-dimensional whole space R 2 and the space case by applying the related entropy method.Later, his result was extended to the isentropic compressible magnetohydrodynamic equations [3,4].Feireisl and Novotný [5] studied the inviscid incompressible limit to the full Navier-Stokes-Fourier system in the whole space.
In this paper, we derive a rigorous quasi-geostrophic equation from the stratified flow of rotational compressible magnetohydrodynamic flows ((1)-( 3)) on the 2-dimensional whole space with the well-prepared initial data.Our contribution of this paper is physically to derive a rigorous quasi-geostrophic equation from the stratified flows of the rotational compressible magnetohydrodynamic equations based on the relative entropy method.Recently, Feireisl and Novotný [6] have studied the asymptotic limit for the models with a rotational term originating from a Coriolis force with the mild stratification and with the well-prepared initial data.This result is based on their paper, but it is more a developed version than their result because we derive a quasi-geostrophic equation from a global weak solution of compressible MHD flows.
Let the density be the solution of the static problem where we have Assume that the initial data have the following property at infinity: Formally, we will investigate the limit as  tends to 0 in the suitable sense such that the given limits (, B) represent the unique local smooth strong solution of the following system on [0, ]: where the notations are defined as follows: Note that the existence of global strong solution of system ( 9) can be proven with the same method of [7].
(iii) The total energy of the system holds: holds for a.e. ∈ (0, ), where (iv) The Maxwell equation ( 3) with div B = 0 and the regularity

Main Results
In this section, we introduce the main results.

Proof of Theorem 3
In this section, we are going to give the rigorous proof of Theorem 3.
Step 1.In this part, we are going to derive some estimates on the sequence {  , u  , B  } >0 .
From the energy inequality (16), we obtain ess sup ess sup ∇u      2 ((0,)×Ω) ≤ where we can see these properties in [8].Following (28) and (29) together with (25), we get ess sup Using (31), we also derive ess sup which gives ess sup Note that The Sobolev embedding also gives and the proof is provided in [9].
Step 2. We introduce the relative entropy in the version of the magnetohydrodynamic flows.Let us set where We define the relative entropy: where we put k fl ∇ ⊥ .Adapting k as a test function to the moment equation ( 2) provides We also use  as a test function to the continuity equation ( 1) to deduce that To compute the relative entropy, we also use B as a test function to the magnetic field equation (3) and insert (9), which yields Adding ( 16), (41), (42), and (43) derives the following inequality: where Step 3. From (19), it is easily seen that it shows that Indeed, using ( 6), (19), and (21), we get where ‖∇ ⊥  0 ‖  ∞ (Ω) ≤  and the constant depends on the support of , and we also have where the Sobolev imbedding theorem  2 ⊂  ∞ implies that ∇ ⊥  ∈  ∞ (Ω).The estimate of the second term of  2 is also given in [10] such that To handle  3 , we multiply  to system (9), which yields while Multiplying B to the magnetic field equation, we obtain the following energy equation: Adding ( 52) and ( 54), it follows that where we have used (6) where we have here used ( 4) and (11).

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Step 4. For this part, we need to estimate the convective term of  5 and the term can be expressed in the following form: The first term  1 can be controlled by where, by (11), ∇k ∈  2 (Ω) together with the Sobolev embedding, which implies that 2 is reformulated by For  3 , employing the estimates of (32) and (34) together with the continuity equation ( 1), we get To handle  4 , we use div k = 0 to obtain where we have used (32) and (33).Thus,  5 is written by where together with using ( 5) and div k = 0. To handle (63), we need the following lemma.
Lemma 4. Let (  , u  , B  ) be global weak solutions of ( 1)-( 3) in the sense of Definition 2.Then, one has the following inequality: for any test function .
Proof.We use the test function  ⊥ to the moment equation ( 2) to deduce In virtue of the estimates in ( 19)-( 21) and ( 24)-( 27), the terms of the right-hand side in (66) can be controlled and so it proved (65).
Making use of Lemma 4 together with k = ∇ ⊥  and  =   ∇ ⊥  + (∇ ⊥  ⋅ ∇)∇ ⊥  and integrating by parts, we obtain that where we have used  = (B⋅∇)B−(1/2)∇|B| 2 as a test function on the last line of (67).Thus, the relative entropy gives Step 5. Finally, we now handle the part of magnetic field.We apply the integration by parts and the facts div B  = 0, div B = 0, and div k = 0 to obtain For the term  6 , making use of (32), (33), and (37), together with the Sobolev embedding and Holder's inequality, it follows that Similarly, the term  7 can also be controlled by Finally, we estimate  2 ,  3 ,  4 , and  5 with the same method.We now rewrite  1 as follows: Thus, from the estimates of (32), (33), and (37), together with the estimate of (24), we get where we have again used the assumption of viscosity 0 <  +  < 2 and so  Using ( 77), ( 78), (79), and (80) and passing to the limit for  → 0, we prove (22).In conclusion, we get the target equation ( 9) by passing to the limits as  → 0, but it is sufficient to show that, for any test function , where we have used div∇ ⊥ = 0 and (5).

Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.