Steady state solutions of ferrofluid flow models

We study two models of differential equations for the stationary flow of an incompressible viscous magnetic fluid subjected to an external magnetic field. The first model, called Rosensweig's model, consists of the incompressible 
Navier-Stokes equations, the angular momentum equation, the 
magnetization equation of Bloch-Torrey type, and the magnetostatic equations. The second one, called Shliomis model, is obtained by assuming that the angular momentum is given in terms of the magnetic field, the magnetization field and the vorticity. It consists of the incompressible Navier-Stokes equation, the magnetization equation and the magnetostatic equations. We prove, for each of the differential systems posed in a bounded domain of $\mathbb{R}^3$ and equipped with boundary conditions, existence of weak 
solutions by using regularization techniques, linearization and the Schauder fixed point theorem.


1.
Introduction. Ferrofluids are suspensions of magnetic nanoparticles in appropriate carrier liquids. These fluids have found a variety of applications in engineering: magnetic liquid seals, cooling and resonance damping for loudspeaker coils, printing with magnetic inks, rotary shaft seals, rotating shaft seals in vacuum chambers used in the semiconductor industry, see for instance [22]. In recent years many investigations were made on the possibility of future biomedical applications of magnetic fluids, such as magnetic separation, drugs or radioisotopes targeted by magnetic guidance, magnetic resonance imaging contrast enhancement, see for instance [13].
A number of works show that ferrofluids can be treated as homogeneous monophase fluids, see [16,19] and the references therein. Consider the flow of an incompressible and viscous, Newtonian ferrofluid occupying a domain D ⊂ R 3 , assumed to be regular, bounded and simply connected with boundary Γ, under the action of an external magnetic field H ext . The magnetic field H ext induces a demagnetizing field H and a magnetic induction B given by the law B = µ 0 (H + M ), where M is the magnetization inside D and µ 0 is the magnetic permeability constant. The 2330 YOUCEF AMIRAT AND KAMEL HAMDACHE flow, in the stationary case, is described by the following system, called Rosensweig's model [14,15,16]: curl H = 0, div µ 0 (H + M ) = F, U = 0, ω = 0, M · n = 0, curl M × n = 0, H · n = 0 on Γ.
Here U is the fluid velocity, ω is the angular momentum (or spin) of the fluid particles, I is the tensor of inertia, p is the pressure, and ρ, η, ζ, µ 0 , κ, η , λ , σ, χ 0 , τ are physical positive constants. In (P), the differential equations are posed in D, (P) 1 is the continuity equation, (P) 2 is the linear momentum equation, (P) 3 is the angular momentum equation, (P) 4 is the magnetization equation of Bloch-Torrey type and σ is a magnetic diffusion coefficient which carries the spins [11,21], (P) 5 are the magnetostatic equations and (P) 6 are the boundary conditions. The term µ 0 (M ·∇)H represents the Kelvin body force due to magnetization, and µ 0 M ×H is the body torque density which causes the magnetic nanoparticles and surrounding fluid to spin. The quantities L, G and F are given source terms; we assume that F satisfies the compatibility condition D F dx = 0 and is linked with the applied magnetic field H ext by the formula F = −div H ext . We also consider the following model deduced from the Rosensweig model (P) by neglecting the inertia and friction effects so that the angular momentum is given (assuming G = 0) by the formula [18,19] With these simplifications, problem (P) becomes where β = µ0 4ζ . Problem (S) is called Shliomis model. For notational convenience we omit in the sequel the parameter µ 0 in the magnetostatic equation by changing F in µ 0 F so that we will write div (H + M ) = F . The aim of this paper is to study the existence of weak solutions to problems (P) and (S). The corresponding time dependent problems have been discussed in [2,3] where it was proved global existence in time of weak solutions with finite energy.
2. Main results. In the sequel we use the following functional spaces. For 1 ≤ q ≤ ∞ and s ∈ R, let L q (D) and W s,q (D) be the usual Lebesgue and Sobolev spaces of scalar functions. We denote by · q the norm in L q (D). If q = 2, W s,2 (D) is denoted by H s (D), and · and (· ; ·) denote the norm and the scalar product in the Hilbert space L 2 (D), respectively. If V is a Banach space we denote by ·; · V ×V (or simply ·; · if no confusion arises) the duality product where V is the dual space of V. For vector valued functions we use the notations L q (D), W s,q (D), L 2 (D), H s (D) and the notations of norms in L q (D) and the scalar product of L 2 (D) are unchanged.
We introduce the functional spaces in the theory of the Navier-Stokes equations, see [9,10,12,20] for example: It is well known that and we have the Stokes formula where ·, · Γ is the duality pairing between H − 1 2 (Γ) and H 1 2 (Γ). Similarly, if V belongs to the space then V has a tangential component V × n ∈ H − 1 2 (Γ) and the following Green's formula holds For solving the Bloch-Torrey equation we introduce the Hilbert space : M · n = 0 on Γ}, equipped with the norm of H 1 (D). It is well known, see [8] for example, that there exists a constant C > 0 such that  (ii) The angular momentum equation holds weakly, i.e. for all Ψ ∈ H 1 0 (D), ρκ (U · ∇)ω; Ψ + η (∇ω; ∇Ψ) + (η + λ ) (div ω; div Ψ) iii) The magnetization equation holds weakly, i.e. for all Λ ∈ H 1 t (D), Here and in the sequel the symbol ; denotes the duality product ; for all Φ ∈ U, Ψ ∈ H 1 t (D) and v ∈ H 1 (D). In the sequel C indicates a generic positive constant that may depend on the domain D and some physical constants. When the constant C depends in addition on some other parameter m we will write C(m). Our first main result is concerned with the existence of weak solutions to problem (P).
Theorem 2.1. Assume that L, G ∈ L 2 (D) and F ∈ L 2 (D). Then: (i) Problem (P) has a weak solution (U, ω, M, H) satisfying the energy estimates where C is positive constant that depends only on the domain D and some physical constants.
(ii) Assume in addition that F ∈ W 1, 3 2 (D). Let p denote the pressure and B the magnetic induction given by B = M + H. Then the weak solution (U, ω, M, H) has the regularity: U, ω, B ∈ W 2, 3 2 (D) and p ∈ W 1, 3 2 (D). Moreover, we have the estimates , where e = F 2 + L 2 + G 2 and C is a positive constant that depends only on the domain D and some physical constants.
In a second part of this work we discuss the Shliomis model. Our second main result is the following Theorem 2.2. Assume that L ∈ L 2 (D) and F ∈ L 2 (D). Then: (i) There is a number r 0 > 0, depending only on the domain D and some physical constants, such that if F ≤ r 0 then problem (S) has a weak solution (U, M, H) satisfying the energy estimates where C is a positive constant that depends only on the domain D and some physical constants.
(ii) Assume in addition that F ∈ W 1, 3 2 (D). Let p denote the pressure and B the magnetic induction given by B = M + H. Then the weak solution (U, M, H) has the regularity: U, B ∈ W 2, 3 2 (D) and p ∈ W 1, 3 2 (D). Moreover we have the estimates where C is a positive constant that depends only on the domain D and some physical constants. 3. Energy estimates for problem (P). We assume in this section that the solutions (U, ω, M, H) of problem (P) are smooth enough. Multiplying the linear momentum equation by U and the angular momentum equation by ω, integrating by parts, adding the results and using the identity we arrive at the equality Using the equations curl H = 0 and div U = 0 we have that Multiplying the magnetization equation by H, using the identity −∆ = curl 2 − ∇div , integrating over D and using (10) we obtain We deduce from the magnetostatic equations that Since H = ∇ϕ, multiplying the magnetostatic equation by ϕ and integrating by parts yields Combining (9)-(13) we obtain Using (1), the Cauchy-Schwarz and Poincaré inequalities we deduce that Multiplying the magnetization equation by M and integrating by parts yields Using (1) and (13) we deduce that Using again (1) we deduce from (13) that Writing the magnetostatic equations in the form with div M ∈ L 2 (D), and applying the regularity results for the second order elliptic equations with homogeneous Neumann boundary condition, we get the estimate 4. The regularized problem (P ε ). Let ε > 0 be a fixed small parameter. We introduce the following regularized problem The energy estimates for problem (P ε ) can be derived as that for problem (P). We obtain The regularization allows to establish some new bounds. We deduce from (16) that U ∈ U, ε∆U ∈ L 2 (D), ω ∈ H 1 0 (D) and ε∆ω ∈ L 2 (D). Using the regularity of the Laplace equation with homogeneous Dirichlet boundary condition, we obtain that U ∈ H 2 (D) ∩ U and ω ∈ H 2 (D) ∩ H 1 0 (D) with the following estimate Note that the constant C does not depend on ε.

5.
Solving problem (P ε ). We establish the following result.
Theorem 5.1. Assume that L, G ∈ L 2 (D) and F ∈ L 2 (D). Then problem (P ε ) has a weak solution , satisfying inequalities (16)- (19). We will solve problem (P ε ) by linearization and use of the Schauder fixed point theorem. The proof consists in four steps. 5.1. Solvability of the magnetization equation. Let (V, w) ∈ U × H 1 0 (D) be fixed. We denote by (M, H) the pair defined to be the solution of the linear differential system We introduce the operator T 1 : where (M, H) is the solution of problem (P 1 ).

Let us introduce the bilinear form
We denote by l the linear form defined on We look for a solution (M, ϕ) ∈ H 1 t (D) × H 1 (D) of the variational equation We establish the result.
where R = C F and B(0, R) is the ball of H 1 t (D) × H 1 t (D) with center 0 and radius R.

Proof. Let (M, Φ) ∈ H 1
We deduce that Consequently, with C > 0, then A is coercive. According to (21) and (22) div (H + M ) = 0, and homogeneous boundary conditions. Multiplying equation (23) by M and (24) by H and integrating by parts yields, respectively, and Using the Young inequality and Sobolev embedding theorems we have, for any α > 0, Choosing α > 0 small enough and applying Proposition 1 we deduce from (25)-(28) the inequalities . Proposition 2 is proved.  ). We introduce the pair (U, ω) defined to be the solution of the coupled linear system We define the map T 2 : where (U, ω) is the solution of problem (P ε 2 ) and denote by T the operator T = T 2 oT 1 . Let We establish the following result.
Proposition 3. Assume that L, G ∈ L 2 (D) and F ∈ L 2 (D). Then problem (P ε 2 ) admits a unique solution (U, ω) ∈ H 2 (D) ∩ U × H 2 (D) ∩ H 1 0 (D) satisfying the estimate where C is a positive constant, depending only on the domain D and some physical constants.

5.3.
Continuity and compactness of the operator T . We establish the following result.
Proposition 4. The operator T is Lipschitz continuous. There is a positive constant C, depending depending only on the domain D and some physical constants, such that G and B(0, R) is the ball of U × H 1 0 (D) with center 0 and radius R.
Using the Hölder inequality, the Young inequality and Propostions 1, 2 and 3, we have for all α > 0, We also have Multiplying the second equation of (35) by U , the third one by ω, integrating by parts, using (8), (36), (37), and choosing α small enough, we get . This estimate implies that the first two claims in Proposition 4 hold.
To prove the third claim, consider a sequence (V n , w n ) ⊂ B(0, R). For each n, let (M n , H n ) = T 1 (V n , w n ), i.e. (M n , H n ) is the weak solution of problem (P 1 ) associated with (V n , w n ), and let (U n , ω n ) = T 2 (M n , H n ) i.e. (U n , ω n ) is the weak solution of problem (P ε 2 ) associated with (M n , H n ). We have (U n , ω n ) = T (V n , w n ). Clearly, estimate (30) holds with U replaced by U n and ω by ω n . Since (U n ) is bounded in U and (ε∆U n ) is bounded in L 2 (D), by using the regularity results of the problem −ε∆U n ∈ L 2 (D), U n ∈ U, we have that (U n ) is bounded in H 2 (D) ∩ U. A similar result holds for the sequence (ω n ), i.e. (ω n ) is bounded in H 2 (D) ∩ H 1 0 (D). Then there exists a subsequence (m) of (n) such that (V m , w m ) and (U m , ω m ) = T (V m , w m ) satisfy Let us show that T (V, w) = (U, ω). To prove that we have to pass to the limit in the nonlinear terms appearing in (P ε 2 ) that is on the terms (V m · ∇)U m , (V m · ∇)ω m , (M m · ∇)H m and M m × H m . By using the weak-strong convergence principle we have , we obtain that (U ε , ω ε , M ε , H ε ) is a solution to problem (P ε ), satisfying the bounds (16)- (18). Moreover, (U ε , ω ε ) belongs to H 2 (D) × H 2 (D) and satisfies inequality (19).
6. End of the proof of Theorem 2.1.
6.1. Existence of a weak solution to problem (P). Let (U ε , ω ε , M ε , H ε ) be the solution to problem (P ε ) given by Theorem 5.1. According to the regularity of (U ε , ω ε , M ε , H ε ), the uniform estimates (16)- (19) stated formally in Section 4 hold: Here C is a positive constant that depends only on the domain D and some physical constants.
We construct a weak solution of problem (P) by passing to the limit, as ε → 0, on the sequence (U ε , ω ε , M ε , H ε ). The weak formulation of problem (P ε ) consists in the following variational equations.
Proof. We use the weak and strong convergences stated in Lemma 6.1. Due to Lemma 6.1, M ε converges to M in L 4 (D) strong and ∇H ε converges to ∇H in According to the Sobolev embedding H 1 (D) → L 4 (D), we have that Φ ∈ L 4 (D). Then (M ε · ∇)H ε ; Φ → (M · ∇)H; Φ . We also have (M ε × H ε ; Ψ) (M × H; Ψ), for all Ψ ∈ H 1 0 (D). By similar arguments we justify the convergence of the other nonlinear terms. Now we can pass to the limit in the weak formulation of problem (P ε ) and obtain that the weak limit (U, ω, M, H) of (U ε , ω ε , M ε , H ε ) is a weak solution of problem (P). Passing to the lower limit in (38)-(40), we obtain that (U, ω, M, H) satisfies estimates (2)-(4). with e = F 2 + L 2 + G 2 .