Singular Limit of the Rotational Compressible Magnetohydrodynamic Flows

Abstract

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.

1. 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 :where is the vector field, is the density, is the magnetic field, , and , and we also assume thatwith , as tends to .

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 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 problemwhere we haveAssume that the initial data have the following property at infinity:Formally, we will investigate the limitas tends to 0 in the suitable sense such that the given limits represent the unique local smooth strong solution of the following system on : for and ,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].

Theorem 1. Let be the 2-dimensional whole space . For the given initial and , there is such that system (9) has the unique local smooth solution on , verifying the following regularity:

The outline of this article is as follows. In Section 2, we present the rigorous result for (8) and (9). In Section 3, we derive a rigorous proof of the rotational compressible magnetohydrodynamic flows ((1)–(3)).

Definition 2. We say that a quantity is a weak solution of the magnetohydrodynamic (MHD) flows ((1)–(3)) supplemented with the initial data provided that the following hold:(i)The density is a nonnegative function, where , the velocity field , , and represents a renormalized solution of equation (1) on ; that is, the integral identity holds for any test function and any such that(ii)The balance of momentum holds in distributional sense; namely, for any test function .(iii)The total energy of the system holds: holds for a.e. , where(iv)The Maxwell equation (3) with and the regularity verifies for all

2. Main Results

In this section, we introduce the main results.

Theorem 3. Let be the 2-dimensional whole space and let be a weak solution to ((1)–(3)) in the sense of Definition 2, verifying viscosity (4) with and the initial data:wherefor . Then, one hasfor sufficiently small , any , and any compact such that verifies (9). Furthermore, the numbers are defined by

3. 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 .
From the energy inequality (16), we obtainWe consider the properties of convex function:where we can see these properties in [8]. Following (28) and (29) together with (25), we getUsing (31), we also derivewhich gives Note thatwhile using (28)–(31) implies thatThe Sobolev embedding also givesand the proof is provided in [9].

Step 2. We introduce the relative entropy in the version of the magnetohydrodynamic flows. Let us setwhereWe define the relative entropy:where we put . Adapting as a test function to the moment equation (2) providesWe also use as a test function to the continuity equation (1) to deduce thatTo compute the relative entropy, we also use 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 thatIndeed, using (6), (19), and (21), we getwhere and the constant depends on the support of , and we also havewhere we have used (21). Similarly, we getBy the regularity of in (11) and (34), we can estimate the first term of as follows:where the Sobolev imbedding theorem implies that The estimate of the second term of is also given in [10] such thatTo handle , we multiply to system (9), which yieldswhileMultiplying to the magnetic field equation, we obtain the following energy equation:Adding (52) and (54), it follows thatwhere we have used (6). Let us next show that the viscosity term vanishes:where we have here used (4) and (11).

Step 4. For this part, we need to estimate the convective term of and the term can be expressed in the following form:The first term can be controlled bywhere, by (11), together with the Sobolev embedding, which implies that is reformulated byFor , employing the estimates of (32) and (34) together with the continuity equation (1), we getTo handle , we use to obtainwhere we have used (32) and (33). Thus, is written bywheretogether with using (5) and To handle (63), we need the following lemma.

Lemma 4. Let 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 deduceIn 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 and and integrating by parts, we obtain thatwhere we have used 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 , , and to obtainFor the term , making use of (32), (33), and (37), together with the Sobolev embedding and Holder’s inequality, it follows thatwhere we have used the assumption of viscosity , (6), and the Sobolev embedding theorem:Similarly, the term can also be controlled byFinally, we estimate , , , and with the same method. We now rewrite as follows:Thus, from the estimates of (32), (33), and (37), together with the estimate of (24), we getwhere we have again used the assumption of viscosity and soConsequently, the relative entropy in (69) is given by

Step 6. Let us apply Grönwall’s inequality to (76) in order to obtainfor any , where the number is defined in (23). Note thatfor any compact subset andwhere is the given time in Theorem 1. For the magnetic field, we obtainUsing (77), (78), (79), and (80) and passing to the limit for , we prove (22). In conclusion, we get the target equation (9) by passing to the limits as , but it is sufficient to show that, for any test function ,Indeed,where we have used and (5).

Conflicts of Interest

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

Acknowledgments

The work of Young-Sam Kwon was supported by the research fund of Dong-A University.