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Renormalized weak solutions to the three-dimensional steady compressible magnetohydrodynamic equations
Boundary Value Problems volume 2015, Article number: 182 (2015)
Abstract
We are concerned with the Dirichlet problem of the three-dimensional steady viscous compressible magnetohydrodynamic (MHD) equations. It is proved that for any specific heat ratio \(\gamma>1\), the Dirichlet problem of the steady compressible MHD equations on a bounded domain \(\Omega\subset\mathbb{R}^{3}\) admits a renormalized weak solution. Our method relies upon the weighted estimates of both pressure and kinetic energy for the approximate system, and the method of weak convergence developed by Lions and Feireisl.
1 Introduction
We consider the steady compressible magnetohydrodynamic equations in a bounded domain \(\Omega\subset\mathbb{R}^{3}\):
where \(\rho\geq0\), \(\mathbf{u}=(u^{1},u^{2},u^{3})\), and \(P(\rho)=\rho ^{\gamma}\) with \(\gamma>1\) being the specific heat ratio are the fluid density, velocity, and pressure, respectively. \(\mathbf {H}=(H^{1},H^{2},H^{3})\) is the magnetic field, \(\nu>0\) is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field. The constant viscosity coefficients μ and λ satisfy
f is a given vector field which models an outer force density.
Equations (1.1)-(1.4) are supplemented the following boundary conditions:
where g is a given function in Ω. Moreover, the total mass is prescribed,
There is huge literature on the studies about global existence of renormalized weak solutions of steady compressible flows. The important progress in the spatial three-dimensional case is due to the seminal work of Lions [1], where he obtained the existence of renormalized weak solutions of the Navier-Stokes equations for \(\gamma >\frac{5}{3}\), i.e., (1.1)-(1.2) with \(\mathbf {H}=\mathbf{0}\). However, as is well known, common fluids and gases in normal conditions (i.e., no high densities or high temperatures) are often well described by the ideal gas model, i.e., (1.1)-(1.2) with \(\mathbf{H}=\mathbf{0}\) for \(\gamma >1\). Furthermore, it is possible to deduce from the kinetic theory of gases that \(\gamma=\frac{5}{3}\) (for a monatomic gas). Hence, the most interesting region for physical applications is \(\gamma\in(1,\frac{5}{3}]\). Later, by combining Lions’ compactness framework of renormalized solutions and Feireisl’s oscillation defect measure on nonsteady compressible Navier-Stokes equations [2], Novo and Novotný [3] improved Lions’ result to \(\gamma>\frac{3}{2}\) for the potential force (i.e., \(\mathbf{f}=\nabla\phi\)). As emphasized in many papers (refer to [4–6] for instance), the condition on γ comes from the integrability of the density ρ in \(L^{p}(\Omega)\). The higher integrability of ρ has, the smaller γ can be allowed. By deriving a new weighted estimate of the pressure, the improved estimates of density have been suggested independently by Plotnikov and Sokolowski [7–9] and by Frehse et al. [10]. Using \(L^{\infty}\) estimates for the inverse Laplacian of the pressure together with the nonlinear potential theory, Březina and Novotný [11] proved existence of weak solutions with \(\gamma>\frac{1+\sqrt{13}}{3}\approx1.53\) for space periodic boundary conditions to avoid the lack of estimates near the boundary. Recently, Frehse et al. [12] treated \(\gamma>\frac{4}{3}\) for Dirichlet boundary conditions, where they relied on the momentum equation by a test function which provides the potential estimates for pressure, and by a bootstrap argument different from that used in [11]. Then, by obtaining weighted estimates for both the pressure P and the kinetic energy \(\rho|\mathbf{u}|^{2}\), Jiang and Zhou [5] showed the existence of spatially periodic weak solutions to the three-dimensional steady compressible Navier-Stokes equations for any \(\gamma>1\). It is worth noticing that the Dirichlet problem in [12] has been revisited very recently by Plotnikov and Weigant [13] for \(\gamma>1\). There are also some studies on existence results for steady compressible full Navier-Stokes equations, refer to [14–16] and references therein.
The relevant background of magnetohydrodynamic fluids can be found in [17–22] and references cited therein. The steady compressible magnetohydrodynamic system is investigated in [6] with \(\gamma>1\) for the periodic case. Recently, Yan [23] considered the three-dimensional full magnetohydrodynamic equations under slip boundary conditions for \(\gamma>\frac{4}{3}\). In fact, one of the important restrictions to the value of γ is due to the a priori estimates. It is a natural and interesting problem to investigate the existence of weak solutions to the Dirichlet problem (1.1)-(1.7) in dimension three. In fact, this is the main aim of the present paper.
Before stating the main result, we first explain the notations and conventions used throughout this paper. For \(1\leq p\leq\infty\) and \(k\geq1\), the standard Lebesgue and Sobolev spaces are defined as follows:
To simplify the notation, in what follows, we do not distinguish between function spaces for scalar and vector valued functions; e.g. both \(L^{p}(\Omega)\) and \(L^{p}(\Omega;\mathbb{R}^{3})\) are denoted \(L^{p}(\Omega)\). Generic constants are denoted by C, their values may vary in the same formula or in the same line.
Next, we give the definition of renormalized weak solutions to the steady MHD equations (1.1)-(1.4) as follows.
Definition 1.1
By a renormalized weak solution of the system (1.1)-(1.7) we mean a triple \((\rho,\mathbf{u},\mathbf{H})\in L^{\gamma}(\Omega )\times W_{0}^{1,2}(\Omega)\times W^{1,2}(\Omega)\) such that:
-
\(\rho\geq0\) a.e. in Ω, \(\int_{\Omega}\rho(x)\,dx=M\).
-
\(\operatorname{div}\mathbf{H}=0\) in Ω.
-
The mass equation (1.1) holds in the sense of a renormalized solution, i.e.,
$$ \operatorname{div}\bigl[b(\rho)\mathbf{u}\bigr]+\bigl[b'(\rho) \rho-b(\rho)\bigr]\operatorname{div}\mathbf{u}=0 \quad \text{in } \mathcal{D}'( \Omega) $$(1.8)for any \(b\in C^{1}(\mathbb{R})\) such that \(b'(z)=0\) when z is big enough.
Our main result can be stated as follows.
Theorem 1.1
Let \(\Omega\subset\mathbb{R}^{3}\) be a bounded domain with \(C^{2}\) boundary. Assume that \(\mathbf{f}\in L^{\infty}(\Omega)\) and \(\mathbf {g}\in H^{3}(\Omega)\) with \(\operatorname{div}\mathbf{g}=0\). Then for any \(\gamma >1\), there exists a renormalized weak solution to the problem (1.1)-(1.7) in the sense of Definition 1.1.
We now make some comments on the key ingredients of the analysis in this paper. The proof of Theorem 1.1 is based on the elaborate a priori uniform estimates of the approximate solutions and the weak convergence method in the framework of Lions [1]. First, inspired by [6, 24], we can construct an approximate scheme to the MHD system (1.1)-(1.7). Then we use a bootstrapping argument to obtain the a priori uniform estimates for the approximate solutions \((\rho_{\delta}, \mathbf{u}_{\delta}, \mathbf {H}_{\delta})\) for any \(\gamma>1\) in the framework of [13]. However, compared with the ones in [1, 13], the main difficulty in the present paper is caused by the magnetic field and its coupling interaction with the fluid variables. To overcome this difficulty, we need some careful analysis to recover all the a priori estimates. To this end, we shall consider the momentum equations and the magnetic field equations together. To pass to the limit and obtain the existence of a weak solution, we cannot directly use the arguments in [1], since \(\rho_{\delta}\in L^{p}(\Omega)\) (\(p>\frac{3}{2}\)) is required in [1] and this is not the case here. In fact, here we only have \(\rho_{\delta}\in L^{\gamma s}(\Omega)\) with some \(s>1\) being very close to 1 when γ is close to 1. Instead, we use the modified method in [5] to get the existence of a weak solution.
The rest of this paper is organized as follows. In Section 2, we collect some known facts that will be needed in later analysis. In Section 3, we construct a sequence of approximate solutions \((\rho _{\delta},\mathbf{u}_{\delta},\mathbf{H}_{\delta})\). In Section 4, the estimates independent of the parameter δ are obtained. Finally, we give a brief proof of our main result in Section 5.
2 Preliminaries
In this section we shall enumerate some auxiliary lemmas used in this paper.
We begin with an auxiliary function φ. To this end, we denote the distance function \(d(x)\) by
For every \(c>0\), the symbols \(A_{c}\) and \(\Omega_{c}\) stand for
Then we have the following consequence, whose proof can be found in Chapter 14.6 of [25].
Lemma 2.1
There is \(t>0\) depending only on Ω such that
Furthermore, there exists a function \(\varphi:\overline{A_{2t}\cup\Omega}\rightarrow\mathbb{R}\) satisfying:
-
\(\varphi\in C^{2}(\overline{A_{2t}\cup\Omega})\), \(\varphi(x)>0\) in Ω and \(\varphi=d(x)\) in \(A_{2t}\).
-
There is \(k>0\) such that \(\varphi(x)>k\) in \(\Omega\backslash\Omega_{2t}\).
Lemma 2.2
([5])
Let \(1< p_{1},p_{2},p<\infty\), \(p\leq p_{1}\), and Ω be a bounded domain in \(\mathbb{R}^{3}\). Suppose that
and
Then there is a subsequence of \(f_{n}g_{n}\) (still denoted by \(f_{n}g_{n}\)), such that
Remark 2.1
If (2.4)-(2.5) hold, combining Lemma 2.2 with Hölder’s inequality, we have
where \(\frac{1}{p_{1}}+\frac{1}{p_{2}}\leq\frac{1}{r}\).
The following Bogovskii lemma will be needed in Lemma 4.3; its proof can be found in [24].
Lemma 2.3
Let Ω be a bounded Lipschitz domain in \(\mathbb{R}^{3}\) and \(1< p<\infty\). Then there exists a positive constant \(C(p,\Omega)\) such that for any \(f\in L^{p}(\Omega)\) with \(\int_{\Omega}f(x)\,dx=0\), there is a vector field \(\boldsymbol{\phi}\in W^{1,p}(\Omega)\) satisfying
and
Finally, let \(W_{0}^{1,2}(\Omega)\) denote the Sobolev space of elements belonging to \(W^{1,2}(\Omega)\) with zero trace at the boundary ∂Ω. Then the following result plays a key role in the proof of Proposition 4.1.
Lemma 2.4
([13])
Let \(\Omega\subset\mathbb{R}^{3}\) be a bounded domain with \(C^{2}\) boundary. Let \(h\in L^{2}(\Omega)\) satisfy
Then there is \(C>0\) depending only on Ω such that for all \(\mathbf{u}\in W_{0}^{1,2}(\Omega)\),
3 Approximation
In this section, we briefly explain how to construct an approximative system to the problem (1.1)-(1.7).
First, we consider the following approximative problem:
with the boundary conditions
where \(\alpha,\varepsilon,\delta>0\), \(\tilde{\mu}\triangleq\lambda+\mu \), \(P_{\delta}(\rho)\triangleq\rho^{\gamma}+\delta\rho^{4}\), and h is a smooth function satisfying \(\int_{\Omega}h \,dx=M\).
Then we follow the same arguments as those in Section 4.3, pp.200-204 of [24].
But in the case of steady compressible MHD flows, the equations for the magnetic field are governed by an elliptic system and we cannot get the bound of \(\|\mathbf{H}\|_{L^{2}}\) and in turn \(\|\mathbf{H}\|_{H^{1}}\) in terms of the velocity coefficient u directly since here the coefficient u we consider is an arbitrary data. Instead, by the special structure of the magnetic field in the system, we can consider the existence of the approximate solutions to both the momentum equations and the magnetic field equations together by the Leray-Schauder fixed point theorem.
Making use of the method of weak convergence, we can pass to the limit \(\varepsilon\rightarrow0^{+}\) and \(\alpha\rightarrow0^{+}\) in (3.1) and (3.2) to get the existence of the weak solutions \((\rho_{\delta},\mathbf{u}_{\delta},\mathbf{H}_{\delta})\) to the following system:
with the boundary conditions (1.6).
More precisely, we have the following.
Lemma 3.1
Let \(\delta\in(0,1]\). Then there exists at least a renormalized weak solution \((\rho_{\delta},\mathbf{u}_{\delta},\mathbf{H}_{\delta})\) to the system (3.6)-(3.9) such that for any \(\boldsymbol{\xi}\in W_{0}^{1,2}(\Omega)\), \(\zeta\in C^{\infty}(\Omega)\), and \(\psi\in C^{1}(\Omega)\) satisfying
we have
and
Proof
The existence of solutions satisfying (3.10) and (3.12) can be found by the method of [1, 24]. Multiplying (3.7) by ξ and integrating over Ω, we obtain (3.11). It remains to show (3.13). Choosing \(\boldsymbol{\xi}=\mathbf{u}_{\delta}\) in (3.11), we get
where we have used
Inserting \(\psi(\rho_{\delta})\triangleq\frac{1}{\gamma-1}\rho_{\delta }^{\gamma} +\frac{\delta}{3}\rho_{\delta}^{4}\) and \(\zeta\equiv1\) into (3.12), we deduce that
Multiplying (3.8) by \(\mathbf{H}_{\delta}\) and integrating by parts, we derive
which combined with (3.14) and (3.15) gives (3.13) and finishes the proof of Lemma 3.1. □
4 Uniform estimates
In this section, we will establish uniform with respect to δ estimates for the approximate solutions \((\rho_{\delta},\mathbf{u}_{\delta },\mathbf{H}_{\delta})\). In what follows, to simplify notations, we omit the subscript δ in \((\rho_{\delta},\mathbf{u}_{\delta},\mathbf {H}_{\delta})\).
Following Plotnikov and Weigant [13], we shall perform a bootstrapping argument through the parameters
where
Then we have the following.
Proposition 4.1
For A defined by (4.1), there exists a positive constant C depending only on γ, λ, μ, ν, M, Ω, and \(\|\mathbf{f}\|_{L^{\infty}}\) such that
where the quantities s, q are denoted by
Here, θ and β are as in (4.2).
The proof of Proposition 4.1 will be postponed at the end of this section.
We begin with the following standard energy estimate for \((\mathbf {u},\mathbf{H})\), which shows that the \(H^{1}\)-norm of \((\mathbf {u},\mathbf{H})\) can be bounded in terms of A and B.
Lemma 4.1
Let θ, β be as in (4.2) and s be as in (4.4), then there exists a positive constant C depending only on γ, λ, μ, ν, M, Ω, and \(\|\mathbf{f}\|_{L^{\infty}}\) such that
Proof
It follows from (3.13) that
Thus, we obtain from (1.6), (1.4), and the Poincaré inequality
Noting that
where all four exponents are positive and their sum is equal to one. Hence, applying Hölder’s inequality and recalling (4.1), we arrive at
which combined with (4.6) yields (4.5) and completes the proof of Lemma 4.1. □
From now on, in what follows, the function φ is as in Lemma 2.1.
Lemma 4.2
Let β be as in (4.2), then there exists a positive constant C depending only on γ, λ, μ, ν, M, Ω, and \(\|\mathbf{f}\|_{L^{\infty}}\) such that
Proof
Motivated by [13], we introduce the vector field
By straightforward computation, we have
Then it follows from (4.2) and Lemma 2.1 that
Hence, we have
By the definition of β in (4.2), it is not hard to see that \(\beta\in(0,\frac{1}{2})\). Then, substituting (4.8) into (3.11), we thus get
We can bound two terms on the left-hand side of (4.9) as
Hence, inserting (4.10)-(4.11) into (4.9), we deduce the desired result (4.7). □
Lemma 4.3
Let β, s be as in (4.2) and (4.4), respectively, then there exists a positive constant C depending only on γ, λ, μ, ν, M, Ω, and \(\|\mathbf{f}\|_{L^{\infty}}\) such that
Proof
Choosing a function \(h\in L^{\frac{s-1}{s}}(\Omega)\), it follows from Lemma 3.1 that the problem
has a solution \(\boldsymbol{\phi}\in W_{0}^{1,\frac{s}{s-1}}(\Omega)\) satisfying
By the definition of s in (4.4), we obtain \(s\in(1,\frac {33}{32})\), thus \(\frac{s}{s-1}>3\). Then Sobolev’s embedding theorem gives \(W_{0}^{1,\frac{s}{s-1}}(\Omega)\hookrightarrow C(\overline{\Omega})\). Hence,
For \(x\in\Omega\), setting
Straightforward calculation yields
Thus, we derive from (4.14) that
Inserting (4.16) into (3.11) and using (4.18), we find that
where in the first inequality we have used
By virtue of (4.16) and (4.13), we see that
which ensures that
By Hölder’s inequality and the boundedness of φ, we have
For the last term on the right-hand side of (4.20), we obtain from (4.15)
which combined with (4.20) and (4.21) gives
Consequently, we deduce from (4.19) and (4.22) that
Then by the duality argument, we get the desired result (4.12). □
Lemma 4.4
Let θ, β be as in (4.2) and s be as in (4.4), then there exists a positive constant C depending only on γ, λ, μ, ν, M, Ω, and \(\|\mathbf{f}\|_{L^{\infty}}\) such that
Proof
It follows from (4.7) and (4.12) that
By Hölder’s and Young’s inequalities, we get
Substituting (4.5), (4.27), (4.28), and (4.26), we arrive at, for every \(\varepsilon>0\),
Recalling (4.2), after a straightforward computation, we find that
Consequently, we obtain from (4.29)
Since \(P_{\delta}\geq\rho^{\gamma}\), we have
Inserting this into (4.30) and choosing ε sufficiently small, we deduce the estimate (4.23).
We now turn to showing that (4.24) and (4.25) hold. First of all, (4.23) and (4.31) imply
which together with (4.5) yields
We infer from the definition of θ in (4.2) that
which along with (4.33) gives (4.24). It remains to prove (4.25). Substituting (4.32) into (4.28), we derive
By the definition of θ in (4.2), it is not hard to show that
which combined with (4.34) leads to (4.25). □
In terms of A, we can derive the following weighted estimates for \(P_{\delta}\) and \(\rho|\mathbf{u}|^{2}\).
Lemma 4.5
Let β, θ be as in (4.2), then there exists a positive constant C depending only on γ, λ, μ, ν, M, Ω, and \(\|\mathbf{f}\|_{L^{\infty}}\) such that for every \(\alpha\in(0,1)\) and \(x_{0}\in\Omega\), we have
Proof
Fix \(\alpha\in(0,1)\) and \(x_{0}\in\Omega\), we denote by
Direct calculus gives
Combining this with \(\beta\in(0,\frac{1}{2})\), we get
Hence, \(\boldsymbol{\xi}\in W_{0}^{1,r}(\Omega)\) for all \(r\in[1,\frac{3}{\alpha})\). In particular,
Thus, substituting ξ into (3.11) yields
The left-hand side of (4.37) can be estimated as follows.
On the one hand, we have
On the other hand, direct computation shows that
Then we obtain from integration by parts and Hölder’s inequality
Inserting (4.38) and (4.39) into (4.37), we derive that
which together with (4.23)-(4.25) implies that
The proof of Lemma 4.5 is finished. □
Lemma 4.6
Let θ, β be as in (4.2), then there exists a positive constant C depending only on γ, λ, μ, ν, M, Ω, and \(\|\mathbf{f}\|_{L^{\infty}}\) such that for every \(x_{0}\in\Omega\), we have
Proof
Note that
Hence, due to \(\rho^{\gamma}\leq P_{\delta}\), we obtain from Young’s inequality
Integrating both sides of this inequality over Ω and using (4.35), we get (4.40) and finish the proof of Lemma 4.6. □
In order to show the boundedness of \(\|\rho|\mathbf{u}|^{2}\|_{L^{s}}\), we have some delicate analysis. In view of Lemma 2.1, there is \(t>0\) such that \(\varphi (x)=d(x)\) in \(A_{2t}\). Introduce the vector field
Fix an arbitrary \(\alpha\in(0,1)\) and \(x_{0}\in A_{t}\). Define the vector field
where
Then we have the following properties of ξ, whose proof can be found in Appendix A of [13].
Lemma 4.7
There is a positive constant C depending only on α and Ω such that for every \(x,x_{0}\in A_{t}\) and for every \(\mathbf{u}\in\mathbb {R}^{3}\), we have
Lemma 4.8
Let \(\alpha\in(0,1)\) and \(x_{0}\in\Omega\). Assume that \(\zeta\in C^{\infty }(\overline{\Omega})\) satisfies
then there exists a positive constant C depending only on γ, λ, μ, ν, M, Ω, \(\|\mathbf{f}\|_{L^{\infty}}\), α, and ζ such that
Proof
If \(x_{0}\in\Omega_{t}\). Let ξ be as in (4.42). Noting that
which combined with (4.46) yields
Hence, replacing ξ by ζ ξ in (3.11) implies
By Hölder’s inequality, it follows that
Substituting the above estimates into (4.48), we obtain, for all \(x_{0}\in A_{t}\),
It follows from (4.44) that, for \(x\in A_{t}\),
which together with (4.45) and (4.49) leads to
Since \(|\varphi(x)-\varphi(x_{0})|\leq|x-x_{0}|\), we have \(\Delta_{-}(x,x_{0})\leq C|x-x_{0}|\). Combining this with (4.50) gives, for all \(x_{0}\in A_{t}\),
If \(x_{0}\in\Omega\backslash A_{t}\). Since ζ vanishes in \(\Omega \backslash A_{t/2}\), the inequality \(2|x-x_{0}|\geq t\) holds for all \(x\in \operatorname{supp}\zeta\) and \(x_{0}\in\Omega\backslash A_{t}\), and hence for all \(x_{0}\in \Omega\backslash A_{t}\),
which combined with (4.51) yields the desired estimate (4.47). □
Lemma 4.9
For every nonnegative function \(\eta\in C_{0}^{\infty}(\Omega)\) and every \(x_{0}\in\Omega\), there exists a positive constant C depending only on γ, λ, μ, ν, M, Ω, \(\|\mathbf{f}\|_{L^{\infty}}\), and η such that
Proof
Fix an arbitrary \(x_{0}\in\Omega\), letting
Obviously, \(\nabla\boldsymbol{\phi}\leq\frac{C}{|x-x_{0}|}\). Thus \(\eta\boldsymbol{\phi }\in W_{0}^{1,2}(\Omega)\) and
Integral identity (3.11) with ξ replaced by η ϕ implies
which combined with (4.53) and \(|\boldsymbol{\phi}|=1\) leads to
On the other hand, direct computations give
Inserting (4.55) into (4.54), we deduce the desired result (4.52). □
Lemma 4.10
Let \(\alpha\in(0,1)\). Then for every \(x_{0}\in\Omega\), there exists a positive constant C depending only on γ, λ, μ, ν, M, Ω, \(\|\mathbf{f}\|_{L^{\infty}}\), and α such that
Proof
Choose a nonnegative function \(\eta\in C^{\infty}(\Omega)\) such that η equals 1 in a neighborhood of ∂Ω and η vanishes in \(\Omega\backslash\Omega_{t/2}\). In particular, we have \(1-\eta\in C_{0}^{\infty}(\Omega)\). We obtain from Lemmas 4.8 and 4.9
By virtue of Young’s inequality, we have
Integrating both sides over Ω and noting that \(1+\alpha<2\), we get
which together with (4.57) implies (4.56) and completes the proof of Lemma 4.10. □
With Lemmas 4.1-4.10 at hand, we are now in a position to prove Proposition 4.1.
Proof of Proposition 4.1
It follows from (4.40) and Lemma 2.4 that
which combined with (4.24) implies that
Since \(1-\frac{\theta}{2}\in(0,1)\), we easily obtain from (4.59)
which together with (4.24) yields
Note that
Making use of Young’s inequality, we derive from (4.23) and (4.60) that
It remains to show that
Set \(\alpha\triangleq\frac{2s}{\gamma(3-s)}\). It follows from (4.4) and (4.2) that \(\alpha\in(0,1)\). Thus, we obtain from Lemma 2.4, (4.56), (4.61), and (4.62)
On the other hand, we have
Hence
Notice that
Then Hölder’s inequality yields
It follows from Sobolev’s embedding theorem that
which combined with (4.65)-(4.66) implies
Since \(\frac{3-s}{2}< s\), (4.63) easily follows from (4.67).
This completes the proof of Proposition 4.1. □
5 Proof of Theorem 1.1
According to (4.3) and the compact embedding \(W^{1,2}(\Omega) \rightarrow L^{p}(\Omega)\) for \(p\in[1,6)\), we can choose a subsequence such that
Combining this with Lemma 2.2 and (4.3), we obtain
Then, passing to the limit \(\delta\rightarrow0\) in the approximate equations (3.6)-(3.9), we get
Hence, to complete the proof of Theorem 1.1, we have to show the strong convergence of \(\rho_{\delta}\) to ρ in \(L^{1}(\Omega)\). This task can be fulfilled following Section 4.11, pp.239-245 in [24] and we will not give the details here.
The proof of Theorem 1.1 is finished.
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Acknowledgements
The authors appreciate the anonymous referees for useful comments and suggestions on our manuscript, which improved the presentation of this article. The research of Xiaoying Wang was supported by NSFC (U1430103), Beijing Higher Education Young Elite Teacher Project (YETP0724) and the Fundamental Research Funds for the Central Universities (13MS35).
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Wang, X., Zhou, L. Renormalized weak solutions to the three-dimensional steady compressible magnetohydrodynamic equations. Bound Value Probl 2015, 182 (2015). https://doi.org/10.1186/s13661-015-0448-8
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DOI: https://doi.org/10.1186/s13661-015-0448-8