Weak-Strong Uniqueness for Compressible Magnetohydrodynamic Equations with Coulomb Force

Abstract

In this paper, we consider the two-dimensional compressible magnetohydrodynamic system with Coulomb force. We apply the method of relative entropy to establish the weak-strong uniqueness property of this system.

1. Introduction

In this paper, we consider the compressible magnetohydrodynamic equations with Coulomb force as follows:

Here and stand for the electron density, electron velocity, magnetic field and electrostatic potential, respectively. denotes the constant background ion density. is the pressure function with the constant and the adiabatic exponent . The constants are the viscosity coefficients satisfying and where is dimension. The constant is the magnetic diffusivity of the fluid. Let be a bounded smooth domain. This system can be applied to describe the dynamics of a charge transport where the compressible electron fluid interacts with its own electric field against a charged ion background together under the influence of the magnetic field. Usually, we denominate the equation (1) as the continuity equation, the equation (2) as the momentum balance equation and the equation (3) as the induction equation. The motion of such fluids is driven by two dominating body forces, namely, the Coulomb force (i.e. the electric field force) and the Lorentz force imposed on the fluid by the magnetic field. This is the reason why the physical phenomena of such fluids is more complicated.

In this paper, we consider the initial-boundary conditions of MHD system (1)–(4) as following: where denotes the outer normal vector of .

When there are no electromagnetic effect and no Coulomb force, the system (1)–(5) reduces to the compressible Navier-Stokes system. Many papers are devoted to the study of Navier-Stokes equations (see [1–5] and the references cited therein). These papers are focused on the existence of classical solution or weak solution and large time asymptotic behavior of solution et al. The weak-strong uniqueness is an important aspect in mathematical theory research of Navier-Stokes equations. The relative entropy method is an important method to research weak-strong uniqueness property of compressible Navier-Stokes equations. With the assistance of the concept of relative entropy, Germain [6] introduced a class of weak solution to the compressible Naiver–Stokes equations satisfying the relative entropy inequality with respect to a strong solution emanating from the same initial condition, and established the weak-strong uniqueness of this weak solution. Unfortunately, the existence of this class of weak solution is not known. Feireisl et al. [7] established the global existence of suitable weak solutions for any finite-energy initial data to compressible Navier-Stokes equations and proved the weak-strong uniqueness property within this class of suitable weak solutions by relative entropy inequality. By means of the relative entropy method, Feireisl et al. not only proved the weak-strong uniqueness of finite-energy weak solution to compressible Navier-Stokes equations with monotone pressure in [8], but also proved the weak-strong uniqueness property for compressible Naiver-Stokes equations with non-monotone pressure in [9]. When the pressure function satisfies a hard sphere law, the weak-strong uniqueness of Navier-Stokes equations is established by Feireisl et al. in [10] and Chaudhur in [11] applying the method of relative entropy.

When there is no electromagnetic effect, the MHD system (1)–(5) reduces to the compressible Navier-Stokes-Poisson system. Many researchers are interested to study the compressible NSP system (see [12–17] and the references cited therein). These papers are focused on the existence and asymptotic behavior of solution et al. For the weeak-strong uniqueness of the compressible Navier-Stokes-Poisson system, we have discussed in [18].

Many researchers are interested in studying MHD system because of its physical importance, mathematical complexity and extensive applications(see [19–23] and references cited therein). There are also many references about the system (1)–(4)(see [19, 24–26]). In [24], Tan and Wang consider global existence and large time behavior of weak solution to the system (1)–(4). The existence of the finite-energy weak solution to the problem (1)–(5) is established by Feireisl et al. in [19] with and Jiang et al. in [25] with . Although the existence of weak solution has been established, the uniqueness problem is still an open problem. So far, there are very few results concerning the uniqueness of weak solutions to the initial-boundary value problem (1)–(5). Only when the dimension is three and the Coulomb force is not taken into account in the system (1)–(5), the weak-strong uniqueness of the finite-energy weak solution to the system (1)–(5) is established by Yang et al. in [26]. Then, our goal of this paper is to consider the uniqueness property of weak solution to two-dimensional MHD system with Coulomb force. In spirit of Feireisl [8], we will establish the uniqueness property of weak solution to the system (1)–(5) by using of the method of relative entropy. Compared with [8], the main difficulty is to deal with the items about magnetic field and electrostatic potential.

The paper is organized as follows. In Section 2, we recall the definition of finite-energy weak solution for the compressible MHD equations (1)–(5) and state the main results. In Section 3, we derive the relative entropy inequality to the system(1)–(5). In last section, we prove the weak-strong uniqueness property of the compressible MHD equations with Coulomb force (1)–(5).

2. Main Results

In this section, we define the finite-energy weak solution to the initial-boundary value problem (1)–(5) and state the main results.

Definition 1. is called as a finite-energy weak solution of (1)–(5), if (i)belongs to the following class (ii) satisfies the energy inequality in , where the total energy is defined as (iii)For any , the pair satisfies the following Poisson equation (iv)Equation (1) holds in , i,e for any and . Moreover, satisfies the renormalized equation where such that (v)Equation Equation (2) (3) holds in , i,e for any .

Theorem 2. Assume that is a smooth domain. Let . Then for any given , there exist a finite-energy weak solution of the system (1)-(5).

Remark 3. (1)When , we can apply the method in [19] without any modifications to prove the existence of finite-energy weak solution(2)When , Jiang proved the existence of finite-energy weak solution of the system (1)–(5) in [25].Let . Then the following equalities hold Now we introduce main results of this paper.

Theorem 4. Let be a finite-energy weak solution to the MHD system (1)-(5) in the sense of Definition 1. Let and . Assume that satisfy Then the following relative entropy inequality holds for a.e where and the remainder is defined as

Theorem 5. Let . Assume that is the finite-energy weak solution of the magnetohydrodynamic system (1)–(5) in the sense of Definition 1. And assume that is the strong solution emanating from the same initial data and satisfy with . Then

3. Relative Entropy Inequality

In this section, we will establish the relative entropy inequality(18) to the system (1)–(5) and extend the admissible class of test function.

3.1. Proof of Theorem 4

Proof. For simplicity, we define symbols , , as Taking as test function in (10), we have Similarly, substituting for as a test function in (13), we can get Taking in (14), we get Multiplying (16) by and integrating over , we can obtain By virtue of satisfying (4) and (17), one has Differentiating (28) with respect to , we get Multiplying (29) by and integrating the resulting equation over , we can obtain From (7), we can deduce Summing up relations (24)–(31), we can infer Note that Then we can rewrite (32) as For simplicity, we abbreviate to .
Replacing with (16), we can infer Finally, we can get The proof of Theorem 4 is completed.☐

3.2. Extending the Admissible Class of Test Function

Making use of density argument, we can extend the class of test function appeared in the relative entropy inequality (18), (20).

For the left hand side of (18) to be well defined, the function must belong at least to the class:

Similarly, a short inspection of the integrals in (20) yields

Moreover, the function must be bounded away from zero, and

It is easy to prove that the relative entropy inequality (18), (20) can be extended to satisfying (37)–(42) by density argument.

4. Weak-Strong Uniqueness

In this section, we will prove weak-strong uniqueness property of the MHD system (1)–(5). In order to prove the weak-strong uniqueness, we firstly rewrite the remainder as another expression. Then we apply Gronwall’s inequality to prove the weak-strong uniqueness property.

4.1. Another Expression of

Lemma 6. Let be a strong solution to MHD system (1)-(5). For , the remainder can be rewritten as

Proof. Because is a strong solution to MHD system (1)-(5), we can get From (44), the remainder is rewritten as Note that Then we can rewrite as Besides, it is easy to infer that On the basis of (48) and (49), we can infer The proof of Lemma 6 is completed.☐

4.2. Proof of Theorem 5

To begin, it is easy to check that where is uniformly bounded for belonging to compact sets in .

Proof. There are two steps to prove Theorem 5.
Step 1: We estimate the remainder .
For and (51), it is easy to infer that By using of Hlder’s inequality and Sobolev’s inequality, we can deduce here is sufficient small and .
For , it is easy to get that Now, we estimate the last three items of . We can make use of Hlder’s inequality, Sobolev’s inequality and (51) to obtain where .
Note that By Hlder’s inequality and (57), we can obtain Therefor, we can get the estimate of with sufficiently small.
Similarly to the estimate of , we write Applying Hlder’s inequality, Sobolev’s inequality, (51) and (57), we can infer that with .
Finally, we estimate the last item of . From (51), we can get Using of Hlder’s inequality and (57), we can infer that with .
Step 2: Making use of Gronwall’s inequality, we can prove the weak-strong uniqueness of MHD system (1)–(5).
Substituting inequalities (52)–(64) into (18), we can obtain with . By Gronwall’s inequality, we can deduce in . Then this implies , , , . The proof of Theorem 5 is completed.☐

Remark 7. The strong solution we work with in the weak-strong uniqueness theorem is only a hypothetical object which we do not know if it even exist. In the future, we will consider the existence of this strong solution.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work of He and Zhou was supported by China NSF grant (11861025).