Existence of weak solutions to the three-dimensional density-dependent generalized incompressible magnetohydrodynamic flows

In this paper we consider the equations of the unsteady viscous, incompressible, and heat conducting magnetohydrodynamic flows in a bounded three-dimensional domain with Lipschitz boundary. By an approximation scheme and a weak convergence method, the existence of a weak solution to the three-dimensional density dependent generalized incompressible magnetohydrodynamic equations with large data is obtained.


Introduction
The study of the dynamics of biological fluid in the presence of magnetic field is very useful in understanding the bioengineering and medical technology. The development of magnetic devices for cell separation, targeted transport of magnetic particles as drug carriers, magnetic wound or cancer tumor treatment causing magnetic hyperthermia, reduction of bleeding during surgeries or provocation of occlusion of the feeding vessels of cancer tumors and the development of magnetic tracers, as well are well-known applications in this domain of research [1,2].
Magnetohydrodynamic flow of a non-Newtonian fluid in a channel of slowly varying cross section in the presence of a uniform transverse magnetic field was studied in [3]. In the recent past, El-Shehawey et al. [13] studied an unsteady flow of blood as an electrically conducting, incompressible, elastico-viscous fluid in the presence of magnetic field through a rigid circular pipe by considering the streaming blood as a non-Newtonian fluid in the axial direction only.
In the present paper, we consider the following system of the three-dimensional incompressible magnetohydrodynamic flows [2,26,34]: (ρu) t + div(ρu ⊗ u) + ∇P = (∇ × H) × H + divS(ρ, θ, D(u)), Φ t + div(u(Φ ′ + P )) = div((u × H) × H + νH × (∇ × H) +uS(ρ, θ, D(u)) + q(ρ, θ, ∇θ)), where ρ, u ∈ R 3 , H ∈ R 3 and θ t; S denote the density, the velocity, the magnetic field and the temperature, respectively; S is the viscous stress tensor depending on the density, the temperature and the symmetric part of the velocity gradient D(u), and the thermal flux q is a function of the density, the temperature and its gradient; Φ is the total energy given by with the internal energy e(ρ, θ), the kinetic energy 1 2 ρ|u| 2 , and the magnetic energy 1 2 |H| 2 ; D(u) = ∇u+∇u T is the symmetric part of the velocity gradient, ∇u T is the transpose of the matrix ∇u, and I is the 3 × 3 identity matrix; ν > 0 is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field.
Equations (1)-(3) describe the conservation of mass, momentum, and energy, respectively. It is well-known that the electromagnetic fields are governed by Maxwell's equations. In magnetohydrodynamics, the displacement current can be neglected [28,29]. As a consequence, Eq. (4) is called the induction equation, and the electric field can be written in terms of the magnetic field H and the velocity u, The basic principles of classical thermodynamics imply that the internal energy e and pressure P are interrelated through Maxwell's relationship: where c ν (θ) denotes the specific heat and Q = Q(θ) is a function of θ. Thus, we assume that the internal energy e can be decomposed as a sum: e(ρ, θ) = P e (ρ) + Q(θ).
For simplicity, we impose the boundary conditions The initial density is supposed to be bounded and the initial total energy is integrable, i.e., and where ρ * , ρ * and θ * are constants. The study of long time and large data existence theory for inhomogeneous incompressible fluids was investigated in several contributions. For the case r = 2 and the viscosity does not depend on |D(u)|, using the concept of renormalized solutions, Lions [30] established a new convergence and continuity properties of the density that may vanish at some parts of the domain where the viscosity depends on the density. Meanwhile, he got rid of the smallness of the data. For the case that the viscosity depends on the shear rate (r = 2), Fernández-Cara et al. [15] proved the existence of weak solutions for that r ≥ 12 5 . Guillén-González [19] (also see [30]) considered the spatially periodic setting by using higher differentiability method. Recently, Frehse et al. [14] established the existence result with non-slip boundary conditions and a viscosity that depends on both the density and the shear rate for r > 11 5 . In [17], Frehse et al. showed the existence result on the full thermodynamic model for inhomogeneous incompressible fluids for r ≥ 11 5 , which improved the result in [14]. For more results about thermal flows of incompressible homogeneous fluids, we refer the reader to [5,6,7,9,32].
Then subtracting above equation from (15), using (6), we obtain the thermal energy equation where S : ∇u denotes the scalar product of two matrices. More precisely, Recently, there have been much work on magnetohydrodynamics because of its physical importance, complexity, and widely application (see [8,28,29,33]).
Magnetohydrodynamics (MHD) is a combination of the compressible Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism. Duvaut and Lions [12], Sermange and Temam [35] obtained some existence and long time behavior results for incompressible case. For compressible magnetohydrodynamic flows of Newtonian fluids, Ducomet and Feireisl [11] proved the existence of global in time weak solutions to a multi-dimensional nonisentropic MHD system for gaseous stars coupled with the Poisson equation with all the viscosity coefficients and the pressure depending on temperature and density asymptotically, respectively. Hu and Wang [20] studied the global variational weak solution to the three-dimensional full magnetohydrodynamic equations with large data by an approximation scheme and a weak convergence method. In [22], by using the Faedo-Galerkin method and the vanishing viscosity method, they also studied the existence and large-time behavior of global weak solutions for the three-dimensional equations of compressible magnetohydrodynamic isentropic flows (1)-(3). They [23] showed that the convergence of weak solutions of the compressible MHD system to a weak solution of the viscous incompressible MHD system. Jiang, et all. [24,25] obtained that the convergence towards the strong solution of the ideal incompressible MHD system in the whole space and periodic domain, respectively. For MHD driven by the time periodic external forces, Yan and Li [38] showed that such system has the time periodic weak solution.
The main difficulty of the study of MHD is the presence of the magnetic field and its interaction with the hydrodynamic motion in the MHD flow of large oscillation. This leads to that many fundamental problems for MHD are still open. For example, the global existence of classical solution to the full perfect MHD equations with large data in one dimensional case is unsolved. But corresponding problem about Navier-Stokes equation was solved in [27] a long time ago. In the present paper, we study the existence of weak solutions for the density-dependent generalized inhomogeneous incompressible Magnetohydrodynamic flows in a bounded three-dimensional domain with Lipschitz boundary. Inspired by the work of [10,17,20,31], we will establish the existence of weak solutions for the density-dependent generalized inhomogeneous incompressible compressible MHD for any r > 2.
These equations, and all functions involved in their descriptions as well, are considered in (0, T ) × Ω, where Ω ⊂ R 3 is an open, connected and bounded set with Lipschitz boundary ∂Ω, and T ∈ (0, ∞).
The paper is organized in the following way: in the next section, by introducing the appropriate function spaces we provide the precise definition of the notion of weak solutions to system (1)-(2), (4) and (16). The main result of this paper is also stated. Then in Section 3, we first introduce the corresponding approximation system whose solvability is established in Appendix (section 5). We also derive some corresponding uniform estimates. We finish the proof of Theorem 1 in section 4 by establishing the strongly convergence of {ρ n }, {u n }, {θ n }, {H n } and {D(u n )}.

Some notations and main result
Before giving the definition of the weak solution to the problem (1)-(2), (4) and (16) with the boundary condition (10), we first state the following notation of relevant Banach spaces of functions defined on a bounded domain Ω ⊂ R 3 . For any p ∈ [1, ∞], L p (Ω) denotes the Lebesgue spaces with the norm · L p (Ω) , W 1,p (Ω) denotes the Sobolev spaces with the norm · W 1,p (Ω) , W 1,p 0 (Ω) denotes the closure of C ∞ 0 (Ω) functions in the norm of W 1,p (Ω). If X is a Banach space of scalar functions, then X 3 , X 4 or X 3×3 , X 4×4 denote the space of vector or tensor-valued functions so that each their component belongs to X. Further, we use the following notation for the spaces of function with zero divergence and their dual (r ′ = r r−1 ) and C(0, T ; X) denote the standard Bochner spaces. We write (a, b) instead of Ω a(x)b(x)dx whenever ab ∈ L 1 (Ω) and use the brackets a, b to denote the duality pairing for a ∈ X * and b ∈ X. We use C([0, T ]; L q weak (Ω)) to denote the space of functions ρ ∈ L ∞ (0, T ; L q (Ω)) satisfying (ρ(t), z) ∈ C([0, T ]) for all z ∈ L q ′ . We do not distinguish between function spaces for scalar and vector valued functions. Generic constants are denoted by M , their values may vary in the same formula or in the same line.
• The temperature non-negative θ function, the velocity function u and the magnetic field H satisfy and the following weak formulations hold: for all b ∈ L 2 (0, T ; W 1,2 (Ω)).
• The initial conditions are attained in the following sense The aim of this paper is to establish the following result.
Theorem 1 Assume that S and q are continuous functions of the form (6) satisfying (7)-(9) with r > 2 and α > − 2 3 , and there are two positive constants c ν and c ν such that Then there exists a weak solution to the problem (1)-(4) in the sense of Definition 1 with initial data satisfying (11)- (13).
Note that we consider in (18) only divergenceless test function, so the pressure does not appear in the definition of weak solutions. The pressure P (zero mean value) can be obtained by comparing two auxiliary Stokes problems (homogeneous Dirichlet boundary conditions) with taking the test function ϕ = χ (0,t) φ (χ (0,t) denotes the characteristic function of (0, t) and φ ∈ W 1,r 0,div (Ω)) in (18). Furthermore, the pressure P has the form The solvability of above Stokes problems can be obtained by a similar proof in [4,18,37]. Since the presence of ∂ t P 2 , we can not know if P is an integrable function on (0, T ) × Ω.
In the following, we give some uniform estimates with respect to n ∈ N. Taking z = |u n | 2 , t 1 = 0 and t 2 = t in (28) we have t 0 (ρ n , |u n | 2 t ) + (ρ n u n , ∇|u n | 2 )ds = (ρ n , |u n | 2 )(t) − (ρ n , |Γ n u 0 | 2 ). (29) Multiplying the jth equation in (23) by a j , then taking the sum over j = 1, . . . , n, using (29), and integrating the equality over (0, t), we get We deal with (25) by the same process as in (30), and have Direct calculation shows that So by (31), we have Summing up (30) and (32), we obtain Then by the first assumption in (8), we derive where M denotes a positive constant depending on the data and maximizes all the estimates. Note that Thus it follows from (34) that By Korn's inequality, (8) and (36), we have The inequality (37) implies that Using (35) and (37), by Hölder inequality and the fact where we require that 1 < p 1 ≤ r and which gives one of the restriction for r in our main result. It follows from Hölder inequality that Combining above estimates, for 1 < p 1 ≤ r, we deduce from (23)-(24) to Let h = 1 in (24). Using (35) and (37), we have By (21) and above estimate, Now we turn to estimate the temperature. Note that Take h = −(θ n ) −λ with 0 < λ < 1 in (24). Then by (7) and (43), By a contradiction argument, we can easily get Thus by these estimates and (44), we have By the standard interpolation of (46) with (43), for α > − 2 3 , we derive Using (9), (45) and (47), for 1 ≤ m < 5+3α 4+3α , we have On the other hand, by (35) and (43), using the interpolation inequality, Hölder's inequality and the standard Sobolev imbedding, we have that for some β > 1 Direct computation shows that Note that 1 ≤ p 4 < 5 3 + α and α > − 2 3 . We get from (49) that where Hence we require the restriction that r > 2. Finally, by (24), (37) and (48), for sufficiently large q, we deduce that It follows from (39) that which implies that u n × H n ∈ L r (0, T ; L 2r r+2 (Ω)).
The existence of solutions for the approximation system (102)-(105) can be proved by modified Faedo-Galerkin method. We first give the solvable of induction equation (105).
Proof Define Note that H k,ǫ (t, x) ∈ Y k . We can write where the coefficients c k,ǫ j (t) are required to solve the system of ordinary differential equations where For given initial data H k,ǫ (0) ∈ Y k , the system (106) has a unique solution c k,ǫ j ∈ C 1 ((0, T ); Y k ) for some T ′ ≤ T . Multiplying both sides (106) by c k,ǫ j , summing over j, integrating by parts, we have Note that by Young inequality, Using the Poincaré inequality, By (107)-(108), we derive Thus, for t ∈ [0, T ], we get This implies that for t ∈ [0, T ], Due to {̟ j (x)} ∞ j=1 be an orthonormal basis of W 1,2 (Ω), so we have |c k,ǫ j (t)| = H k,ǫ (t) 2 L 2 (Ω) , from which we conclude that T ′ = T .
Define the ball B R of radius R and the map Π : B R −→ B R such that Π(H k,ǫ (0)) = H k,ǫ (T ), where the radius R such that Follows [31], we can prove the map Π is continuous. Hence, it has a fixed point. Moreover, from (109), we know that the solution operator u k,ǫ −→ H k,ǫ (u k,ǫ ) maps bounded sets in C([0, T ], X k ) into bounded subsets of the set Y k . Then, as done in [22], the solution operator u k,ǫ −→ H k,ǫ (u k,ǫ ) is a continuity operator. This completes the proof.
First, we summarize the estimates available for (102)-(105) for ǫ > 0 and k ∈ N fixed. Then the behavior of relevant solutions will be studied as ǫ −→ 0.