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Analysis of a three-dimensional phase-ﬁeld model for solidiﬁcation under a magnetic ﬁeld eﬀect André Ferreira e Pereira, Gabriela Planas ∗ Departamento de Matemática, Instituto de Matemática, Estatística e Computação Cientíﬁca, Universidade Estadual de Campinas, Rua Sergio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brazil

a r t i c l e

i n f o

Article history: Received 13 February 2019 Available online 11 September 2019 Submitted by J. Guermond Keywords: Phase-ﬁeld model Solidiﬁcation Convection Magnetic eﬀect Degenerate parabolic systems

a b s t r a c t This paper presents a mathematical analysis to a three-dimensional isothermal model of solidiﬁcation for a binary alloy with melt convection and under a magnetic ﬁeld eﬀect. The model consists of a highly non-linear system of partial diﬀerential equations for the state variables: the velocity ﬁeld, the pressure, the potential function of the electrical ﬁeld, the phase-ﬁeld which represents the solid/liquid phase of the alloy, and the concentration. The well-posedness of the model is discussed. Moreover, the existence of solutions when the diﬀusion coeﬃcient of the concentration equation vanishes for some values of the phase-ﬁeld is investigated. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Phase-ﬁeld models for solidiﬁcation process have attained considerable importance in modeling and simulation of a variety of complex growth structures. This kind of models emerges as an application of the phenomenological Ginzburg-Landau theory [14]. Basically, the model is derived from a free energy or entropy functional that depends on an order parameter, the so-called phase-ﬁeld, which characterizes the diﬀerent phases. The phase-ﬁeld ψ represents an intermediate state when varies smoothly between the liquid state (ψ = 1) and the solid state (ψ = 0). This approach has been applied successfully to modeling solidiﬁcation process for pure materials and binary alloys. We can mention [5,7,8,15,16,18,21,24], among others, where theoretical and numerical analysis were performed. Let us notice that, in the case of binary alloys, the free energy or entropy functional also depends on the concentration c of the solute in the solvent, see for instance [29–31]. Phase-ﬁeld models for solidiﬁcation process have been extended to include melt convection, see, e.g., [3, 4,6,19,20,28]. Indeed, melt convection has important eﬀects on the behavior of the material inﬂuencing the solidiﬁcation pattern; the evolving microstructure can produce unexpected and complicated phenomena. * Corresponding author. E-mail addresses: [email protected] (A. Ferreira e Pereira), [email protected] (G. Planas). https://doi.org/10.1016/j.jmaa.2019.123494 0022-247X/© 2019 Elsevier Inc. All rights reserved.

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Recently, Belmiloudi and Rasheed in [23] proposed a phase-ﬁeld model for binary alloys which incorporates melt convection and the action of a magnetic ﬁeld. Concentration, phase-ﬁeld, and temperature equations are deduced from an entropy functional, the eﬀect of convection is included by coupling the Navier-Stokes equations, and the eﬀect of the magnetic ﬁeld is introduced by using a Lorentz force term in the Navier-Stokes equations. Accordingly, the model turns into a magnetohydrodynamic (MHD) type model where the induced magnetic ﬁeld is negligible in comparison with the imposed magnetic ﬁeld. This happens, for instance, when the magnetic Reynolds number is suﬃciently small (to a more detailed discussion see Davidson [9]). The mathematical analysis of the two-dimensional isothermal problem was done, by the same authors, in [22]. Well-posedness analysis for this kind of models becomes important due to recent experimental results involving solidiﬁcation with ﬂow under the eﬀect of an imposed magnetic ﬁeld, see, [12,13]. These works show the eﬃciency of Alexandrov-Galenko theory ([1,2]) in the description of dendritic growth kinetics with convection. In that theory, new advanced models on crystal growth with convection are introduced, which have been formulated for arbitrary growth Péclet numbers and laminar forced ﬂow. The results presented here provide mathematical support to the model proposed by Belmiloudi and Rasheed in [23] in the three-dimensional case. In this way, the aim of this work is to extend the analysis of [22] to the more interesting physically three-dimensional case. Moreover, we will allow the concentration equation to degenerate. This degeneracy arises when the diﬀusion coeﬃcient vanishes which, from the physical point of view, happens in the solid phase. More precisely, let Ω be an open bounded domain of R3 with a smooth boundary ∂Ω and consider the following system ρ0

∂u

+ (u · ∇)u = −∇p + μΔu + A1 (ψ, c) + b(ψ)((−∇φ + u × B) × B),

∂t div u = 0

(1.1) (1.2)

Δφ = div(u × B), ∂ψ + (u · ∇)ψ = 2 Δψ + A2 (ψ, c), ∂t ∂c + (u · ∇)c = div(D(ψ)∇c + A3 (ψ, c)∇ψ), ∂t

(1.3) (1.4) (1.5)

in Ω × (0, +∞) together with the boundary and initial conditions ∂ψ ∂c ∂φ = = = 0 on ∂Ω × (0, +∞), ∂n ∂n ∂n u(0) = u0 , ψ(0) = ψ0 , c(0) = c0 in Ω, u=

(1.6) (1.7)

where n is the exterior normal to ∂Ω. The unknowns are the velocity ﬁeld u, the pressure p, the potential function φ of the electrical ﬁeld, the phase-ﬁeld ψ which represents the solid/liquid phase of the alloy, and the concentration c of the solute in the solvent. Here, B represents the magnetic-ﬁeld, ρ0 > 0 is the density, μ > 0 is the viscosity, > 0 is a small parameter, D ≥ 0 is the diﬀusion coeﬃcient, and A1 , b, A2 , A3 are nonlinear functions. For further details about the model, the reader is referred to [23] or to the Appendix, where, for the sake of completeness, we present a brief derivation of the model based on a free energy functional. Let us observe that in the two-dimensional case, assuming that the magnetic-ﬁeld B and the movement are in the xz-plane, and all the state variables and data are independent of the variable y, it holds that div(u ×B) = 0. Hence, from equation (1.3) and the non-ﬂux boundary condition, there follows that ∇φ = 0. Consequently, in two-dimensions, one unknown and one equation are missing. Due to this, the arguments

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employed in [22] fail to apply directly here. The major challenges in the mathematical analysis of the problem (1.1)-(1.7) come from the diﬀerent character of equation (1.3) and the lost of the parabolic character of concentration equation (1.5). In order to deal with the degenerate case, we ﬁrst consider the non-degenerate problem (when D ≥ D0 > 0), and investigate the existence, regularity and properties of its solutions. Then we will consider the case where the diﬀusion coeﬃcient D may vanish for some values of the phase-ﬁeld ψ, so allowing the concentration equation to degenerate. From the physical point of view, the diﬀusion coeﬃcient vanishes in the solid phase. As it is expected, for the degenerate case, there is a loss of regularity for the concentration in comparison with the non-degenerate case. This article is organized as follows. In section 2 we set up a more general problem to be study from which system (1.1)-(1.7) is a particular case. We also state the assumptions and introduce the notations. In section 3 we deal with the non-degenerate problem. We show the existence of weak solutions, investigate some regularity properties of solutions and prove the existence of strong solutions. A maximum principle is established as well as the continuity of the strong solution with respect to the data. Consequently, the strong solution will be unique. In section 4, the existence of weak solutions (in a sense to be precise) to the degenerate problem is established. Finally, in the Appendix, we present a brief derivation of the model based on a free energy functional. 2. Statement of the problem and notation In this section we set up a more general problem to be study from which system (1.1)-(1.7) is a particular case. We also state the assumptions and introduce the notation that will be used throughout the work. More precisely, we consider the following system: ρ0

∂u

+ (u · ∇)u = −∇p + μΔu + A1 (ψ, c) + b(ψ)((T (u) + u × B) × B) in Q,

∂t div u = 0 in Q

(2.1) (2.2)

∂ψ + (u · ∇)ψ = 2 Δψ + A2 (ψ, c) in Q, ∂t ∂c + (u · ∇)c = div(D(ψ)∇c + A3 (ψ, c)∇ψ) in Q, ∂t ∂c ∂ψ = = 0 on ∂Ω × (0, +∞), u= ∂n ∂n u(0) = u0 , ψ(0) = ψ0 , c(0) = c0 in Ω,

(2.3) (2.4) (2.5) (2.6)

where Q = Ω × (0, T ), T > 0, and T : (L2 (Ω))3 → (L2 (Ω))3 is a linear operator satisfying T (u)L2 ≤ CuL2

(2.7)

with C > 0 a constant independent of u. Observe that for each u ∈ (H01 (Ω))3 we can deﬁne T (u) = −∇φ where φ ∈ H 2 (Ω)/R is the unique solution to the problem ⎧ ⎨Δφ = div(u × B) in Ω, ⎩ ∂φ = 0 on ∂Ω, ∂n

(2.8)

with B ∈ (W 1,∞ (Ω))3 ([26, Prop. 7.6]). Multiplying equation (2.8) by φ, integrating over Ω, applying integration by parts and Young inequality, we easily see that T satisﬁes (2.7). Thenceforth, once we have obtained a solution to (2.1)-(2.6), by the previous argument, we ﬁnd a solution to (1.1)-(1.7).

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To treat the non-degenerate case we employ the same hypotheses that in [22] for the nonlinear functions and we ask more regularity for the magnetic ﬁeld to deal with the elliptic problem (2.8). So, we assume (H1) A1 : R2 → R3 , Ai : R2 → R, i = 2, 3, and b : R → R are bounded Lipschitz continuous functions, (H2) D : R → R is a bounded Lipschitz continuous function and there exists a constant D0 such that 0 < D0 ≤ D(r), ∀ r ∈ R, (H3) B ∈ (L∞ (0, T ; W 1,∞ (Ω)))3 . Notice that, from the physical point of view, a maximum principle should hold for ψ and c. Thus, we make further assumptions on the behavior of the nonlinear terms which lead to the maximum principle: (H4) A2 (r, s) = 0 for r ∈ (−∞, 0] ∪ [1, +∞), ∀ s ∈ R, (H5) A3 (r, s) = 0 for s ∈ (−∞, 0] ∪ [1, +∞), ∀ r ∈ R. However, if we allow the coeﬃcient D to vanish, it will be required A1 and A2 to be aﬃne in the second variable and the speciﬁc form of A3 given by the modelling, see the Appendix. From the mathematical point of view, due to the low regularity of the concentration when the equation degenerates, the argument employs the fact that the associated Nemytskii operators to A1 and A2 are weakly sequentially continuous which is true only if it is aﬃne, see, for instance, [17, Prop. 3.1]. That means that in the physical model the form of these functions plays an important role. For the sake of clarity, we will assume from now on that ρ0 = μ = = 1. We introduce the usual spaces of divergence free vector ﬁelds H = {v ∈ (L2 (Ω))3 | div v = 0, v · n = 0 on ∂Ω}, V = {v ∈ (H01 (Ω))3 | div v = 0}, and denote H = H × L2 (Ω) × L2 (Ω) and V = V × H 1 (Ω) × H 1 (Ω). The inner product in H and in L2 (Ω) will be denoted by (·, ·) and the duality product between H 1 (Ω) (or V ) and H 1 (Ω) (or V ) by ·, · . Let P be the orthogonal projection from L2 (Ω) onto H and A = −P Δ be the Stokes operator. Let us consider the following continuous trilinear forms

bu (u; v, w) =

3

ui (∂i vj )wj , ∀ (u, v, w) ∈ (V )3 ,

i,j=1 Ω

u · ∇ψz, ∀ (u, ψ, z) ∈ V,

bψ (u; ψ, z) = Ω

u · ∇cz, ∀ (u, c, z) ∈ V,

bc (u; c, z) = Ω

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where u = (u1 , u2 , u3 ), v = (v1 , v2 , v3 ) and w = (w1 , w2 , w3 ). Observe that bu (u; u, u) = 0,

bψ (u; ψ, ψ) = 0,

bc (u; c, c) = 0

for (u, ψ, c) ∈ V.

The following elliptic estimates and interpolation inequalities will be frequently used in the text. Let k ∈ N and ψ ∈ H 1 (Ω) such that Δψ ∈ H k (Ω) and ∂ψ/∂n = 0 on ∂Ω, then it holds that ψ ∈ H k+2 (Ω) and there exists a constant C > 0, independent of ψ, such that the following elliptic estimate is true ψH k+2 ≤ C(ΔψH k + ψL2 ),

(2.9)

(see, for instance, [10, Theorem 7.32]). Another important result that will be used is the Gagliardo-Nirenberg interpolation inequality (see, [11, Theorem 10.1]) 3/4

1/4

f L4 ≤ Cf H 1 f L2 , for any f ∈ H 1 (Ω).

(2.10)

In particular, by the Poincaré inequality, if f = 0 on ∂Ω, we have 3/4

1/4

f L4 ≤ C∇f L2 f L2 , for any f ∈ H01 (Ω).

(2.11)

3. Non-degenerate problem In this section we study problem (2.1)-(2.6) assuming that the diﬀusion coeﬃcient D ≥ D0 > 0. We split the analysis into three subsections where the existence of weak solutions is established, some regularity properties are investigated and ﬁnally, the continuity of the strong solution with respect to the data is proved. 3.1. Existence of a weak solution In this subsection, we discuss the existence of a weak solution to problem (2.1)-(2.6). The proof is based on a Faedo-Galerkin method and follows along the lines of the two-dimensional case considered in [22, Theorem 2.1]. However, in our case, we have to deal with the operator T and use three-dimensional Gagliardo-Nirenberg interpolation inequality, with brings some additional diﬃculties, in particular, to treat the nonlinearities. We ﬁrst introduce the deﬁnition of weak solution to problem (2.1)-(2.6). Deﬁnition 3.1 (Weak solution). We say that (u, ψ, c) is a weak solution to problem (2.1)-(2.6) if (u, ψ, c) ∈ L2 (0, T ; V) ∩ L∞ (0, T ; H) with

∂u ∂ψ ∂c , , ∈ L4/3 (0, T ; V ) ∂t ∂t ∂t

satisﬁes

∂u , v + ∇u · ∇v + bu (u; u, v) ∂t Ω

=

∂ψ ,z ∂t

Ω

(3.1) b(ψ)((T (u) + u × B) × B)v, ∀ v ∈ V,

A1 (ψ, c)v + Ω

∇ψ · ∇z + bψ (u; ψ, z) =

+ Ω

A2 (ψ, c)z, Ω

(3.2)

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∂c ,z ∂t

(D(ψ)∇c + A3 (ψ, c)∇ψ) · ∇z + bc (u; c, z) = 0,

+

(3.3)

Ω

for all z ∈ H 1 (Ω), a.e. in (0, T ), and the initial conditions u(0) = u0 , c(0) = c0 , ψ(0) = ψ0 . Next, we state and prove the theorem about the existence of a weak solution. Theorem 3.2 (Existence of weak solutions). Assume that assumptions (H1)-(H3) are satisﬁed and (u0 ,ψ0 ,c0) ∈ H. Then, there exists at least one weak solution to problem (2.1)-(2.6). Proof. We employ a Faedo-Galerkin method. Since the proof is standard we only sketch it. Let 0 < λ1 ≤ ¯ be the corresponding eigenλ2 ≤ · · · be the eigenvalues of the Stokes operator and (wi )i≥1 ⊂ C ∞ (Ω) functions. These eigenfunctions form a complete orthogonal basis in H, V and V ∩ H 2 (Ω). We denote by Wm = span{w1 , ..., wm } and let Pm be the orthogonal projection from H onto Wm . Similarly, let 0 = μ1 ≤ μ2 ≤ · · · be the eigenvalues of the −Δ operator with homogeneous Neumann ¯ be the corresponding eigenfunctions, such that (ei )i≥1 form a boundary conditions and (ei )i≥1 ⊂ C ∞ (Ω) 2 complete orthogonal basis in L (Ω) and in H 1 (Ω). We denote by Em = span{e1 , ..., em } and by Lm the orthogonal projection from L2 (Ω) onto Em . We introduce the following approximate problem: for each m ∈ N ﬁnd um (t) =

m

um i (t)wi

∈ Wm , ψm (t) =

i=1

m

ψim (t)ei

∈ Em , cm (t) =

i=1

m

cm i (t)ei ∈ Em ,

(3.4)

i=1

such that

∂um , v + ∇um · ∇v + bu (um ; um , v) ∂t Ω

=

∂ψm ,z ∂t

∂cm ,z ∂t

Ω

Ω

A2 (ψm , cm )z,

(3.6)

Ω

(D(ψm )∇cm + A3 (ψm , cm )∇ψm ) · ∇z + bc (um ; cm , z) = 0,

+

(3.5)

∇ψm · ∇z + bψm (um ; ψm , z) =

+

b(ψm )((T (um ) + um × B) × B)v,

A1 (ψm , cm )v + Ω

(3.7)

Ω

(um , ψm , cm )(0) = (Pm u0 , Lm ψ0 , Lm c0 )

(3.8)

for all v ∈ Wm and z ∈ Em . This is a system of ﬁrst order nonlinear ordinary diﬀerential equations for m m (um i , ψi , ci )i≤m which has a unique maximal solution deﬁned on [0, Tm ) with 0 < Tm ≤ T . We next obtain some estimates independent of m for (um , ψm , cm ) that allow to pass to the limit in the approximate problem (3.5)-(3.8) and also to show that Tm = T . We will denote by C a positive constant independent of m which may to change from line to line. By taking um , ψm , cm as test functions in (3.5)-(3.7), respectively, integrating by parts, using assumptions (H1)-(H3), and Hölder and Young inequalities, we arrive at d um 2L2 + 2∇um 2L2 ≤ C(1 + um 2L2 + T (um )2L2 ), dt d ψm 2L2 + 2∇ψm 2L2 ≤ C(1 + ψm 2L2 ), dt

(3.9) (3.10)

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d cm 2L2 + D0 ∇cm 2L2 ≤ C∇ψm 2L2 . dt

7

(3.11)

By multiplying equation (3.11) by 1/C and adding (3.10) and (3.9), noting that, by (2.7), T (um )L2 ≤ Cum L2 , we can apply Gronwall Lemma and conclude that (um , ψm , cm ) is bounded in L∞ (0, Tm ; H) ∩ L2 (0, Tm ; V) independently of m

(3.12)

and also that Tm = T . From equations (3.5)-(3.7), we have that

∂u

m

≤ C(∇um L2 + um 2L4 + 1 + T (um )L2 + um L2 ), ∂t V

∂ψ

m ≤ C(∇ψm L2 + um L4 ψm L4 + 1),

∂t (H 1 )

∂c

m ≤ C(∇cm L2 + ∇ψm L2 + um L4 cm L4 ).

∂t (H 1 ) By using Gagliardo-Nirenberg inequality (2.10) and (2.7), we obtain

∂u

m 3/2 1/2

≤ C(um V + um V um L2 + 1),

∂t V

∂ψ

m 3/4 1/4 3/4 1/4 ≤ C(∇ψm L2 + um V um L2 ψm H 1 ψm L2 + 1),

∂t (H 1 )

∂c

m 3/4 1/4 3/4 1/4 ≤ C(∇cm L2 + ∇ψm L2 + um V um L2 cm H 1 cm L2 ).

∂t (H 1 ) Thenceforth, from estimates (3.12) we infer that ∂u

m

∂t

,

∂ψm ∂cm , is bounded in L4/3 (0, T ; V ) independently of m. ∂t ∂t

(3.13)

Since the embedding V into H is compact, it follows from Corollary 4 in [25] and estimates (3.12)-(3.13), that there exists (u, ψ, c) ∈ L∞ (0, T ; H) ∩L2 (0, T ; V) ∩W 1,4/3 (0, T ; V ), and a subsequence of (um , ψm , cm ), relabelled the same, such that (um , ψm , cm ) (u, ψ, c) weakly in L2 (0, T ; V),

(3.14)

(um , ψm , cm ) → (u, ψ, c) strongly in L2 (0, T ; H).

(3.15)

We observe that from assumptions (H1)-(H3), the nonlinearities are bounded Lipschitz continuous functions, thus it is not diﬃcult to pass to the limit on them (cf. [22, Lemma 2.4]). Moreover, from (2.7) it holds that T (um ) → T (u) strongly in L2 (0; T ; L2 (Ω)). Therefore, these convergences allow us to pass to the limit in (3.5)-(3.8) and conclude that there exists at least one weak solution for (2.1)-(2.6). 2 As a consequence of Theorem 3.2 we obtain the existence of a weak solution to (1.1)-(1.7). Corollary 3.3. Let (u0 , ψ0 , c0 ) ∈ H and assumptions (H1)-(H3) be fulﬁlled. Then, there exists a weak solution (u, φ, ψ, c) to problem (1.1)-(1.7) in the following sense (u, ψ, c) ∈ L2 (0, T ; V) ∩ L∞ (0, T ; H), φ ∈ L∞ (0, T ; H 1 (Ω)/R) ∩ L2 (0, T ; H 2 (Ω)/R),

8

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∂u ∂ψ ∂c , , ∈ L4/3 (0, T ; V ), ∂t ∂t ∂t satisﬁes

∂u , v + ∇u · ∇v + bu (u; u, v) ∂t Ω

=

∂ψ ,z ∂t

∂c ,z ∂t

(u × B) · ∇z,

(3.18)

Ω

∇ψ · ∇z + bψ (u; ψ, z) =

+ Ω

A2 (ψ, c)z,

(3.19)

Ω

(D(ψ)∇c + A3 (ψ, c)∇ψ) · ∇z + bc (u; c, z) = 0,

+

(3.17)

Ω

b(ψ)((−∇φ + u × B) × B)v,

A1 (ψ, c)v +

∇φ · ∇zdx = Ω

Ω

(3.16)

(3.20)

Ω

for all v ∈ V and z ∈ H 1 (Ω) in the distributional sense on (0, T ) and u(0) = u0 , c(0) = c0 , ψ(0) = ψ0 . Proof. Let (u, ψ, c) be a weak solution given by Theorem 3.2, with T (u) = −∇φ, where φ(t) ∈ H 2 (Ω)/R is the unique satisfying (2.8). In particular,

∇φ · ∇zdx = Ω

(u × B) · ∇z

for all z ∈ H 1 (Ω).

Ω

By taking φ(t) as test function, it follows that there exists a constant C > 0, such that T (u)L2 = ∇φL2 ≤ CuL2 , whence ∇φ ∈ L∞ (0, T ; L2 (Ω)). Moreover, by applying [27, Prop. 1.2, Ch. I], we deduce that φ(t) ∈ L2 (Ω) and φ(t)L2 /R ≤ C∇φ(t)L2 . Therefore, φ ∈ L∞ (0, T ; H 1 (Ω)/R). Additionally, due to the fact that u ∈ L2 (0, T ; V) and B ∈ L∞ (0, T ; W 1,∞ (Ω)), it follows that div(u×B) ∈ 2 L ((0, T ) × Ω). Therefore, by multiplying equation (2.8) by Δφ, integrating over (0, T ) × Ω and applying Young inequality, we obtain that Δφ2L2 ((0,T )×Ω) ≤ Cdiv(u × B)2L2 ((0,T )×Ω) . Thus, φ ∈ L2 (0, T ; H 2 (Ω)/R).

2

3.2. Regularity of the solutions In this subsection, we ﬁrst improve the regularity of the velocity ﬁeld and with this improved regularity, we show that the phase-ﬁeld is more regular. Diﬀerently, to the two-dimensional case, a strong solution of the three-dimensional Navier-Stokes equations is local in time, which in turn implies local regularity

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for the phase-ﬁeld and the concentration. Then, we will prove the existence of local strong solutions to problem (2.1)-(2.6). Finally, with additional assumptions on the nonlinearities, a maximum principle for the phase-ﬁeld and the concentration is established. Theorem 3.4 (Local regularity). Let (u0 , ψ0 , c0 ) ∈ V ×H 1 (Ω) ×L2 (Ω) and assumptions (H1)-(H3) be fulﬁlled. Then, there exists T∗ > 0 such that the weak solution given by Theorem 3.2 has the following regularity u ∈ L∞ (0, T∗ ; V ) ∩ L2 (0, T∗ ; H 2 (Ω)) ∩ H 1 (0, T∗ ; H), ψ ∈ L∞ (0, T∗ ; H 1 (Ω)) ∩ L2 (0, T∗ ; H 2 (Ω)) ∩ H 1 (0, T∗ ; L2 (Ω)). Proof. To establish the regularity of a solution, we derive further a priori estimates for the approximate solution. First, as the initial data is more regular, it holds that Pm u0 V ≤ Cu0 V and ∇Lm ψ0 L2 ≤ C∇ψ0 L2 , where C is independent of m. For the velocity ﬁeld, we notice that A1 (ψm , cm )+b(ψm )((T (um )+um ×B)×B) is bounded in L∞ (0, T ; H) independently of m. So we can proceed as in Theorem 3.11 Chapter III in [27] to obtain the existence of T∗ > 0 such that (um ) is bounded in L∞ (0, T∗ ; V ) ∩ L2 (0, T∗ ; H 2 (Ω)) ∩ H 1 (0, T∗ ; H)

(3.21)

independently of m, whence it follows the regularity for the limit u. It worth to observe that T∗ does not depend on m. Next, we will show the regularity of ψ. To this end, we take −Δψm ∈ Em as test function in (3.6) and use Hölder inequality to obtain 1 d ∇ψm 2L2 + Δψm 2L2 2 dt

≤ |bψm (um ; ψm , Δψm )| + A2 (ψm , cm )Δψm Ω

≤ C (um L4 ∇ψm L4 Δψm L2 + Δψm L2 ) . Since H 1 (Ω) ⊂ L4 (Ω) together with estimate (3.21) and Gagliardo-Nirenberg inequality (2.10), there follows, for all t ∈ [0, T∗ ], d 3/4 1/4 ∇ψm 2L2 + 2Δψm 2L2 ≤ C(∇ψm H 1 ∇ψm L2 Δψm L2 + Δψm L2 ). dt By applying the Young inequality and using elliptic estimates (2.9) we ﬁnd d 3/2 1/2 ∇ψm 2L2 + 2Δψm 2L2 ≤ C(ψm H 2 ∇ψm L2 + 1) + Δψm 2L2 dt 3/2

1/2

≤ C(Δψm L2 ∇ψm L2 + ∇ψm 2L2 + 1) + Δψm 2L2 . By using again the Young inequality and arranging terms we arrive at d ∇ψm 2L2 + 1 + Δψm 2L2 ≤ C ∇ψm 2L2 + 1 . dt

(3.22)

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Thus, Gronwall Lemma implies (∇ψm ) is bounded in L∞ (0, T∗ ; L2 (Ω)) independently of m, which, together with estimate (3.12), give us (ψm ) is bounded in L∞ (0, T∗ ; H 1 (Ω)) independently of m.

(3.23)

By integrating (3.22) over [0, T∗ ], using (3.23) and elliptic estimates (2.9) we deduce that (ψm ) is bounded in L2 (0, T∗ ; H 2 (Ω)) independently of m.

(3.24)

Consequently, the limit ψ ∈ L∞ (0, T∗ ; H 1 (Ω)) ∩ L2 (0, T∗ ; H 2 (Ω)). ∂ψ m ∈ Em as test function in (3.6) and using Hölder and Young inequalities, we have Finally, by taking ∂t that

∂ψm 2

+ 1 d ∇ψm 2 2 ≤ C um L4 ∇ψm L4 ∂ψm + ∂ψm

L

∂t 2 2 dt

∂t 2 ∂t 2 L L L

2 1

∂ψm . ≤ C um 2V ψm 2H 2 + 1 + 2 ∂t L2 By integrating over [0, T∗ ] and using (3.21) and (3.24), there follows that ∂ψ m

∂t

is bounded in L2 (0, T∗ ; L2 (Ω)).

(3.25)

The proof of the Theorem is then completed. 2 Remark 3.5. With the regularity obtained, from the De Rham Theorem there exists a distribution p ∈ L2 (0, T∗ ; H 1 (Ω)/R) such that ∂u + (u · ∇)u − Δu(t) − A1 (ψ, c) − b(ψ)((T (u) + u × B) × B) = −∇p ∂t a.e. in Ω × (0, T∗ ). To obtain higher order regularity for the concentration is necessary to improve the regularity of the phase-ﬁeld. In this way, the existence of a local strong solution is obtained. Theorem 3.6 (Existence of strong solutions). Let (u0 , ψ0 , c0 ) ∈ V × H 2 (Ω) × H 1 (Ω) with and assumptions (H1)-(H3) be satisﬁed. Then, there exists T∗ > 0 such that u ∈ L∞ (0, T∗ ; V ) ∩ L2 (0, T∗ ; H 2 (Ω)) ∩ H 1 (0, T∗ ; H), p ∈ L2 (0, T∗ ; H 1 (Ω)/R), ψ ∈ L∞ (0, T∗ ; H 2 (Ω)) ∩ L2 (0, T∗ ; H 3 (Ω)) ∩ H 1 (0, T∗ ; H 1 (Ω)), c ∈ L∞ (0, T∗ ; H 1 (Ω)) ∩ L2 (0, T∗ ; H 2 (Ω)) ∩ H 1 (0, T∗ ; L2 (Ω)), satisfy (2.1)-(2.6) a.e. in Ω × (0, T∗ ).

∂ψ0 ∂n

= 0 on ∂Ω

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Proof. The regularity of u is given by the previous Theorem. Next, we derive further a priori estimates for the approximate solution. By taking Δ2 ψm ∈ Em as test function in (3.6), integrating by parts and using m that ∂Δψ = 0 on ∂Ω, we obtain ∂n d Δψm 2L2 + 2∇Δψm 2L2 dt ≤ 2 |bψm (um ; ψm , Δ2 ψm )| + |∇A2 (ψm , cm )||∇Δψm | .

(3.26)

Ω

To estimate the ﬁrst term on the right hand side, we re-write by integrating by parts, bψm (um ; ψm , Δ ψm ) = −

(∇um · ∇ψm + um · D2 ψm ) · ∇Δψm ,

2

Ω

|bψm (um ; ψm , Δ ψm )| ≤ ∇um L4 ∇ψm L4 + D2 ψm L4 um L4 ∇Δψm L2 1/4 3/4 ≤ C um H 2 ψm H 2 + D2 ψm L2 D2 ψm H 1 um V ∇Δψm L2 1/4 3/4 ≤ C um H 2 ψm H 2 + ψm H 2 ψm H 3 ∇Δψm L2 , 2

where we have used the Sobolev embedding and the fact that um is bounded in L∞ (0, T∗ ; V ) (see (3.21)). Now, by using Young inequality and elliptic estimates (2.9), one has |bψm (um ; ψm , Δ2 ψm )| 1 1/2 3/2 ≤ C um 2H 2 + 1 ψm 2H 2 + ψm H 2 ∇Δψm L2 + ∇Δψm 2L2 4 1 ≤ C um 2H 2 Δψm 2L2 + ψm 2L2 + ψm 2H 2 + ∇Δψm 2L2 . 2 The bound for the second term on the right hand side of (3.26) follows from Hölder and Young inequalities,

1 |∇A2 (ψm , cm )||∇Δψm | ≤ C ∇ψm 2L2 + ∇cm 2L2 + ∇Δψm 2L2 . 2

Ω

Plugging the previous estimates in (3.26), we arrive at d Δψm 2L2 + ∇Δψm 2L2 ≤ C um 2H 2 Δψm 2L2 + um 2H 2 + ψm 2H 2 + cm 2H 1 . dt

(3.27)

By (3.12) and improved regularity (3.21), (3.24), there follows that um and ψm are bounded in L2 (0, T∗ ; H 2 (Ω)) and cm in L2 (0, T∗ ; H 1 (Ω)). Hence, from Gronwall Lemma (Δψm ) is bounded in L∞ (0, T∗ ; L2 (Ω)), 0 2 2 where we have used that ψ0 ∈ H 2 (Ω) and ∂ψ ∂n = 0 on ∂Ω imply that Δψm (0)L ≤ Δψ0 L . Then, by elliptic estimates together with (3.12), we conclude that

(ψm ) is bounded in L∞ (0, T∗ ; H 2 (Ω)).

(3.28)

12

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By integrating (3.27) over [0, T∗ ], using the previous bound and elliptic estimate (2.9) we infer that (ψm ) is bounded in L2 (0, T∗ ; H 3 (Ω)).

(3.29)

Now, we show the regularity of cm . By taking −Δcm ∈ Em as test function in (3.7), integrating by parts and using assumptions (H1)-(H3), we have that 1 d 2 ∇cm L2 + D0 2 dt

|Δcm |2 Ω

≤C

|∇ψm ||∇cm | + |∇ψm |2 + |Δψm | |Δcm | + bc (um ; cm , Δcm ),

Ω

and applying Hölder inequality and Sobolev embedding it follows d 2 ∇cm L2 + 2D0 Δcm 2L2 dt ≤ C ∇ψm L4 ∇cm L4 + ∇ψm 2L4 + Δψm L2 + um L4 ∇cm L4 Δcm L2 ≤ C ψm H 2 ∇cm L4 + ψm 2H 2 + ψm H 2 + um V ∇cm L4 Δcm L2 . Notice that um is bounded in L∞ (0, T∗ ; V ) and ψm in L∞ (0, T∗ ; H 2 ), thus, by applying Young inequality we infer that d 2 ∇cm L2 + D0 Δcm 2L2 ≤C(∇cm 2L4 + 1). dt By applying Gagliardo-Nirenberg (2.10) and Young inequalities, and elliptic estimates, we have d 1/2 3/2 2 ∇cm L2 + D0 Δcm 2L2 ≤ C ∇cm L2 ∇cm H 1 + 1 dt D0 ≤ C ∇cm 2L2 + 1 + Δcm 2L2 . 2 By applying Gronwall lemma together with (3.12), and using that c0 ∈ H 1 (Ω), we deduce (cm ) is bounded in L∞ (0, T∗ ; H 1 (Ω)) ∩ L2 (0, T∗ ; H 2 (Ω)).

(3.30)

Once we have improved the regularity of the solution, by using the equations, it can be proved that 2 2 ∈ L2 (0, T∗ ; H 1 (Ω)) and ∂c ∂t ∈ L (0, T∗ ; L (Ω)) as in Theorem 2.3 in [22] because the proof does not depend on the dimension. This completes the proof. 2 ∂ψ ∂t

Now, we turn to the original problem (1.1)-(1.7) and discuss the regularity of the potential function φ. We have two situations depending on the regularity of the magnetic ﬁeld B. (1) If B ∈ L∞ (0, T ; W 1,∞ (Ω)), as u ∈ L∞ (0, T∗ ; V ), it holds that div(u(t) × B(t)) ∈ L∞ (0, T∗ ; L2 (Ω)). Thus, for a.e. t ∈ (0, T∗ ) the solution of (2.8) satisﬁes φ(t) ∈ H 2 (Ω)/R. Therefore, multiplying equation (2.8) by Δφ(t) and integrating over Ω, it follows that Δφ2L2 ≤ Cdiv(u × B)2L2 . It implies that φ ∈ L∞ (0, T∗ ; H 2 (Ω)/R)).

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(2) If B ∈ L∞ (0, T ; W 2,∞ (Ω)), as u ∈ L2 (0, T∗ ; H 2 (Ω)), we have that div(u(t) × B(t)) ∈ L2 (0, T∗ ; H 1 (Ω)). Then, for a.e. t ∈ (0, T∗ ), φ(t) ∈ H 3 (Ω)/R. By applying the gradient to equation (2.8), multiplying the resulting equation by Δ∇φ(t) and integrating over Ω, we obtain ∇Δφ2L2 ≤ C∇div(u × B)2L2 , whence φ ∈ L2 (0, T∗ ; H 3 (Ω)/R)). We ﬁnish this subsection by noting that, under physical assumptions on the nonlinearities, a maximum principle for the phase-ﬁeld and the concentration is true. This will be useful to treat the degenerate case. Thus, by assuming that the nonlinear terms A2 and A3 satisfy the extra assumptions (H4) and (H5), that is, (H4) A2 (r, s) = 0 for r ∈ (−∞, 0] ∪ [1, +∞), ∀ s ∈ R, (H5) A3 (r, s) = 0 for s ∈ (−∞, 0] ∪ [1, +∞), ∀ r ∈ R, there holds Proposition 3.7 (Maximum principle). In addition to assumptions in the Theorem 3.6 suppose that (H4)-(H5) are fulﬁlled and that 0 ≤ ψ0 ≤ 1, 0 ≤ c0 ≤ 1, a.e. in Ω. Then 0 ≤ ψ ≤ 1, 0 ≤ c ≤ 1,

a.e. in Ω × (0, T∗ ).

The proof is standard, so we omit the details. See, for instance, [22]. 3.3. Continuous dependence In this subsection we prove the continuity of the strong solution with respect to the data which yields the uniqueness of the local strong solution. ∂ψ i

Theorem 3.8. Let (u0i , ψ0i , ci0 ) ∈ V × H 2 (Ω) × H 1 (Ω) satisfying ∂n0 = 0 on ∂Ω, i = 1, 2. Assume that (H1)-(H3) are satisﬁed. Let (ui , φi , ci ) be the local strong solution of (2.1)-(2.6) with initial data (u0i , ψ0i , ci0 ) and magnetic ﬁeld Bi , i = 1, 2. Then, it holds u1 − u2 L∞ (0,T∗ ;L2 ) + ψ1 − ψ2 L∞ (0,T∗ ;H 1 ) + c1 − c2 L∞ (0,T∗ ;L2 ) ≤ C u01 − u02 L2 + ψ01 − ψ02 H 1 + c10 − c20 L2 + B1 − B2 L∞ (Q)

(3.31)

u1 − u2 L2 (0,T∗ ;H 1 ) + ψ1 − ψ2 L2 (0,T∗ ;H 2 ) + c1 − c2 L2 (0,T∗ ;H 1 ) ≤ C u01 − u02 L2 + ψ01 − ψ02 H 1 + c10 − c20 L2 + B1 − B2 L∞ (Q)

(3.32)

and

where C > 0 is a constant. Consequently, the local strong solution given by Theorem 3.6 is unique.

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14

Proof. We introduce u = u1 − u2 , φ = φ2 − φ1 , c = c1 − c2 , B = B1 − B2 . We will derive a diﬀerential inequality which allows to apply the Gronwall Lemma and conclude the result. Observe that u, ψ and c satisfy, for all v ∈ V and z ∈ H 1 (Ω),

∂u ·v+ ∂t

Ω

∇u · ∇v + bu (u; u1 , v) + bu (u2 ; u, v) Ω

[(A1 (ψ1 , c1 ) − A1 (ψ2 , c2 ))v + (b(ψ1 ) − b(ψ2 ))T (u1 )v + b(ψ2 )T (u)v]

= Ω

(3.33) (b(ψ1 ) − b(ψ2 ))((u1 × B1 ) × B1 )v

+ Ω

+

Ω

∂ψ z+ ∂t

Ω

b(ψ2 ) (u × B1 ) × B1 + (u2 × B) × B1 + (u2 × B2 ) × B v,

∇ψ · ∇z + bψ (u; ψ1 , z) + bψ (u2 ; ψ, z) =

Ω

∂c z+ ∂t

Ω

(A2 (ψ1 , c1 ) − A2 (ψ2 , c2 ))z,

(3.34)

Ω

[D(ψ2 )∇c + (D(ψ1 ) − D(ψ2 ))∇c1 ] · ∇z + bc (u; c1 , z) + bc (u2 ; c, z) Ω

(3.35) [(A3 (ψ1 , c1 ) − A3 (ψ2 , c2 ))∇ψ1 · ∇z + A3 (ψ2 , c2 )∇ψ · ∇z]

= Ω

with u(0) = u01 − u02 ,

ψ(0) = ψ01 − ψ02

and

c(0) = c10 − c20 .

By taking u ∈ V as test function in (3.33), using assumptions (H1)-(H3), (2.7), and Hölder inequality, we can estimate 1 d 2 uL2 +∇u2L2 2 dt ≤C u2L4 ∇u1 L2 + (ψL2 + cL2 )uL2 + ψL4 uL4 T (u1 )L2 + T (u)L2 uL2 + ψL4 u1 L4 uL2 + u2L2 + u2 L2 uL2 BL∞ . Observe that ui ∈ L∞ (0, T∗ ; V ). So, by using Gagliardo-Nirenberg (2.11) and Young inequality, there follows d 2 uL2 + ∇u2L2 ≤ C u2L2 + ψ2H 1 + c2L2 + B2L∞ . dt

(3.36)

By taking ψ ∈ H 1 (Ω) as test function in (3.34), noting that ψ1 ∈ L∞ (0, T∗ ; H 1 (Ω)), similarly as before we have that d 2 ψL2 + 2∇ψ2L2 ≤ C uL4 ∇ψ1 L2 ψL4 + ψ2L2 + cL2 ψL2 dt 1 ≤ C ψ2H 1 + c2L2 + ∇u2L2 . 4

(3.37)

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Now, we take −Δψ ∈ H 1 (Ω) as test function in (3.34) and use Hölder inequality to ﬁnd 1 d ∇ψ2L2 + Δψ2L2 ≤ C uL4 ∇ψ1 L4 + u2 L4 ∇ψL4 + ψL2 + cL2 ΔψL2 . 2 dt As ψ1 ∈ L∞ (0, T∗ ; H 2 (Ω)) and u2 ∈ L∞ (0, T∗ ; V ), by using Gagliardo-Nirenberg inequality (2.10), we obtain 1 d ∇ψ2L2 + Δψ2L2 2 dt 1/4 3/4 1/4 3/4 ≤ C uL2 ∇uL2 + ∇ψL2 ∇ψH 1 + ψL2 + cL2 ΔψL2 . By applying Young inequality and elliptic estimates (2.9), we infer that 1 d ∇ψ2L2 + Δψ2L2 ≤ C u2L2 + ψ2H 1 + c2L2 + ∇u2L2 . dt 4

(3.38)

Finally, we take c ∈ H 1 (Ω) as test function in (3.35) and use assumptions (H1)-(H3), as before 1 d 2 cL2 + D0 ∇c2L2 2 dt ≤ C ψL∞ ∇c1 L2 + ψL4 ∇ψ1 L4 + cL4 ∇ψ1 L4 + ∇ψL2 ∇cL2 + uL4 ∇c1 L2 cL4 . Recalling that ψ1 ∈ L∞ (0, T∗ ; H 2 (Ω)) and c1 ∈ L∞ (0, T∗ ; H 1 (Ω)), using Gagliardo-Nirenberg inequality (2.10), and elliptic estimate (2.9) one has 1 d 2 cL2 + D0 ∇c2L2 2 dt 1/4 3/4 ≤ C ΔψL2 + cL2 cH 1 + ψH 1 ∇cL2 + C∇uL2 cH 1 . By applying Young inequality we arrive at d 2 cL2 + D0 ∇c2L2 ≤ C c2L2 + ψ2H 1 + Δψ2L2 + ∇u2L2 . dt

(3.39)

By adding (3.36), (3.37), (3.38) and (3.39) multiplied by η > 0, we ﬁnd 1 d (u2L2 + ψ2H 1 + ηc2L2 ) + ∇u2L2 + ΔψL2 + ηD0 ∇c2L2 dt 2 ≤ C u2L2 + ψ2H 1 + c2L2 + B2L∞ + ηC Δψ2L2 + ∇u2L2 . We choose ηC =

1 4

(3.40)

and apply Gronwall Lemma to obtain

u(t)2L2 + ψ(t)2H 1 + c(t)2L2 ≤ C u01 − u02 2L2 + ψ01 − ψ02 2H 1 + c10 − c20 2L2 + B2L∞ , for all t ∈ [0, T∗ ], which is (3.31). By integrating (3.40) over (0, T∗ ) we deduce (3.32). This ﬁnishes the proof. 2 Remark 3.9. Let us observe that for a given u and B, the solution φ of problem (2.8) is unique and since the operator T (u) = −∇φ is linear, there follows that problem (1.1)-(1.7) also has a unique local strong solution.

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4. Degenerate problem In this section we consider an extended version of the problem (2.1)-(2.6) when the concentration equation degenerates. As mentioned in the Introduction, this degeneracy arises when the diﬀusion coeﬃcient D vanishes which, from the physical point of view, happens in the solid phase. More precisely, instead of assumption (H2) we will assume that D(r) ≥ 0, ∀ r ∈ R. In this way, the parabolic character of the concentration equation (2.4) is lost. As it is expected, the regularity of the concentration will be very weak. In order to be able to deal with the nonlinearities of the problem we have to employ the speciﬁc form for the functions A1 , A2 and A3 given by the model. Thus, we will assume the following hypotheses for the degenerate case: (H1 ) A1 : R2 → R3 and A2 : R2 → R are given by A1 (r, s) = E1 (r) + sE2 (r) and A2 (r, s) = F1 (r) + sF2 (r), where Ei : R2 → R3 , Fi : R → R, i = 1, 2, are bounded Lipschitz continuous functions, (H2 ) D : R → R is a bounded Lipschitz continuous function such that

0 ≤ D(r), ∀ r ∈ R, (H3 ) A3 : R2 → R is given by A3 (r, s) = D(r)D2 (r, s), where D2 : R2 → R and b : R → R are bounded Lipschitz continuous functions, (H4 ) B ∈ L∞ (0, T ; W 1,∞ (Ω)).

As in the non-degenerate case, we make further assumptions on the behavior of the nonlinear terms which lead to the maximum principle: (H5 ) Fi (r) = 0, for all r ∈ (−∞, 0] ∪ [1, +∞), (H6 ) D2 (r, s) = 0 for all s ∈ (−∞, 0] ∪ [1, +∞) and r ∈ R. The maximum principle is important to overcome the diﬃculties coming from the low regularity of the concentration. As before, we assume that T : (L2 (Ω))3 → (L2 (Ω))3 is a linear transformation satisfying (2.7). Then, we have the following result. Theorem 4.1 (Existence of solution to the degenerate problem). Let assumptions (H1 )-(H6 ) be fulﬁlled and (u0 , ψ0 , c0 ) ∈ V ×H 1 (Ω) ×L2 (Ω) such that 0 ≤ ψ0 , c0 ≤ 1 a.e. in Ω. Then, there exist T∗ > 0 and (u, p, ψ, c) such that u ∈ L∞ (0, T∗ ; V ) ∩ L2 (0, T∗ ; H 2 (Ω)) ∩ H 1 (0, T∗ ; L2 (Ω)), p ∈ L2 (0, T∗ ; H 1 (Ω)/R), ψ ∈ L∞ (0, T∗ ; H 1 (Ω)) ∩ L2 (0, T∗ ; H 2 (Ω)) ∩ H 1 (0, T∗ ; L2 (Ω)), c ∈ L∞ (0, T∗ ; L2 (Ω)) ∩ H 1 (0, T∗ ; (H 1 (Ω)) ), 0 ≤ ψ, c ≤ 1 a.e. in Ω × (0, T∗ ),

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satisfy ∂u + (u · ∇)u = −∇p + Δu + A1 (ψ, c) + b(ψ)((T (u) + u × B) × B) a.e. in Ω × (0, T∗ ), ∂t div(u) = 0 a.e. in Ω × (0, T∗ ), ∂ψ + (u · ∇)ψ = Δψ + A2 (ψ, c) a.e. in Ω × (0, T∗ ), ∂t ∂ψ = u = 0 a.e. on ∂Ω × (0, T∗ ), ∂n

∂c ,z + ∇(D(ψ)c) − c∇D(ψ) + D(ψ)D2 (ψ, c)∇ψ − uc · ∇z = 0 ∂t

(4.1) (4.2) (4.3) (4.4)

Ω

a.e. t ∈ (0, T∗ ), for all z ∈ H 1 (Ω),

(4.5)

u(0) = u0 , ψ(0) = ψ0 , a.e. in Ω, c(0) = c0 in H 1 (Ω) .

(4.6)

4.1. Proof of Theorem 4.1 In order to prove this theorem we will regularize the coeﬃcient D following [16,24]. More precisely, let Dλ (r) = D(r) + λ

where 0 < λ ≤ 1.

Therefore, Dλ is a bounded Lipschitz continuous function such that 0 < λ ≤ Dλ (r) for all r ∈ R. Moreover, to apply the results from the non-degenerate case, we deﬁne 1 (r, s) =E1 (r) + Π(s)E2 (r), A 2 (r, s) =F1 (r) + Π(s)F2 (r), A where Π is a truncation function ⎧ ⎪ ⎨ 1, Π(s) = s, ⎪ ⎩ 0,

s>1 0≤s≤1 s < 0.

1 and A 2 are bounded Lipschitz continuous functions. Let us observe that once that the In this way A 1 and A 2 will be aﬃne in the second variable. maximum principle is proved, A For a ﬁxed λ > 0, we consider system (2.1)-(2.6) by replacing D by Dλ only in front of ∇c in the concentration equation: ∂uλ 1 (ψλ , cλ ) + b(ψλ )((T (uλ ) + uλ × B) × B), + (uλ · ∇)uλ = −∇pλ + Δuλ + A ∂t

(4.7)

div(uλ ) = 0,

(4.8)

∂ψλ 2 (ψλ , cλ ), + (uλ · ∇)ψλ = Δψλ + A ∂t ∂cλ + (uλ · ∇)cλ = div(Dλ (ψλ )∇cλ + D(ψλ )D2 (ψλ , cλ )∇ψλ ), ∂t

(4.9) (4.10)

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uλ =

∂cλ ∂ψλ = = 0 on ∂Ω × (0, +∞), ∂n ∂n

(4.11)

uλ (0) = u0 , ψλ (0) = ψ0λ , cλ (0) = cλ0 in Ω,

(4.12)

where ψ0λ ∈ H 2 (Ω) and cλ0 ∈ H 1 (Ω) are such that ψ0λ → ψ0 ∈ H 1 (Ω) and cλ0 → c0 in L2 (Ω). Notice that ψ0λ H 1 and cλ0 L2 are bounded independently of λ. This fact will be used when obtaining estimates uniform in λ. Theorems 3.6 and 3.8 furnish the existence of a unique local strong solution for this regularized system and, from their proofs, it is clear that the local time of existence does not depend on λ. Moreover, a maximum principle holds true for the phase-ﬁeld and the concentration (see Proposition 3.7). We collect this results in the next proposition. Proposition 4.2 (Local solution for the regularized system). Let assumptions (H1 )-(H6 ) be fulﬁlled and (u0 , ψ0λ , cλ0 ) ∈ V × H 2 (Ω) × H 1 (Ω) with 0 ≤ ψ0λ , cλ0 ≤ 1 a.e. in Ω. Then, for each λ ∈ (0, 1], there exist T∗ > 0 (independent of λ) and (uλ , pλ , ψλ , cλ ) such that uλ ∈ L∞ (0, T∗ ; V ) ∩ L2 (0, T∗ ; H 2 (Ω)) ∩ H 1 (0, T∗ ; L2 (Ω)), pλ ∈ L2 (0, T∗ ; H 1 (Ω)/R), ψλ ∈ L∞ (0, T∗ ; H 2 (Ω)) ∩ L2 (0, T∗ ; H 3 (Ω)) ∩ H 1 (0, T∗ ; H 1 (Ω)), cλ ∈ L∞ (0, T∗ ; H 1 (Ω)) ∩ L2 (0, T∗ ; H 2 (Ω)) ∩ H 1 (0, T∗ ; L2 (Ω)), 0 ≤ ψλ , cλ ≤ 1

a.e. in Ω × (0, T∗ ),

satisfy (4.7)-(4.12) a.e. in Ω × (0, T∗ ). Notice that, as 0 ≤ cλ ≤ 1, it holds Π(cλ ) = cλ , so 1 (ψλ , cλ ) = E1 (ψλ ) + cλ E2 (ψλ ) = A1 (ψλ , cλ ) and A 2 (ψλ , cλ ) = F1 (ψλ ) + cλ F2 (ψλ ) = A2 (ψλ , cλ ). A Our aim now is to obtain some estimates uniform in λ and convergences for the solution of the regularized problem in order to pass to the limit in system (4.7)-(4.12), as λ → 0+ , and show the existence of solution for the degenerate problem. Lemma 4.3 (Energy estimates). There exists C > 0, independent of λ, such that uλ L∞ (0,T∗ ;H) + uλ L2 (0,T∗ ;V ) ≤ C,

(4.13)

ψλ L∞ (0,T∗ ;L2 (Ω)) + ψλ L2 (0,T∗ ;H 1 (Ω)) ≤ C,

(4.14)

T∗ cλ L∞ (0,T∗ ;L2 (Ω)) +

Dλ (ψλ )|∇cλ |2 ≤ C. 0 Ω

(4.15)

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Proof. By multiplying (4.7) by uλ , integrating over Ω, and using assumptions (H1 )-(H6 ), there follows 1 d uλ 2L2 + ∇uλ 2L2 = 2 dt

b(ψλ )((T (uλ ) + uλ × B) × B)uλ

A1 (ψλ , cλ )uλ + Ω

Ω

≤ C uλ L1 (Ω) + T (uλ )L2 uλ L2 + uλ 2L2 ≤ C 1 + uλ 2L2 .

Therefore, by Gronwall Lemma we obtain (4.13). Similarly, by multiplying (4.9) by ψλ , integrating over Ω, and using that A2 is bounded, one has 1 d ψλ 2L2 + ∇ψλ 2L2 = 2 dt

A2 (ψλ , cλ )ψλ Ω

≤ Cψλ L1 (Ω) ≤ C 1 + ψλ 2L2 . Gronwall Lemma gives (4.14). Finally, by multiplying (4.10) by cλ , integrating over Ω, and using that D2 is bounded and that D < D + λ = Dλ , yield 1 d cλ 2L2 + 2 dt

Dλ (ψλ )|∇cλ |2 = −

Ω

D(ψλ )D2 (ψλ , cλ )∇ψλ · ∇cλ Ω

≤C

Dλ (ψλ )|∇ψλ ||∇cλ | Ω

≤

1 2

Dλ (ψλ )|∇cλ |2 + C∇ψλ 2L2 . Ω

By arranging terms and integrating in time lead to (4.15). 2 The previous estimates furnish only weak convergences. In order to pass to the limit in the nonlinear terms it is necessary some strong convergences. To this end, we perform more estimates independent of λ. Lemma 4.4 (Additional estimates). There exists C > 0, independent of λ, such that uλ L∞ (0,T∗ ;V ) + uλ L2 (0,T∗ ;H 2 (Ω)) ≤ C,

(4.16)

ψλ L∞ (0,T∗ ;H 1 (Ω)) + ψλ L2 (0,T∗ ;H 2 (Ω)) ≤ C,

∂ψλ

∂uλ

+ ≤ C,

∂t 2

∂t 2 L (0,T∗ ;L2 (Ω)) L (0,T∗ ;L2 (Ω))

∂cλ

≤ C.

∂t 2 L (0,T∗ ;(H 1 (Ω)) )

(4.17) (4.18) (4.19)

Proof. Estimates (4.16)-(4.18) are proved in an analogous way as in Theorem 3.4 since they do not depend on the regularity of the concentration.

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By multiplying (4.10) by z ∈ H 1 (Ω), integrating over Ω, and using Hölder inequality we arrive at ∂cλ λ z ≤ D (ψλ )|∇cλ ||∇z| + C |∇ψλ ||∇z| + |uλ cλ · ∇z| ∂t Ω Ω Ω Ω ⎛⎛ ⎞ ⎞1/2 ⎜ ⎟ ≤ C ⎝⎝ Dλ (ψλ )|∇cλ |2 ⎠ + ∇ψλ L2 + uλ L2 cλ L∞ ⎠ ∇zL2 . Ω

By using estimates of Lemma 4.3 and the fact that 0 ≤ cλ ≤ 1, we conclude (4.19). 2 From the uniform estimates obtained in the previous lemmas, we will deduce some convergences for the regularized solution. Lemma 4.5. There exist (u, ψ, c) such that u ∈ L∞ (0, T∗ ; H 1 (Ω)) ∩ L2 (0, T∗ ; H 2 (Ω)) ∩ H 1 (0, T∗ ; L2 (Ω)), ψ ∈ L∞ (0, T∗ ; H 1 (Ω)) ∩ L2 (0, T∗ ; H 2 (Ω)) ∩ H 1 (0, T∗ ; L2 (Ω)), c ∈ L∞ (0, T∗ ; L2 (Ω)) ∩ H 1 (0, T∗ ; (H 1 (Ω)) ), 0 ≤ ψ, c ≤ 1 a.e. in Ω × (0, T∗ ), and a subsequence of (uλ , ψλ , cλ ) (relabelled the same), such that, as λ → 0+ , (uλ , ψλ ) (u, ψ)

weakly in L2 (0, T∗ ; (H 2 (Ω))2 ),

(uλ , ψλ ) → (u, ψ)

strongly in L (0, T∗ ; V × H (Ω)) ∩ C([0, T∗ ]; H × L (Ω)),

cλ c cλ → c

2

1

weakly in L2 (Ω × (0, T∗ )), 1

(4.20) 2

(4.21) (4.22)

strongly in C([0, T∗ ]; H (Ω) ),

∗

cλ c weakly-* in L∞ (Ω × (0, T∗ )).

(4.23) (4.24)

Proof. Convergences (4.20) and (4.22) follow directly from to the previous lemmas. Moreover, since the embeddings from (H 2 (Ω) ∩ V ) × H 2 (Ω) into V × H 1 (Ω) and from V × H 1 (Ω) into H × L2 (Ω) are compact, by applying Corollary 4 of [25], we conclude that (uλ , ψλ ) is relatively compact in L2 (0, T∗ ; V × H 1 (Ω)) ∩ C([0, T∗ ]; H × L2 (Ω)) and (4.21) holds. Similarly, by estimates (4.14) and (4.19) and since the embedding from L2 (Ω) into (H 1 (Ω)) is compact, we have that (cλ ) is a relatively compact in C([0, T∗ ]; (H 1 (Ω)) ) and (4.23) follows. Convergence (4.24) is a direct consequence of the maximum principle. 2 1 = A1 and A 2 = A2 are aﬃne in the second variable. So, from now Recall that 0 ≤ cλ , c ≤ 1, then A on, A1 (r, s) = E1 (r) + sE2 (r) and A2 (r, s) = F1 (r) + sF2 (r). Comparing with the passage to the limit in the proof of the existence of weak solution, the main diﬀerence here is to treat nonlinear terms involving the concentration. We collect these convergences in the next lemma. Lemma 4.6. As λ → 0+ , it holds A1 (ψλ , cλ ) A1 (ψ, c) weakly in L2 (Ω × (0, T∗ ))3 ,

(4.25)

A2 (ψλ , cλ ) A2 (ψ, c) weakly in L2 (Ω × (0, T∗ )),

(4.26)

A. Ferreira e Pereira, G. Planas / J. Math. Anal. Appl. 482 (2020) 123494

Dλ (ψλ )∇cλ ∇(D(ψ)c) − c∇D(ψ) weakly in L2 (Ω × (0, T∗ ))3 ,

21

(4.27)

D(ψλ )D2 (ψλ , cλ )∇ψλ D(ψ)D2 (ψ, c)∇ψ weakly in L (Ω × (0, T∗ )) ,

(4.28)

uλ cλ uc weakly in L2 (Ω × (0, T∗ )).

(4.29)

2

3

Proof. The fact that A1 and A2 are aﬃne in second variable is crucial to show convergences (4.25) and (4.26) due to the weak convergence of cλ . Indeed, since Ei and Fi , i = 1, 2, are bounded Lipschitz continuous functions, it holds Ei (ψλ ) → Ei (ψ) strongly in (Lp (Ω × (0, T∗ )))3 and Fi (ψλ ) → Fi (ψ) strongly in Lp (Ω × (0, T∗ )), for any p ∈ [1, ∞). Moreover, as the associated Nemytskii operator to A1 and A2 is weakly sequentially continuous, by using (4.22), there follows (4.25) and (4.26). The proof of convergences (4.27) and (4.28) are more technical and analogous to [24, Lemmas 2,3,4], so we omit the details. We just observe that in [24] was proved that D(ψλ )D2 (ψλ , cλ )∇ψλ D(ψ)D2 (ψ, c)∇ψ weakly in L1 (Ω × (0, T∗ ))3 . However, since D(ψλ )D2 (ψλ , cλ )∇ψλ is uniformly bounded in L2 (Ω × (0, T∗ ))3 , by the uniqueness of the limit follows (4.28). Finally, for v ∈ L2 (Ω × (0, T∗ )), we have T∗

T∗ (uλ cλ − uc)v =

0 Ω

T∗ cλ (uλ − u)v +

0 Ω

(cλ − c)uv. 0 Ω

Since 0 ≤ cλ ≤ 1, by the strong convergence (4.21), the ﬁrst term on the right-hand side goes to zero as λ → 0+ . The second one goes to zero by (4.24). 2 Convergences from Lemmas 4.5 and 4.6 allow to pass to the limit in system (4.7)-(4.12), as λ → 0+ , and show the existence of solution for the degenerate problem. This completes the proof of Theorem 4.1. Acknowledgments We would like to express our gratitude to the referee for his/her comments and suggestions that improved the presentation of the paper. This work was ﬁnanced in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior Brasil (CAPES) - Finance Code 001. A.P. Ferreira was partially supported by Centro Federal de Educação Tecnológica de Minas Gerais. G. Planas was partially supported by CNPq-Brazil, grants 308093/2018-6 and 402388/2016-0. Appendix A We present here a brief derivation of the model based on a free energy Ginzburg-Landau functional. Let us observe that in [23] a description of the evolution equations of phase-ﬁeld and concentration is given based on an entropy functional. Let Ω be an open bounded domain of R3 with a smooth boundary ∂Ω. The region Ω is occupied by a binary alloy, where B is the solute and A is the solvent. We consider an isothermal process where the

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A. Ferreira e Pereira, G. Planas / J. Math. Anal. Appl. 482 (2020) 123494

transition occurs at the critical temperature TM . The alloy is under a magnetic ﬁeld eﬀect that induces movement in the liquid part. We denote by u the velocity of the ﬂuid, c the concentration of solute B in the solvent A and ψ the phase-ﬁeld. The free energy Ginzburg-Landau functional can be expressed in the form (see [31]): F (ψ, c) =

f (ψ, c) +

ε2 |∇ψ|2 , 2

Ω

where ε > 0 is a constant and f is the free energy density given by f (ψ, c) = (1 − c)f A (ψ) + cf B (ψ) +

RTM [(1 − c) ln(1 − c) + c ln(c)], vm

where R is the Boltzmann constant and vm is the molar volume. The free energy density f A and f B of materials A and B, respectively, can be expressed as f A (ψ) = WA ψ 2 (ψ − 1)2 , f B (ψ) = WB ψ 2 (ψ − 1)2 , where WA and WB are constants. As the phase-ﬁeld is a non-conserved quantity and the concentration is a conserved quantity, we have that ∂ψ δF + (u · ∇)ψ = −M1 , ∂t δψ

(A.1)

∂c + (u · ∇)c = −∇ · Jc , ∂t

(A.2)

where M1 > 0 is a positive constant and Jc = M (ψ, c)∇

δF , δc

and M is chosen so that in the solid/liquid states it becomes the classical diﬀusion equation. In particular, it can have the form (see [30]) M (ψ, c) = −

vm D(ψ)c(1 − c) RTM

where D is an increasing function such that D(0) = Ds and D(1) = Dl , where Ds and Dl are the solid and liquid diﬀusion coeﬃcients, respectively. The Gâteaux’s derivative of F with respect to ψ in direction of η ∈ C ∞ (Ω) is given by δηψ F (ψ, c)

ε2 d d 2 F (ψ + λη, c) f (ψ + λη, c) + |∇(ψ + λη)| = = dV dλ dλ 2 λ=0 λ=0 = Ω

Ω

∂f (ψ, c)η + ε2 ∇ψ · ∇ηdV = ∂ψ

∂f (ψ, c) − ε2 Δψ ηdV ∂ψ

Ω

where we have integrated by parts and used that ∇ψ · n = 0 on ∂Ω × [0, ∞), being n the exterior normal to ∂Ω. Thus,

A. Ferreira e Pereira, G. Planas / J. Math. Anal. Appl. 482 (2020) 123494

23

∂f δF = (ψ, c) − ε2 Δψ. δψ ∂ψ Analogously, ∂f δF = (ψ, c). δc ∂c Therefore, (A.1) and (A.2) imply that ∂f ∂ψ + (u · ∇)ψ =M1 ε2 Δψ − (ψ, c) , ∂t ∂ψ ∂f ∂c + (u · ∇)c = − ∇ · M (ψ, c)∇ (ψ, c) . ∂t ∂c

(A.3) (A.4)

We can see that ∂f A ∂f B ∂f A ∂f = (1 − c) +c = +c ∂ψ ∂ψ ∂ψ ∂ψ

∂f B ∂f A − ∂ψ ∂ψ

,

thus, −

∂f = F1 (ψ) + cF2 (ψ), ∂ψ

where F1 and F2 are cubic polynomials in the variable ψ such that F1 (0) = F2 (0) = F1 (1) = F2 (1) = 0. In addition, ∇

∂2f ∂2f ∂f RTM 1 = ∇ψ + 2 ∇c = −F2 (ψ)∇ψ + ∇c. ∂c ∂c∂ψ ∂ c vm c(1 − c)

By replacing the previous expressions in (A.3) and (A.4) give us ∂ψ + (u · ∇)ψ = M1 ε2 Δψ + F1 (ψ) + cF2 (ψ) , ∂t RTM 1 ∂c + (u · ∇)c = −∇ · M (ψ, c) −F2 (ψ)∇ψ + ∇c . ∂t vm c(1 − c) The evolution equations for the ﬂuid ﬂow are derived from the laws of conservation of momentum and mass as in [23]. Due to the magnetic eﬀect, the Lorentz force appears in the Navier-Stokes equations. In this way, we coupled the following equations to the previous ones: ρ0

∂u ∂t

+ (u · ∇)u = −∇p + μΔu + A1 (ψ, c) + b(ψ)((−∇φ + u × B) × B), div u = 0, Δφ = div(u × B),

where A1 (ψ, c) = E1 (ψ) + cE2 (ψ) is a Boussinesq approximation such that A1 (0, c) = 0, b is a function satisfying b(0) = 0, p is the pressure, φ is the potential function to the electrical ﬁeld intensity, ρ0 > 0 is the mean density of the ﬂuid, μ > 0 is the viscosity, and B is the magnetic ﬁeld. √ Therefore, by denoting = M1 ε, A2 (ψ, c) = M1 (F1 (ψ) + cF2 (ψ)), and A3 (ψ, c) = M (ψ, c)F2 (ψ), we have that time evolution of (u, p, φ, ψ, c) is given by the following system:

A. Ferreira e Pereira, G. Planas / J. Math. Anal. Appl. 482 (2020) 123494

24

ρ0

∂u

+ (u · ∇)u = −∇p + μΔu + A1 (ψ, c) + b(ψ)((−∇φ + u × B) × B),

∂t div u = 0

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