On a heated incompressible magnetic fluid model

In this paper we study the equations describing the dynamics of heat transfer in an incompressible magnetic fluid under the action of an applied magnetic field. The system consists of the Navier-Stokes equations, the magnetostatic equations and the temperature equation. We prove global-in-time existence of weak solutions to the system posed in a bounded domain 
of $R^3$ and equipped with initial and boundary conditions. 
The main difficulty comes from the singularity of the term representing 
the Kelvin force due to magnetization.


1.
Introduction. The purpose of the present work is the mathematical analysis of a model describing heat transfer in an incompressible magnetic fluid heated from below. Magnetic fluids (also called ferrofluids) are colloidal suspensions of nanoscale magnetic particles in a carrier fluid. These fluids have found a wide variety of applications in technology, industry and medicine, see for instance [22,35] and the references therein.
We consider an electrically non-conducting incompressible ferrofluid occupying a bounded domain D ⊂ R 3 and heated from a part of its boundary ∂D. We assume that the domain D is a cylinder D = Ω × (−d/2, d/2) of height d, the cross section Ω is a regular open bounded subset of R 2 . The generic point of D is denoted x = ( x, z) with x = (x 1 , x 2 ) ∈ Ω. The fluid is subjected to an external magnetic field and gravity acts in the negative z-direction g = −|g|e 3 , with e 3 = (0, 0, 1). The resulting magnetization M is assumed to be parallel to the demagnetizing field H, namely M = χH where χ, the total magnetic susceptibility, is a function depending on the temperature τ of the fluid and the intensity H = |H| of the magnetic field H. Thus, χ = χ(τ, H) = M/H where M = |M| is the magnetization intensity. The magnetization intensity is then a function of both the temperature τ and the magnetic intensity H; we write M = M (τ, H). The upper and lower boundaries of D, z = d/2 and z = −d/2, are maintained at constant temperature τ + and τ − , 676 YOUCEF AMIRAT AND KAMEL HAMDACHE respectively; we denote τ a = (τ + +τ − )/2. Following Finlayson [9], the magnetization intensity has the relationship where M a , H a are the intensity of magnetization and magnetic field of reference, respectively, and k 1 , k represent the susceptibility and pyromagnetic coefficients, respectively. Relation (1) is deduced by linearization of the Langevin magnetization law about an equilibrium state, see [9,13] for example. Thus the magnetization reads The state variables of the model are the fluid velocity U and the pressure p satisfying the Navier-Stokes equations (with the Kelvin force as external force), the magnetic field H satisfying the magnetostatic equations and the temperature τ satisfying the heat equation. We assume that the Oberbeck-Boussinesq approximation is valid. Since we expect that the temperature τ is less than τ = max(τ + , τ − ) we rewrite (1) in the form with M = M a − k 1 H a + k(τ a − τ ) and (2) becomes The Kelvin force, represented by µ 0 M ∇H where µ 0 is the magnetic permeability of vacuum, can be rewritten as where p m = µ 0 (M H + k1 2 H 2 ) is the magnetic pressure which we add to the hydrostatic pressure p in the Navier-Stokes equations. We assume that M ≥ 0; this means that the reference parameters M a and H a are linked by the condition M a ≥ k 1 H a + k(τ − τ a ). This condition seems to be necessary for the solvability of the magnetostatic equation, see Lemma 2.2 below.
Note that relation (2) is valid in the framework of the quasi-stationary approximation of Neuringer and Rosensweig [20]. The magnetization M is assumed to be independent of the fluid velocity U, and the relaxation time of magnetization is 0, i.e. the magnetization is collinear with the magnetic field H. Collinearity is a good approximation when the internal rotation of colloidal particles can be neglected. In models of magnetic fluid, including the intrinsic rotation of colloidal particles, the magnetization is described by a dynamic equation. Neglecting the spin diffusion terms, the magnetization equation reads [25]: where Ω is the ferrofluid spin velocity which is an average particle rotation angular velocity, t m is the Brownian relaxation time, and M eq is the equilibrium magnetization which is in the form (2). When the spin diffusion terms are included, equation (12), which is of Bloch type, is replaced by the equation [33] which is of Bloch-Torrey type, λ > 0 being a diffusion coefficient that carry spins. See [13,14] where the authors consider the convective instability problem in the horizontal layer of a magnetic fluid with internal rotation with vortex viscosity. Let us also mention some recent papers [1,2,3,4,5,30,34] addressing the question of existence, uniqueness and stability of the equations describing the motion of magnetic fluids in the isothermal case, the magnetization obeying a dynamic equation. In a forthcoming work we investigate a model describing heat transfer in an incompressible magnetic fluid where the magnetization obeys a Bloch-Torrey equation.
Our aim here is to study the global-in-time existence of solutions to system (3)-(10), denoted problem (P) in what follows. Let L q (D), H s (D) and W s,q (D) (1 ≤ q ≤ ∞, s ∈ R) be the usual Lebesgue and Sobolev spaces of scalar-valued functions, respectively. We denote L q (D) = (L q (D)) 3 , H s (D) = (H s (D)) 3 , W s,q (D) = (W s,q (D)) 3 . We denote by · the L 2 -norm on D. If X is a Banach space, the duality product between X (the dual space of X ) and X is denoted by ·; · X ×X or simply by ·; · when there is no confusion of notation. We denote by C([0, T ]; X weak) the space of functions v : [0, T ] −→ X which are continuous with respect to the weak topology. We introduce the classical function spaces in the theory of the Navier-Stokes equations, see [10,11,16,18,19,31,32], , and recall that We suppose that Without lost of generality we also suppose that ρ 0 = c p = 1, the parameter k 1+k1 will be replaced by k and F 1+k1 by F . (ii) the magnetic field H is such that H = ∇ϕ where ϕ ∈ L ∞ (0, T ; W 1,q (D)) and D ϕ dx = 0; (iii) the temperature τ belongs to L ∞ (D T ) ∩ L 2 (0, T ; H 1 (D)) and satisfies 0 ≤ τ ≤ τ a.e. in D T ; (iv) the external body force S, defined by (7), belongs to L 2 (0, T ; H −1 (D)); (v) the momentum equation (4) holds weakly, in the sense that, for every v ∈ U, (vi) the function ϕ is a weak solution of problem (11), i.e. for a.e. t ∈ (0, T ), where V = {v ∈ H 1 (D) : (vii) the function τ is in the form τ = θ + ξ(z) with ξ(z) = 2β d z + τ a , β = τ+−τ− 2 , the function θ belongs to L 2 (0, T ; H 1 (D)), with θ| Γ ± T = 0, and satisfies the integral identity Our main result is the following one.
We organize the rest of the paper as follows. The next section deals with a priori estimates. We show that the temperature satisfies the uniform bounds 0 ≤ τ ≤ τ a.e. in D T ; this allows to establish suitable a priori estimates satisfied by any smooth solution of problem (P). Section 3 is devoted to the proof of Theorem 1.2. We introduce a regularized problem (P ε ) depending on a small parameter ε by regularizing the magnetic force in the momentum equation. We construct approximate solutions (U n , H n , τ n ) of problem (P ε ) by using the semi-Galerkin method, then show the existence of a weak solution to problem (P ε ) by passing to the limit, as n → ∞, in the sequence of approximate solutions. Passing to the limit, as ε → 0, in the sequence of solutions to problem (P ε ) we obtain the existence of a weak solution (in the sense of Definition 1.1) to problem (P). In Section 4 we consider the differential system (3)-(7) with other boundary conditions on Γ ± T for the temperature; we show the existence of a global weak solution to the corresponding boundary-value problem.
In the paper, C indicates a generic constant, depending only on some bounds of the physical data, which can take different values in different occurrences.

2.
A priori estimates. We assume in this section that the solutions (U, H, τ ) of problem (P) are regular enough.
2.1. Bounds for the temperature. Consider the heat equation where U is a given function. By a weak solution of problem (19) we mean a function τ in the form the function θ belongs to L 2 (0, T ; H 1 (D)), with θ| Γ ± T = 0, and satisfies the integral identity (18).
We have: Proof. The existence and uniqueness of weak solutions to problem (19) can be proved as in [17] (Chapter III). Indeed, it suffices to show that any smooth solution of (19) satisfies estimates (21)- (23). Let us prove that that τ ≥ 0. Multiplying equation (19) by the negative part τ − of τ , integrating by parts and using the conditions τ − (±d/2) = 0, we deduce that Multiplying equation (24) by ψ + , the positive part of ψ, and integrating by parts yields 1 2 which implies ψ + = 0 and then τ ≤ τ a.e. in D T . Hence (21). Consider now the function θ given by (20). Inequalities (22) follow directly from (21). We easily verify that θ satisfies Multiplying equation (25) by θ, integrating by parts and using the Poincaré and Young inequalities yields Integrating the latter inequality from 0 to t we obtain (23). Lemma 2.1 is proved.
Proof. Since M ≥ 0, the operator A : V → V given by is strictly monotone, i.e.
Integrating by parts and using the Sobolev embedding H 1 (D) → L 6 (D), we have, a.e. in (0, T ), Here , denotes the duality pairing between H −1 (D) and H 1 0 (D). We estimate each integral in the right-hand side of (31) as follows: then, together with the Sobolev embedding H 1 (D) → L 6 (D) and (28), (29), we deduce that Using (23), we deduce from the latter inequality the estimate of Lemma 2.3.
We have: Lemma 2.4. Assume that the hypotheses of Lemma 2.3 hold true. Then, any weak solution to problem (32) satisfies the estimates where the constants A and B are given in Corollary 1.
Proof. The usual energy estimate for the Navier-Stokes equations, together with the estimate in Corollary 1, implies that Then Gronwall's inequality gives and then we easily deduce (33). The lemma is proved.
3. Proof of the main theorem. The proof consists in three steps: -Step 1. We introduce a regularized problem (P ε ) depending on a small parameter ε > 0 by regularizing the magnetic force in the momentum equation.
-Step 2. We prove the existence of a weak solution (U ε , H ε , τ ε ) to problem (P ε ) (see Proposition 3 below) by using the semi-Galerkin method and the Schauder fixed point theorem.
-Step 2. We prove the existence of a weak solution (U, H, τ ) to problem (P) by passing to the limit, as ε → 0, in the sequence (U ε , H ε , τ ε ) of solutions of problem (P ε ).
3.1. The regularized problem. We introduce the standard smoothing operators based on convolution with a sequence of regularizing kernels. Let σ ∈ D(R 3 ), σ ≥ 0 with support in the unit ball and with finite mass where v has been extended by 0 outside D.
The parameter ε > 0 being fixed, we consider the system formed by the following coupled equations. -Magnetostatic equation: We denote H = ∇ϕ and -Momentum equation: -Temperature equation: In what follows problem (34)-(37) is referred as problem (P ε ).

YOUCEF AMIRAT AND KAMEL HAMDACHE
3.2. The semi-Galerkin approximation. In Sections 3.2 and 3.3, the parameter ε is fixed and we are concerned with the existence of a weak solution to problem (P ε ). For notational convenience we omit to index by ε the variables which generally depend on this parameter (except σ ε ).
To solve problem (P ε ) we use the semi-Galerkin method. Let (a j ) j≥1 be a smooth basis of the Hilbert space U. We define an approximate solution (U n , H n , τ n ) of problem (P ε ) as follows. We look for U n , H n and τ n in the form U n = n j=1 α n j (t) a j , H n = ∇ϕ n , τ n = θ n + ζ(z) (where ζ(z) is given by (20)) and the functions ϕ n , U n and θ n are defined by the following scheme.
-For every t ∈ (0, T ), the function ϕ n solves the problem div (A n (t, x, ∇ϕ n )) = F in D T , where A n (t, x, ξ) = ξ + M + k(τ − τ n (t, x)) ξ |ξ| , (t, x) ∈ D T , ξ ∈ R 3 . We then define H n = ∇ϕ n and -The function U n solves the equation and the initial condition where U 0n is the orthogonal projection in U 0 of U 0 onto the space spanned by a 1 . · · · , a n .
-The function θ n is the solution of In what follows, problem (38)-(41) is referred as problem (P n ). We denote by X n the space spanned by a 1 , · · · , a n .
3.2.1. Solving problem (P n ). We shall solve problem (P n ) by using the Schauder fixed point theorem. Introduce the closed bounded convex subset of L 2 (D T )
Taking ϕ 2 − ϕ 1 as a test function in the variational formulation associated with this equation and integrating over (0, T ) yields Since 0 ≤ τ − τ , M ≥ 0, and a is a monotone vector field, the second term in the previous equality is nonnegative, we have and, using the Cauchy-Schwarz inequality we obtain hence (45). The proof of the lemma is complete.
Here the constatnt C ε > 0 depends only on some bounds of the physical data and on the L 2 -norm of ∇σ ε .
Proof. Clearly, the L 2 -norm of the first term of S(ω) is bounded by C g . The L 2norm of the second term is bounded by C ε H L 2 (0,T ;L 2 (D)) which, according to (43), is bounded by C ε F L 2 (D T ) . Hence (47).
To prove (48) we write Clearly, Using the Hölder inequality and (44), we have, for a.e. t ∈ (0, T ), from which follows For the last term we have We conclude that and using the Sobolev embedding L 6/5 (D) → H −1 (D) we obtain (48). The lemma is proved.
Let us now consider problem (39), (40). We have: where the constant C ε > 0 depends only on some bounds of the physical data and on the L 2 -norm of ∇σ ε . Moreover, the map F : L 2 (0, T ; H −1 (D)) → L 2 (0, T ; L 2 (D)) which associate with S n the solution U n of problem (39), (40), is continuous.
Let us now prove the continuity of F. Consider a sequence (S m ) in the space L 2 (0, T ; H −1 (D)) converging to S in L 2 (0, T ; H −1 (D)). Let U m , U ∈ H 1 (0, T ; X n ) the solution of problem (39), (40), with S n = S m and S n = S, respectively. Arguing as above one shows that the sequence (U m ) is bounded in H 1 (0, T ; X n ). Then one can extract a subsequence still indexed by m and there is W ∈ H 1 (0, T ; X n ) such that (U m ) converges to W strongly in L 2 (0, T ; L 2 (D)) and weakly in L 2 (0, T ; H 1 (D)). Note that all norms are equivalent in X n . Passing to the limit, as m → ∞, in the equation of U m we obtain that W ∈ H 1 (0, T ; X n ) is the solution of (39), (40), with S n = S. Since the solution of this equation is unique we conclude that W = U and that the whole sequence (U m ) converges to U strongly in L 2 (0, T ; L 2 (D)). The function F is then continuous. The proof of the lemma is complete.
We can now prove the following result. Proof. Consider the map R : H → H defined by R(ω) = θ where θ is the weak solution of problem (41), as constructed above. By virtue of Lemmas 3.1-3.4, R is a continuous map. By Lemma 3.4, θ ∈ H ∩ L 2 (0, T ; H 1 (D)), with θ| Γ ± T = 0, and we deduce from equation (41) that ∂ t θ ∈ L 2 (0, T ; H −1 (D)). Consequently, one can use the Aubin-Lions lemma to deduce that R has a precompact image. Applying the Schauder fixed point theorem, we get the existence of a function θ n ∈ H ∩ L 2 (0, T ; H 1 (D)), with θ n | Γ ± T = 0, such that R(θ n ) = θ n . Considering τ n = θ n + ξ(z), H n = ∇ϕ n (where ϕ n is the solution of (42)) and U n (the solution of (39), (40)), we obtain the result. The proof of Proposition 1 is complete.
Here all constants are independent of n and ε.
Proof. Let (U n , H n , τ n ) be the approximate solution constructed above and let θ n = τ n − ξ(z). We have (see (50)) Arguing as in the proof of Lemma 2.4 and using Lemma 2.3 and Corollary 1 one shows that, for all t ∈ (0, T ), where the constants A and B are given in Corollary 1. Hence (55). Estimate (56) holds according to Lemma 3.1, and (57) is obtained in Proposition 1. Using Lemma 2.1 and estimate (55) we deduce (58). Proposition 2 is proved.
3.3. Passing to the limit as n → ∞. Our goal now is to obtain an existence result for problem (P ε ). We have: Proposition 3. Problem (P ε ) admits one weak solution in the following sense.
We will obtain the existence of a weak solution to problem (P ε ) by passing to the limit, as n → ∞, in the sequence of approximate solutions (U n , H n , τ n ) constructed above. According to estimates (55)-(58), there are subsequences (still indexed by n) and functions U, H, τ such that U n U weakly in L 2 (0, T ; U) and in L ∞ (0, T ; L 2 (D) weak-, τ n τ in L ∞ (D T ) weak-and in L 2 (0, T ; H 1 (D)) weak.
Moreover, τ = θ+ξ(z), withθ ∈ L 2 (0, T ; H 1 (D)), θ| Γ ± T = 0, and 0 ≤ τ ≤ τ a.e. in D T . (66) 3.3.1. Strong convergence of (τ n ) and (H n ). It follows from equation (41) and the estimates obtained in Proposition 2 that ∂ t θ n is bounded in L 2 (0, T ; H −1 (D)). Consequently, one can use the Aubin-Lions lemma to deduce that the sequence (θ n ) contains a subsequence (still indexed by n) such that θ n → θ in L 2 (D T ) and a.e. in D T , and then τ n → τ in L 2 (D T ) and a.e. in D T . Using Lemma 3.1 one shows that the sequence H n = ∇ϕ n contains a subsequence which converges strongly in L 2 (0, T ; L 2 (D)), and a.e. in D T , to H = ∇ϕ where ϕ is the unique solution of problem (61).
Consider now We have, for any v ∈ L 2 (0, T ; H 1 0 (D)), Here , denotes the duality pairing between H −1 (D) and H 1 0 (D). Using the strong convergence of (τ n ) and (H n ), the convergence of (σ ε |H n |) to σ ε |H| in L 2 (0, T ; L 2 (D)) and a.e. in D T , and the convergence in L 2 (0, T ; L 2 (D)) weak of (∇θ n ), we obtain We conclude that S n S in L 2 (0, T ; H −1 (D)) weak, 3.3.2. Strong convergence of (U n ). To prove the compactness of (U n ) we need an estimate on the derivative with respect to t of U n . We use the method introduced in [18] (pp. 64-79) for the study of weak solutions to the Navier-Stokes equations.
Lemma 3.5. The sequence (U n ) belongs to a compact set of L 2 (0, T ; L 2 (D)).
Proof. LetǓ n : R → U, the function defined by U n = U n on [0, T ], 0 outside this interval.
We denote by U n the Fourier transform ofǓ n . Let us show that, for 0 < γ < 1/4, We first rewrite (39) in the form where δ 0 and δ T are Dirac distributions at 0 and T andǦ n is defined by G n = G n on [0, T ], 0 outside this interval, with G n = −(U n · ∇)U n + µ∆U n + S n .
Using also (67), (68), we easily pass to the limit in equation (39). We obtain that the limit function U satisfies the integral identity (59). Moreover, U ∈ C([0, T ]; U 0 weak) and satisfies the initial condition (60). Using the strong convergence in L 2 (0, T ; L 2 (D)) of the sequence H n = ∇ϕ n to H = ∇ϕ we easily pass to the limit in problem (38) and obtain that ϕ satisfies (34).
Using Lemma 3.1, one shows that the sequence H ε = ∇ϕ ε contains a subsequence which converges strongly in L 2 (0, T ; L 2 (D)), and a.e. in D T , to H = ∇ϕ where ϕ is the unique solution of problem (17).
The proof of the main theorem is achieved.
4. Concluding remarks. In this section we consider the differential system (3)-(7) with other boundary conditions; we show the existence of a weak solution to the corresponding boundary-value problem. For notational convenience we continue to denote by (P) the corresponding boundary-value problem.