Abstract
We consider in an infinite horizontal layer the stationary motion of a viscous compressible fluid in a magnetic field subject to the gravitational force, where the Dirichlet boundary condition for the velocity and similar but non-homogeneous and large enough conditions for the magnetic field are assumed. Existence of a stationary solution in a neighborhood close to the equilibrium state is obtained in Sobolev spaces as limit of a sequence of fixed points of some suitable operators.
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1 Introduction
Magnetohydrodynamics (MHD) describes the dynamics of conducting fluids, such as gases, under the in influence of an electromagnetic field. If a conducting fluid moves in a magnetic field, electric fields are induced and an electric current flow is developed. The magnetic field exerts forces on these currents which considerably modify the hydrodynamic motion of the fluid. There is a complex interaction between the magnetic field and fluid dynamic phenomena, and both hydrodynamic and electrodynamic effects have to be considered. The applications of magnetohydrodynamics cover a very wide range of physical areas from liquid metals to cosmic plasmas, for example, the intensely heated and ionized fluids in an electromagnetic field in astrophysics, geophysics, high-speed aerodynamics and plasma physics. The set of equations which are obtained by neglecting the motion of electrons and consider only heavy ions, are a combination of the compressible Navier–Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism, connecting the magnetic field, plasma velocity, pressure and density.
This paper is concerned with the steady compressible magnetohydrodynamic system with the gravitational force on \( \Omega = \mathbb {R}^2\times ]0,h[\)
with the following boundary conditions
where \({\bar{B}}\in \mathbb {R}^3\) is a given constant field while \(\delta (x')\in \mathbb {R}\) is small perturbation of magnetic field \({\bar{B}}\) on the upper plane \(x_3 = h\) and \(x'= (x_1,x_2)\) is a generic point in \(\mathbb {R}^2\) so that \(x=(x',x_3)\in \Omega = \mathbb {R}^2\times ]0,h[\). As to \(B_{\infty }(x_3)\in \mathbb {R}^3\) and \(\varrho _{\infty }(x_3)>0\) will be defined below (see (1.10)). We will assume that \(\varrho _{\infty }\in L^1([0,h])\) and
where \(M>0\) is given.
In the system (1.1)–(1.3), the unknowns \(v,B:\Omega \rightarrow \mathbb {R}^3\) and \(\varrho :\Omega \rightarrow (0,\infty )\) stand for the velocity field, magnetic field and density respectively. The physical constants \(\mu \) and \(\lambda \) are the shear viscosity and bulk viscosity of the fluid respectively and satisfy \(\mu >0\) and \(2\mu +3\lambda >0\), while the other physical constant \(\nu > 0\) is the magnetic diffusivity coefficient, g the gravitational acceleration and \(e_3 =(0,0,1)\) is the horizontal axis.
We assume that the pressure p is given by the law of viscous barotropic fluid
We look for a stationary solution in a neighborhood close to the equilibrium state \((0,B_{eq},\varrho _{eq})\) corresponding to the perturbation \(\delta \) in (1.4) identically zero. As can easily be seen, this rest state is given by the following density and magnetic field distributions
where the only positive constant \(C_M\) is determined by the conditions
This being, we can now specify the boundary conditions (1.5) for magnetic field and density. We then set (see (1.8))
From now on we make the assumptions
where \(B_0>0\) and \(M_0>0\) are given large enough.
As to the small perturbation \(\delta \) of the given magnetic field \({\bar{B}}\) on the upper plane \(x_3=h\), we assume that
where \(\delta _0>0\) is given small enough. According to (1.9) and (1.8), we have
Hence, (see (1.11))
where \(M_0\) is large enough so that we definitely get
For the equations (1.1)–(1.3), Zhou [48] proved the existence of a spatially periodic weak solution to the steady compressible isentropic MHD equation with \(p(\varrho ) = a\varrho ^\gamma \) (\(a>0\)) in \(\mathbb {R}^3\) for any specific heat ratio \(\gamma >1\) and a periodic external force \(f\in L^\infty (\mathbb {R}^3)\). To the best our knowledge, this is the first result concerning the steady MHD equations of compressible flows. For the incompressible case, we refer the interested reader to the articles [1,2,3,4, 26, 34] for the existence of strong and weak solutions. In [23], Gerbeau et al. also considered several kinds of unsteady problems and gave some numerical analysis (see also [41]). Because of its physical importance, complexity and mathematical challenges, there have been numerous studies on unsteady MHD by physicists and mathematicians in the recent years (see e.g., [16, 18,19,20,21, 31, 32, 45] and the references cited therein). In particular, the one-dimensional problem has been studied in many papers, see, for examples, [6, 10, 15, 33]. On the other hand, there has been considerable interest in developing accurate and reliable numerical methods for the study of MHD system of equations (see e.g., [5, 12, 24, 29, 35, 40, 42]).
Almost all the literature mentioned above is concerned with the Cauchy problem for (1.1)–(1.3) or the initial boundary value problem for compressible MHD equations, with the homoegenous Dirichlet condition on the magnetic field. In contrast with the extensive researches on unsteady MHD flow, we find that there are only few results concerning the steady flow. In [46], Yang et al. have established the existence and uniqueness of a strong solution to the steady magnetohydrodynamic equations for the compressible barotropic fluids in a bounded smooth domain with a perfectly conducting boundary, under the assumption that the external force field is small. This solution is obtained in a neighborhood close to a constant equilibrium state.
In [7] the authors have investigated the question of a stationary motion in an horizontal layer of a viscous compressible fluid in the presence of a magnetic field subjected to gravitational force. By imposing periodic conditions in the horizontal directions, they showed that a non-homogeneous distribution of the magnetic field given around a large constant on the upper plane can generate a stationary motion in a state close to the equilibrium state \((0,B_{eq},\varrho _{eq})\) (see (1.8)).
In this work, we consider in an infinite horizontal layer the stationary motion of a compressible viscous fluid in a magnetic field subjected to the gravitational force with boundary conditions on the magnetic field close to a arbitrarily large constant. As will be seen later, the mathematical analysis of the equations describing this motion in question presents serious challenges.
Remark 1.1
If the perturbation \(\delta \) of magnetic field \({\bar{B}}\) on the upper plane \(x_3 = h\) (see (1.4)) is not identically zero, and if there exists a solution \((v,B,\varrho )\) of system equations (1.1)–(1.5), then necessarily v is not identically equal to zero (see Remark 7.1 section 5). This is obtained from the main result (Theorem 2.1 and see also Remark 2.1) in the next section.
2 Main Result and Preliminary of Proof
We recall that we look for a solution \((v,B,\varrho )\) in a small neighborhood of the stationary profile \((0,\widehat{B},\widehat{\varrho })\) close to the equilibrium state \((0,B_{eq},\varrho _{eq})\) given in (1.8). The stationary profile is defined by
where \(\widehat{b}\) is a solution of the boundary value problem
According to (1.12)\(_3\), there exists (see e.g. [11, 22]) at least one solution \(\widehat{b}\in H^1_0(\Omega )\). Moreover we have
Note that, considering (2.2) and (2.1), we have
This being so, we set then
and, since as we seek \(\varrho \) close to \(\widehat{\varrho }\), we can assume that
By considering the set of identities:
and (2.5) (see also (2.4)), we rewrite the problem (1.1)–(1.5) with unknowns v, B and \(\varrho \) into a new problem with unknowns v, b and \(\sigma \) as follows:
with the boundary conditions
where the functions \(F = F(v,b,\sigma )\) and \(G=G(v,b,\sigma )\) are given by
where \(0<\xi <1\) is a regular function depending on \(\widehat{\varrho }\) and \(\widehat{\varrho } + \sigma \).
Notice that the Eq. (2.9) is obtained from (1.3) which is written simply
because \(\nabla \times (\nabla \times B) = \nabla \nabla \cdot B - \Delta B\), \(\nabla \cdot B = 0\), \(\nabla \cdot \widehat{B} = 0\) (see (2.4)) and so \(\nabla \cdot b= 0\). Thus the problem (1.1)–(1.5)) is reduced to find
satisfying the problem (2.7)–(2.11). Our main result is the following theorem.
Theorem 2.1
Under (1.11), the problem (1.1)–(1.5) admits at least one solution \((v,B,\varrho )\) such that
where \(\Omega '_h = \Omega '\times ]0,h[\) and \(\Omega '\) is an arbitrary large bounded domain of \(\mathbb {R}^2\).
Remark 2.1
This result means that the non homogeneous repartition of the magnetic field around the constant field \({\bar{B}}\) in the upper plane \(x_3 = h\) can generate a stationary motion close to the equilibrium state \((0,B_{eq},\varrho _{eq})\) given by (1.8).
Remark 2.2
We look for a solution
of the system of equations (1.1)–(1.3) with boundary conditions (1.4) and (1.5)), where
is the appropriate solution of problem (2.7)–(2.11).
By recalling here (see (2.1), (1.8) and (2.2)) the expressions of \( \widehat{\rho }\) and \( \widehat{B}\) namely,
we see that \(\widehat{\varrho }\) is only in \(H^2_{\text{ loc }}(\Omega )\) because it only depends on \(x_3\) and \(\Omega =\mathbb {R}^2\times ]0,h[\). We also have \( \widehat{B}\notin L^2(\Omega )\) otherwise, we would get that the constant field \(\bar{B}=\widehat{B}-\displaystyle {x_3\over h}\delta e_3 - \widehat{b}\in L^2(\Omega )\) (with \(\Omega \) being unbounded). However, we have \(\nabla \widehat{B}\in H^1(\Omega )\). This justifies why we only have local regularity for B and \(\varrho \) in the theorem 2.1.
The solution \((v,b,\sigma )\) will be obtained as limit of a sequence fixed points of some operators built starting from a suitable linearization of the system of equations (2.7)–(2.9).
3 Linearized Equations
Let D be an open set of \(\Omega = \mathbb {R}^2\times ]0,h[\). We put
Furthermore, we set
Note that if D is bounded the embedding \(\mathbb {H}(D)\hookrightarrow \mathbb {H}_0(D)\) is compact.
Now, let \(u'=(v',\vartheta ',\sigma ')\in V\) and \(k>0\). We consider the equations system
with the conditions (2.10) and (2.11) where (see (2.12) and (2.13)),
We obtain in the following lemma, an existence result for the system (3.7)–(3.9) with the boundary conditions (2.10) and (2.11). In the sequel we note \(\left\Vert {\cdot } \right\Vert _{H^k}\) and \(\left\Vert {\cdot } \right\Vert _{L^2}\) the norm of \(H^k(\Omega )\) (\(k\ge 1\)) and \(L^2(\Omega )\). As for the constant \(c_\Omega \), it designates any constant depending only on \(\Omega \), \(\widehat{B}\) and \(\widehat{\varrho }\).
Lemma 3.1
Let \(u' = (v',b',\sigma ')\in V\). If \(k>0\) is large enough, the system (3.7)–(3.9) with the conditions (2.10) and (2.11) has a unique solution \(u=(v,b,\sigma )\in V\) such that
Proof
Let us start by showing that the system of Eqs. (3.7) and (3.8) with the boundary conditions (2.10) and (2.11) has a unique weak solution
(see (3.1)) such that
where
Indeed, by endowing \(H^1_0(\Omega )\times \widehat{H}^1_0(\Omega )\) with the norm \(\Vert (v,b)\Vert = (\Vert \nabla v\Vert _{L^2}^2 + \Vert \nabla b\Vert _{L^2}^2)^{1\over 2}\) which makes it a Hilbert space, we can easily see that the bilinear form a is continuous on \({H}^1_0(\Omega )\times \widehat{H}^1_0(\Omega )\). Moreover, since
we have
So, the bilinear form a is coercive and, since the linear form L is obviously continuous over \(H^1_0(\Omega )\times \widehat{H}^1_0(\Omega )\), according to the Lax-Milgram theorem, the problem (3.14) has a unique solution \((v,b)\in H^1_0(\Omega )\times \widehat{H}^1_0(\Omega )\) such that
Now, by standard \(L^2\)-regularity theory of linear elliptic systems, according to (3.15) we have (3.11) and (3.12). As for the solution \(\sigma \in {H}^2(\Omega )\) of equation (3.9) and its estimate (3.13) we refer the interested reader to [6, 37] for instance. \(\square \)
According to the lemma 3.1, we then define the nonlinear operator S as follows
where \(u= (v,b,\sigma )\in V\) is the unique solution of the system (3.7)–(3.9) with the boundary conditions (2.10) and (2.11).
Lemma 3.2
The nonlinear operator \(S: V_0\rightarrow V_0\) is continuous. More precisely, for any bounded subset D of V, there exists \(c_D>0\) such that
Proof
Let us note firstly that S transforms a bounded set of V into a bounded set of V. Indeed, taking into account (3.10) (see also (2.6), (2.3), (2.1) and (1.12)\(_2\)), it is easy to see that, for every \(u'=(v',\vartheta ',\sigma ')\in V\), we have
Hence, by recalling the V norm (see (3.6))
Let D be a V-bounded set. From (3.11)–(3.13) and the previous inequalities, if k is large enough, one sees easily that
this means that S(D) is a bounded set of V. Now, let
We write
By the definition (3.16) of S, we have
From (3.10) (see also (2.6), (2.12) and (2.13)), it is easy to see that
where P is polynomial function of two variables. With the same arguments as those of the Lemma 3.1, we can see that the system of Eqs. (3.22) and (3.23) has a unique weak solution (v, b) such that (see (3.21) and (3.6))
It remains us to estimate in \(H ^1\) norm the solution \(\sigma \) of Eq. (3.24). We first multiply that equation by \(\sigma \) and we integrate over \(\Omega \). Using
we easily obtain
We next apply the nabla operator \(\nabla \) to the Eq. (3.24), multiply the new equation by \(\nabla \sigma \) and integrate over \(\Omega \). Integration by parts and the use of the inequality
give us
Adding (3.26) and (3.27) we obtain according to (3.25) (see also (3.20) and (3.21)),
According now to (3.28) and (3.25), we obtain
If k is large enough so that (see (3.19)) \(k - c_\Omega \left\Vert {S(u'_1)} \right\Vert _{V}\ge k - c_\Omega c_{D}\ge 1\) from the previous inequality it follows
from which (see (3.21)) follows (3.17) and this completes the proof of the Lemma 3.2. \(\square \)
Remark 3.3
The \(V_0\)-Continuous operator S (i.e continuous in the norm of \(V_0\)) will be viewed as realizing the hypotheses of the Schauder fixed point theorem. Following the work done in [7, 8], we can show that for a sufficiently small, we have that \(S(B^a_V)\subseteq B^a_V\) where \(B^a_V = \overline{B(0,a)}\) is the closed ball of V with radius a. However, \(\Omega = \mathbb {R}^2\times ]0,h[\) being unbounded, the closed ball \(B^a_V\) is not \(V_0\)-compact (because the embedding \(H^k(\Omega )\hookrightarrow H^{k-1}(\Omega )\)) is not compact when \(\Omega \) is unbounded). In order to overcome this lack of compactness, our strategy here largely inspired by the work of [8], is to construct a sequence \(V_n\) of closed subspaces of V, closed convex subsets \(B^a_n\) of \(V_n\) which are \(V_0\)-compact and also operators \(S_n\) such that \(S_n: B^a_n\rightarrow B^a_n\) is \(V_0\)-continuous. We then apply the Schauder fixed point theorem to each operator operator \(S_n\) and get that \(S_n\) has a fixed point \(u_n\) in \(B^a_n\). Then, we will show that the sequence \((u_n)\) of fixed points converges to some \(u = (v,b,\sigma )\in V\) which solves the boundary problem (2.7)–(2.11).
The following section is devoted to the construction of the subspaces \(V_n\), the bounded closed convex subsets of \(V_n\) relatively \(V_0\)-compact and finally the operators \(S_n\).
4 Operators \(\mathbf{S_\textbf{n}}\) and their Properties
Let \(\tau \in C^\infty (\mathbb {R}^2)\) such that for all \(x'=(x_1,x_2)\in \mathbb {R}^2\),
We recall that \(\Omega =\mathbb {R}^2\times ]0,h[\) and we set
and (see (3.2))
Lemma 4.1
For any \(n\ge 1\) integer, \(V_n\) is a closed subspace of V.
Proof
It is clear that \(V_n\) is a subspace of V (see (3.5)). Let now \((u_p)\) be a sequence of \(V_n\) strongly convergent in V to some u. It is clear that \(u= 0\) outside \(\Omega _n\). It remains to show that \(u = \tau _n w\) with \(w\in {\mathbb {H}}_n\). Indeed, since \(u_p = \tau _n w_p\) in \(\Omega _n\), where \(w_p=(v_p,b_p,\sigma _p)\in {\mathbb {H}}_n \), we have (see (3.3) and (3.5))
so the sequence \((u_p)\) strongly converges in \({\mathbb {H}}(\Omega _n)\). On the other hand, since
because \(H^k(\Omega _n)\hookrightarrow C^{k-2}({\bar{\Omega }}_n)\) (\( k = 2,3,\cdots \)), it follows that each component of the sequence \((u_p)\) converges in \(C^0({\bar{\Omega }}_n)\) and hence, \((w_p)\) converges pointwise on \(\Omega _n\) to some \(w = (v,b,\sigma )\). Therefore \(u = \tau _n w\). To complete the proof, let us show that \(w\in {\mathbb {H}}_n\). In fact, notice first that \(w\in {\mathbb {H}}(\Omega _n)\) since \(u=\tau _n w\in {\mathbb {H}}(\Omega _n)\) (being in V). On the other hand, since \(u_p =\tau _n w_p\) satisfies the conditions (2.10) (because \(w_p\in {\mathbb {H}}_n\)), taking into account the continuous embedding (4.6) and the strong convergence of the sequence \((u_p)\) in \({\mathbb {H}}(\Omega _n)\), we deduce that u verifies (2.10). So \(w\in {\mathbb {H}}_n\) and therefore \(u\in V_n\). \(\square \)
Lemma 4.2
Let
Then \(B_n^a\) is a closed bounded convex set of \(V_n\).
Proof
Since (see (4.1))
notice that
where \( \nu ^0_n = 0\) and for all \(k\ge 1\) fixed integer \(\nu ^k_n\downarrow 0\). Hence (see (3.5)),
where \( \nu _n\downarrow 0\). Since \(V\subset {\mathbb {H}_n}\) it follows that \(B_n^a=\tau _nB_V^a\subset V_n\) and is convex and bounded (see (4.10)).
Let us now show that \(B_n^a\) is closed in V. To this aim, it sufficient to show that it is weakly closed in V. So, let \((u_p)\) with \(u_p = \tau _n w_p\), be a sequence of \(B_n^a\) which converges weakly in V to some \(u\in V\). Hence, we get that
Indeed, from the fact that \(w_p\in B_V^a \) we can extract a subsequence \((w_{p'})\) which converges weakly in V to \(w\in B_V^a\) and hence, \((u_{p'})\) converges weakly to u in V since \((u_{p'})\) with \(u_{p'}= \tau _n w_{p'}\), is a subsequence of \((u_p) \). On the other hand, as the linear mapping \( L: V \rightarrow V\) defined by \(L (w): = \tau _n w\) is (see (4.10)) strongly continuous, it follows that L is also weakly continuous (see e.g., [13, Theorem 3.10 page 61]). Therefore, \(L(w_{p'})\rightharpoonup L(w)\) i.e. \(u_{p'}\rightharpoonup \tau _n w = u\) and hence, (4.11) holds. \(\square \)
Lemma 4.3
\(B_n^a\) is \(V_0\)-compact.
Proof
Let \((u_p)\) with \( u_p =\tau _nw_p \), be a sequence of \(B_n^a = \tau _nB_V^a\). Given (4.10) (see also (3.5)), we have
and hence, \(\tau _n w_p \) is bounded in \({\mathbb {H}}(\Omega _n)\). So from the compact embedding \({\mathbb {H}}(\Omega _n) \hookrightarrow {\mathbb {H}}_0(\Omega _n)\), there exists a subsequence \(u_{p'} = \tau _n w_ {p'}\) of \(u_p \) which converges strongly to some u in \({\mathbb {H}}_0(\Omega _n) \) and weakly in V. Since \( w_{p '} \in B_V^a \), passing if necessary to a further subsequence, we can assume that \((w_{p'}) \) converges weakly in V to some \(w\in B_V ^a\). Arguing as (4.11), we find that \( u = \tau _n w\). Since (see (3.6) and (3.4))
recalling that \(\tau _n w_{p'}\) converges strongly in \({\ H_0(\Omega _n)}\) to \(\tau _n w\), we see that from any sequence \(\tau _n w_{p}\in \tau _n B_V^a\) on can extract subsequence \(\tau _n w_{p'} \) strongly convergent in \(V_0\) so \(\tau _nB_V^a\) is \(V_0 \)-compact. \(\square \)
Now we are in a position to define the operators \(S_n\)
Lemma 4.4
Set \(S_n:= \tau _n S\). Then, \(S_n: V_n\rightarrow V_n\) is \(V_0\)-continuous.
Proof
Notice first that similarly to (4.10), we have
Since \(S:V\rightarrow V\) is \(V_0\)-continuous (from Lemma 3.2), its restriction \(S:V_n\rightarrow V\) to the closed subspace \(V_n\) is also continuous. Moreover, taking into account (4.12), it is clear that \(\tau _n S: V_n\rightarrow \tau _nV\) is \(V_0\)-continuous and hence, it follows from \(\tau _nS(V_n)\subset \tau _nV\subset \tau _n{\mathbb {H}}_n = V_n,\) that \(S_n:V_n\rightarrow V_n\) is \(V_0\)-continuous. \(\square \)
Remark 4.5
From the Lemmas 4.2, 4.3 and 4.4, to show that \(S_n\) possess a fixed point via the Schauder theorem, it remains to prove that
which will be established in the proof of Lemma 6.1.
Lemma 4.6
Let \( a > 0\) be small enough. If \( S_n \) has a fixed point in \(\tau _nB^a_V\) then the boundary value problem (2.7)–(2.11) has at least one solution \(u\in V\).
Proof
Indeed, if (4.13) is satisfied, according to Remark 4.5, \(S_n\) has a fixed point \(\tau _nu'_n\in \tau _nB^a_{V}\), i.e,
Recalling the definition of \(S_n\) in Lemma 4.4, we have that \(\tau _nS(\tau _n u'_n)=\tau _n u'_n\) and hence,
Set
According to the definition (see (3.16)) of the operator S, we get that \(u_n\in V\) and satisfies in \(\Omega \) the system of equations
where
As (see (4.15) and (4.14)) \( u_n = u'_n\) in \({\mathbb {H}}(\Omega _n)\), then \(u_n=(v_n,b_n,\sigma _n)\) verifies in \(\Omega _n\) the following system
Let now fix N be large enough. Since for any \( n> 2 N \) (see (4.2)) \(\tau _n =1\) in \(\Omega _N\), from the previous system of equations, it follows that \(u_n=(v_n,b_n,\sigma _n)\) verifies in \(\Omega _N\) the following system
Since \(u_n = S(\tau _nu'_n)\) then (see (3.19) and (4.10)), \(u_n\) is bounded in V so in \(\mathbb {H}(\Omega _N)\), passing if necessary to a subsequence, we can assume that \(u_n\) converges weakly in V and \({\mathbb {H}}(\Omega _N)\) so strongly in \( {\mathbb {H}}_0 (\Omega _N)\). Let u be the weak limit of \(u_n\) in V and \(u_N\) is weak limit in \({\mathbb {H}}(\Omega _N)\) of \(u_n|_{\Omega _N}\). Therefore
Indeed, as the linear mapping \( R: V \rightarrow {\mathbb {H}}(\Omega _N)\) defined by \(R(w): = w|_{\Omega _N}\) is of course strongly continuous, it follows that R is also weakly continuous (see e.g., [13, Theorem 3.10 page 61]). Therefore, \(R(u_n)\rightharpoonup R(u)\) i.e. \(u_n|_{\Omega _N}\rightharpoonup u|_{\Omega _N}\) and hence, (4.22) holds.
On the other hand, considering the expressions (see (2.12) and (2.13)) of F and G, one can easily see that
and
Hence, by recalling the \(V_0\)-norm (see (3.6) and (3.4)), we get
This being, in passing to \(L^2(\Omega _N)\)-weak limit in the Eqs. (4.19)–(4.21), given the inequalities (4.23), the strong convergence in \({\mathbb {H}}_0(\Omega _N)\) and weak convergence in \({\mathbb {H}}(\Omega _N)\) of the sequence \( (u_n) \) to \( u=(v,b,\sigma )\), we obtain
in \(\Omega _N\) for any N large enough and so, also in \(\Omega \). Therefore \(u\in V\) as a weak limit of \(u_n\) satisfies the Eqs. (2.7)–(2.9). \(\square \)
Now, to conclude on the existence of a solution of the Eqs. (2.7)–(2.9) with the boundary conditions (2.10) and (2.11), it remains, considering the Lemma 4.6, to show the crucial point (4.13). For this, we need suitable estimates of \(S(\tau _nu') = u_n\) (see (3.16)).
From now on, for any generic function \(\varphi \), we denote
Let us bear in mind that
is solution of the system
with the boundary conditions (2.10)–(2.11) and (see (2.12) and (2.13))
We recall that our goal is to establish the estimates of the solutions (4.25) from which we will obtain (4.13) in Remark 4.5. These estimates will require a more elaborate treatment and hence will be discussed in the next technical section. We will need some estimates of the nonlinear terms
which appear in the Eqs. (4.26)–(4.28). These estimates in terms of the norm \( \left\Vert {u'^{(n)}} \right\Vert _V \) of \(u'^{(n)}\in \tau _nB_V^a\) and therefore in terms of a, will be very useful in the proof of (4.13).
Indeed, let \( u'^{(n)} = (v'^{(n)},b'^{(n)},\sigma '^{(n)})\in \tau _n B_{V}^a\). If a is small enough then
In fact, notice first that, from (3.19) and (3.18) we have
Now by considering (3.12) and (3.13), we have
From (4.31) (see also (4.10)) it follows that
If k is large enough so that
from (4.32) and (4.33), it follows that
whence, given (4.32)
where (see (4.31))
From (4.37), (4.36), and (4.34) we get
By choosing (see (1.12)) \(\delta _0\) so that \(\left\Vert {\delta } \right\Vert _{H^2(\mathbb {R}^2)}\le a^2\), it is easy to see that for all \(a\in ]0,1[\), we have
whence follows (4.30). \(\square \)
5 Estimates of the Solutions
This section is devoted to solutions \(u_n=(v_n,b_n,\sigma _n)\) the system of equations (4.26)–(4.28) with the boundary conditions (2.10) and (2.11). These estimate which will be obtained in the following lemmas are based largely on the ideas developed in [8]. We remind here that these lemmas will be proven under the assumption (1.11). As for the positive number k which appears in Eq. (4.28) it can, as in Lemma 3.1, be chosen arbitrarily large. It will be convenient to set
where \(\bar{k}\) is large positive number satisfying in particular
We will denote in the statements of the following lemmas, by \(C'_i\) \( (i= 1, \cdots 10) \) the constants which depend on \(\Omega \) and \({\bar{B}}\) but not on M (possibly on its lower bound \(M_0\) see (1.11)) and by \({\tilde{C}}_i\) \( (i = 1,2,3) \) the constants which depend on \(\Omega \), \({\bar{B}}\) and M. In addition, in the proof of each lemma, if it is not necessary to specify them, one will denote by \(C_\Omega \) the constants independent of M and by \( \widehat{C}\) those which depend on M.
Lemma 5.1
Let \(u'=(v',b',\sigma ')\in V\) and \(u_n=S(u'^{(n)}) = (v_n,b_n,\sigma _n)\in V\) the solution of the system (4.26)–(4.28) with the boundary conditions (2.10) and (2.11) guaranteed by Lemma 3.1. Let k be given by (5.1) and suppose that the hypothesis (1.11)–(1.12) holds, then we have
Proof
Let us first observe that
Multiply (4.26) by \((\gamma \vartheta )^{-1}v_n\), (4.27) by \((\gamma \vartheta )^{-1}b_n\) and (4.28) by \( \widehat{\varrho }^{\gamma -2}\sigma _n\) and integrate the resulting equations over \(\Omega \). By using the above identities, we find after integration by parts that
where
Recalling the expression of \(\widehat{\varrho }\) (see (2.1) and (1.8)), thanks to (1.11)–(1.14), one can easily see that \( \left\Vert {\nabla \log \widehat{\varrho }} \right\Vert _{L^\infty }\) is small enough so that
thus, by using (5.1) and (5.2), we obtain
As for the last term, we have
Combining these estimates and using (5.4), we get
Next, let us establish the \(L^2\)-estimate of \(\sigma '^{(n)}\) which appears in the right-hand side of (5.6). To this aim, we introduce the following boundary value problem
where (see (4.3))
with \(\varepsilon =\varepsilon (x')\) being a given positive function in \(L^2(\mathbb {R}^2)\setminus L^1(\mathbb {R}^2)\). There exists (see e.g. [11, 22, 39]) at least one solution \(\varphi \in H^1(\Omega )\) of the problem (5.7)–(5.8) such that
Now, we rewrite the Eq. (4.26) as
We multily this new equation by \((\gamma \vartheta )^{-1}\varphi \) and integrate over \(\Omega \). By integrating by parts, we obtain taking into account (5.9) and (5.7)
Given (5.10) and (5.9) (see also (2.1) and (1.14)), we get from (5.11) that
If we multiply now the above inequality by \((4C_\Omega )^{-1}\) and add it to (5.6) we obtain the estimate (5.3) and this completes the proof the Lemma 5.1. \(\square \)
Lemma 5.2
Under the same the hypotheses of Lemma 5.1, we have
Proof
Let us remark that, since \(\partial _{x_i}v_n\) and \(\partial _{x_i}b_n\) (\(i=1,2\)) satisfy the same boundary conditions (2.10), then we have
Now, we apply the operator \(\partial _{x_i}\) to both sides of each Eqs. (4.26)–(4.28), then we multiply the resulting equations by \( (\gamma \vartheta )^{-1}\partial _{x_i}v_n\), \((\gamma \vartheta )^{-1}\partial _{x_i}b_n\) and \(\widehat{\varrho }^{\gamma -2}\partial _{x_i}\sigma _n\) respectively and we integrate over \(\Omega \). The integration by parts gives us
where
By the similar arguments used in the proof of the Lemma 5.1 (particulary for the term \(I_1\)), we obtain
As for the term \(I_6\), we have
By adding these estimates to (5.13) and summing on \(i=1,2\), we get the estimate (5.12) and this completes the proof the lemma 5.2. \(\square \)
Lemma 5.3
Under the same hypotheses of Lemma 5.1, we have
where
Proof
Using the identity
from the Eq. (4.26) it follows that
Next, we apply the differential operator \(\partial _ {x_3}\) to both sides of (4.28), multiply the resulting equation by \(\partial _{x_3}\sigma _n\) and integrate over \(\Omega \). Integrating by parts and taking into account of (5.17), we get
where
Using the identity
with
and taking into account the expressions of \(\widehat{\varrho }\) and \(k_1\) (see (2.1), (1.8) and (5.1)), we obtain
where
Moreover, given (2.1) (see also, (1.8) and (1.14), we have
As for the last \(I_6\) term, one has
Combining these estimates with (5.18) and taking into account of (5.19), we obtain
By multiplying now this inequality by
we get the estimate (5.15) using (5.1), and this completes of the Lemma 5.3. \(\square \)
Lemma 5.4
Under the assumptions of the Lemma 5.1, we have
Proof
We first rewrite Eq. (4.26) as a Stokes problem in \(\Omega \)
with the boundary conditions (2.10)\(_1\). From the classical estimates (see [10]) of the Stokes problem, one has
Since
using (5.17), we have
Moreover, thanks to (1.14) and (1.11), we have
We then obtain the estimate (5.20) from (5.23) using (1.7) and (5.24), which achieves the proof of the lemma 5.4. \(\square \)
Lemma 5.5
We have
Proof
Since \(\left\Vert {b_n} \right\Vert _{H^2}\le C_\Omega \left\Vert {\Delta b_n} \right\Vert _{L^2}\) for all \(b_n\in H^2(\Omega )\cap H_0^1(\Omega )\), from the Eq. (3.8) and the boundary condition (1.4) we easily get (5.26). \(\square \)
Lemma 5.6
Under the same assumptions of the Lemma 5.1, we have.
Proof
As in the Lemma 5.2, since \(\partial _{x_i}\partial _{x_j}v_n\) (\(i,j=1,2\)) satisfies the same boundary conditions (2.10), then we have
Now if we apply the differential operator \(\partial _{x_i}\partial _{x_j}\), \((i, j=1,2)\) to both sides of the Eqs. (4.26) and (4.28) and we multiply the resulting equations by \((\gamma \vartheta )^{-1}\partial _{x_i}\partial _ {x_j}v_n\) and \(\widehat{\varrho }^{\gamma -2}\partial _{x_i}\partial _{x_j}\sigma _n\) respectively, then we integrate over \(\Omega \). We then obtain through integratin by parts
where
By estimating the terms \(I_i\) \((i = 1, \ldots 4)\) as in the proof of Lemma 5.2, we obtain
As for the terms \(I_i\) \((i = 5,\ldots 7)\), we have
By adding now the estimates of \( I_i \) \( (i = 1, \ldots 7)\) to (5.28) and summing over \( i = 1,2 \), we obtain (5.27), and the lemma is proved. \(\square \)
Lemma 5.7
Under the assumptions of the Lemma 5.1, we have
where \(C'_M\) is given by (5.16).
Proof
We apply the differential operator \(\partial _{x_3}\partial _{x_i}\), \((i=1,2)\) to both sides of the Eq. (4.28) and we multiply the resulting equations by \(\partial _ {x_3}\partial _{x_i}\sigma _n\) and integrate over \(\Omega \). We then obtain through integration by parts taking into account (5.17)
where
Just like in the proof of Lemma 5.3 we use the inequality
and we estimate similarly the terms \(I_i\) \((I = 1, \ldots , 5)\). By combining these estimates with (5.18), we obtain
Multiplying now this inequality by
and taking into account of (5.1), we obtain by summing on \(i=1,2\), the inequality (5.29). \(\square \)
Lemma 5.8
Under the assumptions of Lemma 5.1, we have
Proof
We apply to Eq. (4.26) the differential operator \(\partial _{x_i}\) \( (i = 1,2) \) and we rewrite the obtained equation as Stokes problem in \( \Omega \)
with the same boundary conditions on \(\partial _{x_i}v\) as (2.10)\(_1\). We obtain by the same arguments of proof of Lemma 5.4 that
In addition, let us note that
and, in applying \(\partial _{x_i}\) \((i=1,2) \) to (5.17), it follows that
By substituting (5.33) and (5.34) into (5.32) and taking into account the estimate
(obtained similarly to (5.25) through (1.14)), we get the estimate (5.31). This completes the proof of the lemma. \(\square \)
Lemma 5.9
Under the assumptions of the Lemma 5.1 one has
Proof
We apply the Laplacian operator to Eq. (3.9), multiply the resulting equation by \(\Delta \sigma _n\) and integrate over \(\Omega \). Following integration by parts, on account of (5.17) we get
where
In a similar way to the proof of the lemmas 5.3 and 5.7, we have
and by estimating similarly the terms \(I_i\) (\(i=1,\ldots ,4)\), we obtain
Multiplying now this inequality by
and taking into account (5.1), we obtain the estimate (5.35). \(\square \)
Lemma 5.10
Under the same assumptions of Lemma 5.1 one has
Proof
As in Lemma 5.4, according the well-known theory estimates of the stokes problem, one deduced from (5.21) and (5.22) with the boundary conditions (2.10)
Let us first notice that
In addition, taking into account (5.17), we have
Since (see (2.1)\(_1\) and (1.14))
by adding these inequalities to (5.38), one obtains (5.37). \(\square \)
6 Fixed Point of the Operator \(\textbf{S}_n\)
Having proved the lemmas 5.1–5.10 one is now in a position to establish that, for every n, the operator \(S_n\) has a fixed point. Following Remark 4.5 and Lemma 4.6, it remains for us to establish the crucial point (4.13), which is the goal of the following lemma.
Lemma 6.1
There is a norm \(|\cdot |_V\) equivalent (see (3.6) and (3.3)) to \(\Vert \cdot \Vert _V\) such that
Proof
Notice that, it suffices to prove that
Let then \(\lambda _1, \lambda _2, \ldots , \lambda _9\) be positive numbers which will be suitably chosen later. We set
where \(C'_M\) is given in (5.16) and \({\bar{k}}\) in (5.2). For any \(\varphi \in H^2\), we set
It is clear that \( |\cdot |_{2,\Omega }\) is equivalent to the \(H^2\)-norm. Moreover (see (4.8)), an easy computation shows that
If we set
we obtain (see (3.5)) a equivalent norm to \(\Vert \cdot \Vert _V\). In addition, given (6.4) and (4.9), we can see that
where \((\tilde{\nu }_n)\) is a decreasing sequence of positive numbers converging to zero. Let now
If we multiply the estimates on solutions \(u_n\) obtained in the lemmas 5.1–5.10 by the positive numbers \(\lambda _1, \lambda _2, \ldots , \lambda _9\) and \(\lambda _{10}= 1\) respectively and adding them, we obtain
with some positive constant \({\tilde{C}}\). As for \(N(\sigma '^{(n)})\) it is given by
and the numbers \(\Lambda _i\) (\(i=1,2, \ldots ,8)\) are given by
Since (see (1.13) and (1.11)) \(C_M\) is large enough, we can see easily that it is possible to choose
so that
Indeed for \(j = i+1,\ldots ,9\) and \(i=8,\ldots ,1\) given, as can be seen easily, considering the constraints (6.10) we can choose a \(\lambda _i\) large enough so that the inequalities containing only \(\lambda _i\) and \(\lambda _j\) are verified (and so we can proceed to the choice of \(\lambda _i\) starting from \(\lambda _9\) and then choosing successively \(\lambda _i\) for \(i=8,\ldots ,1\)). Such a choices of the numbers \(\lambda _i= \lambda _i(M_0)\) (\(i=1, \ldots , 9)\) imply in particular that \(\Lambda _7 > \)0 and \(\Lambda _8 > 0\), because \(C_M\) is large enough.
If we recall now the inequality (6.8), considering (6.10), we obtain
and given (4.30) (see also (6.5)–(6.7)), from (6.11) it follows that for all \(u'^{(n)}\in \tau _nB_{V}^a\)
Let n be large enough and a be small enough so that
From (6.12) it follows that
which means that \(S(u'^{(n)})\in B_{V}^a\) for every \(u'^{(n)}\in \tau _nB_{V}^a\). Hence, we get (6.2) and this completes the proof of the lemma 6.1. \(\square \)
7 The Proof of the Theorem 2.1
After explaining the strategy to follow in order to obtain the proof of our existence result, we were able to establish some crucial non trivial estimates in the previous lemmas 5.1–5.10. We are now in a position to prove our main result from the estimates in those lemmas.
In fact, recalling Remark 4.5, we obtain from Lemma 6.1 together with Lemma 4.6 that the system of Eqs. (2.7)–(2.11) has a solution \(u=(v,b,\sigma )\). More precisely, the solution \(u = (v,b,\sigma )\) belongs to V and verifies in \(\Omega \) the system of equations
with the boundary conditions (2.10) and (2.11). Now, if we set
(see (2.5)), then the triple \((v,B,\varrho )\) solves the system of Eqs. (1.1)–(1.3) with the boundary conditions (1.4)–(1.5) and this completes the proof of Theorem 2.1. \(\square \)
Remark 7.1
It is clear that if \(\delta (x') = 0\) then \((v,B,\varrho ) = (0,B_{eq},\varrho _{eq})\) where (see (1.8))
is the unique stationary solution to the boundary value problem (1.1)–(1.5). However, it is important to establish whether this system can still have a stationary solution \((v,B,\varrho ) = (0,B_{st},\varrho _{st})\) when \(\delta (x') \ne 0\).
Indeed suppose that
and that there exists a stationary solution \((0,B_{st},\varrho _{st})\) to the boundary value problem (1.1)–(1.5). Hence, the pair \((B_{st},\varrho _{st})\) solves in the domain \(\Omega =\mathbb {R}^2\times ]0,h[,\) the system of equations
with the boundary conditions
where, we recall here that \(e_3=(0,0,1)\), \({\bar{B}}\in \mathbb {R}^3\) is a given constant field and \(\delta (x')\) is a small perturbation of magnetic field \({\bar{B}}\) on the upper plane \(x_3=h\). It is obvious that
is the unique solution of the boundary value problem (7.3)–(7.4), where A is the unique solution in \(\Omega \) of the system of equations
with the boundary conditions
As can be easily seen, the only solution A of (7.6)\(_1\)–(7.7) is given by
where \(\widehat{a}\) is such that for every \(\xi \in \mathbb {R}^2\), the function \(a(\xi ,\cdot )\) solves the second order boundary value problem
and
which, considering (7.6)\(_2\), is
Thus, as we can see now, the only solution of (7.9) is given by
As for the Eq. (7.6)\(_3\), it is satisfied by A (see (7.10 and (7.8)) if and only if
By considering the boundary conditions (7.9), it follows from (7.11)\(_2\) that
and therefore,
which contradicts (7.1). Thus the system of Eqs. (1.1)–(1.5) does not have a stationary solution \((v,B,\varrho )=(0,B_{st},\varrho _{st})\) when the function \(\delta \) is not identically equal to 0.\(\square \)
8 Conclusion and Remarks
In our recent contribution [8], we developed a strategy through which we studied the stationary convective motion in an infinite layer of a viscous compressible and heat-conducive fluid and an existence result for a solution close to the hydrostatic state was obtained in Sobolev spaces. Subsequently, we studied in [7] the stationary motion of a viscous compressible fluid in a magnetic field under the action of a large external force field, and with non-homogeneous and rather large Dirichlet boundary conditions on the magnetic field. Two questions naturally arose: On the one hand, the question about the extension of the study in [7] in the case of the motion in an infinite layer, through the approach developed in [8]. The current work is precisely an answer to that question. On the other hand, we are currently investigating on the stability of the stationary solution with a large potential force field and large enough non-homogeneous Dirichlet boundary conditions on the magnetic field. In the current state, there are no such results in the literature. Let us mention that a stability result was obtained for instance in [30], for the stationary solution of the viscous compressible MHD equations for a barotropic fluid with a large potential force and rather a homogeneous Dirichlet condition on the magnetic field.
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Benabidallah, R., Ebobisse, F. On the Steady Flows of Viscous Compressible Magnetohydrodynamic Equations in an Infinite Horizontal Layer. J. Math. Fluid Mech. 26, 47 (2024). https://doi.org/10.1007/s00021-024-00881-4
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DOI: https://doi.org/10.1007/s00021-024-00881-4