() CUBO A Mathematical Journal Vol.17, No¯ 02, (73–87). June 2015 On an anisotropic Allen-Cahn system Alain Miranville Laboratoire de Mathématiques et Applications, Université de Poitiers, UMR CNRS 7348 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France alain.miranville@math.univ-poitiers.fr ABSTRACT Our aim in this paper is to prove the existence and uniqueness of solutions for an Allen-Cahn type system based on a modification of the Ginzburg-Landau free energy proposed in [11]. In particular, the free energy contains an additional term called Willmore regularization and takes into account anisotropy effects. RESUMEN Nuestro propósito en este trabajo es probar la existencia y unicidad de soluciones para un Sistema de tipo Allen-Cahn basados en una modificación de la enerǵıa libre Ginzburg-Landau propuesta en [11]. En particular, la enerǵıa libre contiene un término adicional llamado regularización de Willmore y considera efectos de anisotroṕıa. Keywords and Phrases: Allen-Cahn equation, Willmore regularization, anisotropy effects, well- posedness 2010 AMS Mathematics Subject Classification: 35B45, 35K55 74 Alain Miranville CUBO 17, 2 (2015) 1 Introduction The Allen-Cahn equation, ∂u ∂t − ∆u + f(u) = 0, (1.1) where u is the order parameter and f(s) = s3 −s, describes important processes related with phase separation in binary alloys, namely, the ordering of atoms in a lattice (see [1]). This equation is obtained by considering the Ginzburg-Landau free energy, ΨGL = ∫ Ω ( 1 2 |∇u|2 + F(u)) dx, (1.2) where Ω is the domain occupied by the material and, typically, F(s) = 1 4 (s2 − 1)2. Assuming a relaxation dynamics, i.e., writing ∂u ∂t = − DΨGL Du , (1.3) where D Du denotes a variational derivative, we obtain (1.1). In [11] (see also [2]), the authors introduced the following modification of the Ginzburg-Landau free energy: ΨAGL = ∫ Ω (δ( ∇u |∇u| )( 1 2 |∇u|2 + F(u)) + β 2 ω2) dx, β > 0, (1.4) ω = −∆u + f(u), (1.5) where G(u) = 1 2 ω2 is called nonlinear Willmore regularization, β is a small regularization parame- ter and the function δ accounts for anisotropy effects. The Willmore regularization is relevant, e.g., in determining the equilibrium shape of a crystal in its own liquid matrix, when anisotropy effects are strong. Indeed, in that case, the equilibrium interface may not be a smooth curve, but may present facets and corners with slope discontinuities (see, e.g., [9]), which can lead to an ill-posed problem and requires regularization. The Allen-Cahn equation associated with (1.4) has been studied in [6] in the particular cases δ ≡ 1 (isotropic case) and δ ≡ −1 (in that case, ΨAGL is also called functionalized Cahn-Hilliard energy in [8]). In particular, well-posedness results have been obtained. The Cahn-Hilliard equation associated with (1.4) (obtained by writing ∂u ∂t = ∆DΨAGL Du ) has been studied in [4], again, in the isotropic case δ ≡ 1; we also refer the reader to [2] and [12] for numerical studies. In this paper, we actually consider the following regularization of ΨAGL: CUBO 17, 2 (2015) On an anisotropic Allen-Cahn system 75 ΨRAGL = ∫ Ω (δ( ∇u (ǫ + |∇u| 2 ) 1 2 )( 1 2 |∇u|2 + F(u)) + β 2 ω2) dx, ǫ > 0. (1.6) We have, in that case and formally, DΨRAGL = ∫ Ω (δ( ∇u (ǫ + |∇u| 2 ) 1 2 )(∇u · ∇Du + f(u)Du) + βωf′(u)Du − βω∆Du) dx + ∫ Ω δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇Du (ǫ + |∇u| 2 ) 1 2 ( 1 2 |∇u|2 + F(u)) dx − ∫ Ω δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u ∇u.∇Du (ǫ + |∇u| 2 ) 3 2 ( 1 2 |∇u|2 + F(u)) dx, where δ′ denotes the differential (gradient) of δ. Therefore, assuming proper boundary conditions, DΨRAGL Du = −div(δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u) + δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u) (1.7) − 1 2 div( |∇u|2 (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) − div( F(u) (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) + 1 2 div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u |∇u|2∇u (ǫ + |∇u| 2 ) 3 2 ) + div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u F(u)∇u (ǫ + |∇u| 2 ) 3 2 ) +βωf′(u) − β∆ω. Assuming again a relaxation dynamics, we finally obtain the following regularized anisotropic Allen-Cahn system: ∂u ∂t − div(δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u) + δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u) (1.8) − 1 2 div( |∇u|2 (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) − div( F(u) (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) + 1 2 div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u |∇u|2∇u (ǫ + |∇u| 2 ) 3 2 ) + div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u F(u)∇u (ǫ + |∇u| 2 ) 3 2 ) 76 Alain Miranville CUBO 17, 2 (2015) +βωf′(u) − β∆ω = 0, ω = −∆u + f(u). (1.9) We proved in [5] the existence and uniqueness of solutions to (1.8)-(1.9), but only in one space dimension, due to a lack of regularity on ∂u ∂t . Thus, in order to handle the problem in higher space dimensions, we consider in this paper the following further regularized Allen-Cahn system: ∂u ∂t − α ∂∆u ∂t − div(δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u) + δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u) (1.10) − 1 2 div( |∇u|2 (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) − div( F(u) (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) + 1 2 div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u |∇u|2∇u (ǫ + |∇u| 2 ) 3 2 ) + div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u F(u)∇u (ǫ + |∇u| 2 ) 3 2 ) +βωf′(u) − β∆ω = 0, α > 0, ω = −∆u + f(u). (1.11) A term of the form −α∂∆u ∂t appears in generalizations of the Allen-Cahn equation proposed in [3], based on a separate balance law for internal microforces, i.e., for interactions at a microscopic level (we can note that the derivation proposed in [3] is strongly based on the usual Ginzburg-Landau free energy; it would thus be interesting to go back to the arguments in [3] and see whether/how they can be adapted to a more general free energy). Such a regularization is also similar to the viscous Cahn-Hilliard equation proposed in [7]. Actually, the approach in [3], applied to the Cahn-Hilliard equation, allows to recover the viscous Cahn-Hilliard equation. We prove, in the next sections, the existence and uniqueness of solutions to (1.10)-(1.11). It is important to note however that our estimates are not independent of ǫ, so that we cannot pass to the limit as ǫ goes to 0. This is not surprising, as the problem formally obtained by taking ǫ = 0 cannot correspond to the (Allen-Cahn) problem associated with the free energy (1.4) (see also [2] and [11]). Actually, this is related with a proper functional setting for the limit problem and, more precisely, for the Allen-Cahn system associated with (1.4). This will be studied elsewhere. CUBO 17, 2 (2015) On an anisotropic Allen-Cahn system 77 2 Setting of the problem We consider the following initial and boundary value problem (for simplicity, we take α = β = 1): ∂u ∂t − ∂∆u ∂t − div(δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u) + δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u) (2.1) − 1 2 div( |∇u|2 (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) − div( F(u) (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) + 1 2 div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u |∇u|2∇u (ǫ + |∇u| 2 ) 3 2 ) + div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u F(u)∇u (ǫ + |∇u| 2 ) 3 2 ) +ωf′(u) − ∆ω = 0, ω = −∆u + f(u), (2.2) u = ω = 0 on Γ, (2.3) u|t=0 = u0, (2.4) in a bounded and regular domain Ω ⊂ Rn, n = 1, 2 or 3, with boundary Γ. As far as the nonlinear terms δ and f are concerned, we assume that δ is of class C2, (2.5) f is of class C2, f(0) = 0, f′ ≥ −c0, c0 ≥ 0, (2.6) sf′(s)f(s) − f(s)2 ≥ −c1, c1 ≥ 0, s ∈ R, (2.7) sf′′(s) ≥ −c2, c2 ≥ 0, s ∈ R, (2.8) 78 Alain Miranville CUBO 17, 2 (2015) |F(s)| ≤ σf(s)2 + cσ, ∀σ > 0, s ∈ R, (2.9) |F(s)| ≤ c3(|s| p + 1), c3 > 0, p ≥ 0 if n = 1 or 2, p ∈ [0, 7] if n = 3, (2.10) where F is an antiderivative of f. These assumptions are satisfied by polynomials of the form f(s) = ∑q i=1 ais i, q ≥ 3 odd (q ≤ 5 when n = 3), aq > 0, and, in particular, by the usual cubic nonlinear term f(s) = s 3 − s. We denote by ((·, ·)) the usual L2-scalar product, with associated norm ‖ · ‖, and we denote by ‖ · ‖X the norm in the Banach space X. Throughout the paper, the same letter c (and, sometimes, c′) denotes constants which may vary from line to line. Similarly, the same letter Q denotes monotone increasing (with respect to each argument) functions which may vary from line to line. 3 A priori estimates We multiply (2.1) by u and have, integrating over Ω and by parts and owing to (2.2), 1 2 d dt (‖u‖2 + ‖∇u‖2) + ((δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u, ∇u)) + ((δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u), u)) (3.1) + 1 2 ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ), |∇u|2∇u (ǫ + |∇u| 2 ) 1 2 )) + ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ), F(u)∇u (ǫ + |∇u| 2 ) 1 2 )) − 1 2 ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u, |∇u|4 (ǫ + |∇u| 2 ) 3 2 )) − ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u, F(u)|∇u|2 (ǫ + |∇u| 2 ) 3 2 )) +‖ω‖2 + ∫ Ω (uf′(u)f(u) − f(u)2) dx + ((uf′′(u)∇u, ∇u)) = 0. We note that |s| (ǫ + |s|2) 1 2 ≤ 1, s ∈ Rn, so that CUBO 17, 2 (2015) On an anisotropic Allen-Cahn system 79 |δ( ∇u (ǫ + |∇u| 2 ) 1 2 )| ≤ c, |δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )| ≤ c′. Therefore, |((δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u, ∇u))| ≤ c‖∇u‖2, (3.2) |((δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u), u))| ≤ σ‖f(u)‖2 + cσ‖∇u‖ 2, ∀σ > 0, (3.3) |((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ), |∇u|2∇u (ǫ + |∇u| 2 ) 1 2 ))| ≤ c‖∇u‖2 (3.4) ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ), F(u)∇u (ǫ + |∇u| 2 ) 1 2 )) ≤ c ∫ Ω |F(u)| dx (3.5) ≤ (owing to (2.9)) ≤ σ‖f(u)‖2 + cσ, ∀σ > 0, |((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u, |∇u|4 (ǫ + |∇u| 2 ) 3 2 ))| ≤ c‖∇u‖2 (3.6) and, as above, |((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u, F(u)|∇u|2 (ǫ + |∇u| 2 ) 3 2 ))| ≤ c ∫ Ω |F(u)| dx ≤ σ‖f(u)‖2 + cσ, ∀σ > 0. (3.7) It thus follows from (2.7)-(2.8) and (3.1)-(3.7) that d dt (‖u‖2 + ‖∇u‖2) + 2‖ω‖2 ≤ c‖∇u‖2 + σ‖f(u)‖2 + cσ, ∀σ > 0. (3.8) We then note that, owing to (2.6), ‖ω‖2 ≥ ‖∆u‖2 + ‖f(u)‖2 − 2c0‖∇u‖ 2 (3.9) and it follows from (3.8)-(3.9) that, taking σ = 1, 80 Alain Miranville CUBO 17, 2 (2015) d dt (‖u‖2 + ‖∇u‖2) + c(‖u‖2 H2(Ω) + ‖f(u)‖2) ≤ c′(‖u‖2 H1(Ω) + 1), c > 0. (3.10) We then multiply (2.1) by ∂u ∂t and obtain, owing to (2.2), ‖ ∂u ∂t ‖2 + ‖∇ ∂u ∂t ‖2 + ((δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u, ∇ ∂u ∂t )) + ((δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u), ∂u ∂t )) (3.11) + 1 2 ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ), |∇u|2∇∂u ∂t (ǫ + |∇u| 2 ) 1 2 )) + ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ), F(u)∇∂u ∂t (ǫ + |∇u| 2 ) 1 2 )) − 1 2 ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u, |∇u|2∇u · ∇∂u ∂t (ǫ + |∇u| 2 ) 3 2 )) − ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u, F(u)∇u · ∇∂u ∂t (ǫ + |∇u| 2 ) 3 2 )) + 1 2 d dt ‖ω‖2 = 0. We have, proceeding as above, |((δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u, ∇ ∂u ∂t ))| + |((δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u), ∂u ∂t ))| (3.12) + 1 2 |((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ), |∇u|2∇∂u ∂t (ǫ + |∇u| 2 ) 1 2 ))| + 1 2 |((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u, |∇u|2∇u · ∇∂u ∂t (ǫ + |∇u| 2 ) 3 2 ))| ≤ c(‖∇u‖ + ‖f(u)‖)‖∇ ∂u ∂t ‖ ≤ 1 2 ‖∇ ∂u ∂t ‖2 + c(‖∇u‖2 + ‖f(u)‖2). Furthermore, for the most difficult case n = 3 and p = 7 and owing to (2.10) and Agmon’s inequality (see, e.g., [10]), |((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ), F(u)∇∂u ∂t (ǫ + |∇u| 2 ) 1 2 ))| + |((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u, F(u)∇u · ∇∂u ∂t (ǫ + |∇u| 2 ) 3 2 ))| (3.13) ≤ cǫ− 1 2 ∫ Ω |F(u)||∇ ∂u ∂t | dx ≤ 1 2 ‖∇ ∂u ∂t ‖2 + c′ǫ−1 ∫ Ω (|u|14 + 1) dx CUBO 17, 2 (2015) On an anisotropic Allen-Cahn system 81 ≤ 1 2 ‖∇ ∂u ∂t ‖2 + cǫ−1(‖u‖8L∞(Ω) + 1)(‖u‖ 6 L6(Ω) + 1) ≤ 1 2 ‖∇ ∂u ∂t ‖2 + cǫ−1(‖u‖4 H1(Ω) ‖u‖4 H2(Ω) + 1)(‖u‖6 H1(Ω) + 1) ≤ 1 2 ‖∇ ∂u ∂t ‖2 + cǫ−1(‖u‖10 H1(Ω) + 1)(‖u‖4 H2(Ω) + 1). We thus deduce from (3.11)-(3.13) that d dt ‖ω‖2 + ‖ ∂u ∂t ‖2H1(Ω) ≤ cǫ −1(‖u‖10H1(Ω) + 1)(‖u‖ 4 H2(Ω) + 1). (3.14) Recalling that ‖ω‖2 ≥ ‖∆u‖2 + ‖f(u)‖2 − 2c0‖∇u‖ 2, we finally deduce that d dt ‖ω‖2 + ‖ ∂u ∂t ‖2H1(Ω) ≤ cǫ −1(‖u‖10H1(Ω) + 1)(‖u‖ 2 H2(Ω) + 1)(‖ω‖ 2 + ‖u‖2H1(Ω) + 1). (3.15) We now multiply (2.1) by −∆u and have, owing to (2.2), 1 2 d dt (‖∇u‖2 + ‖∆u‖2) + ((divϕ1(∇u), ∆u)) − ((ϕ2(∇u)f(u), ∆u)) (3.16) +((divϕ3(∇u), ∆u)) + ((div(F(u)ϕ4(∇u)), ∆u)) −((divϕ5(∇u), ∆u)) − ((div(F(u)ϕ6(∇u)), ∆u)) −((ωf′(u), ∆u)) + ((∆f(u), ∆u)) + ‖∇∆u‖2 = 0, where ϕ1(s) = δ( s (ǫ + |s|2) 1 2 )s, ϕ2(s) = δ( s (ǫ + |s|2) 1 2 ), ϕ3(s) = 1 2 |s|2 (ǫ + |s|2) 1 2 δ′( s (ǫ + |s|2) 1 2 ), 82 Alain Miranville CUBO 17, 2 (2015) ϕ4(s) = 1 (ǫ + |s|2) 1 2 δ′( s (ǫ + |s|2) 1 2 ), ϕ5(s) = 1 2 δ′( s (ǫ + |s|2) 1 2 ) · s |s|2s (ǫ + |s|2) 3 2 , ϕ6(s) = δ ′( s (ǫ + |s|2) 1 2 ) · s s (ǫ + |s|2) 3 2 . Noting that divϕi(∇u) = ϕ ′ i(∇u) · ∇∇u, i = 1, 3, 5, div(F(u)ϕi(∇u)) = F(u)ϕ ′ i(∇u) · ∇∇u + f(u)ϕi(∇u) · ∇u, i = 4, 6, it follows from the continuous embedding H2(Ω) ⊂ C(Ω) and (2.2) that |((divϕ1(∇u), ∆u))| + |((ϕ2(∇u)f(u), ∆u))| +|((divϕ3(∇u), ∆u))| + |((div(F(u)ϕ4(∇u)), ∆u))| +|((divϕ5(∇u), ∆u))| + |((div(F(u)ϕ6(∇u)), ∆u))| +|((ωf′(u), ∆u)) + ((∆f(u), ∆u))| ≤ Q(ǫ−1, ‖u‖H2(Ω)) (here, we have used the facts that the ϕ′i’s are bounded and that 1 ǫ+|s|2 ≤ ǫ−1 and |s| 2 ǫ+|s|2 ≤ 1), hence, d dt (‖∇u‖2 + ‖∆u‖2) + c‖u‖2H3(Ω) ≤ Q(ǫ −1, ‖u‖H2(Ω)), c > 0. (3.17) We finally multiply (2.1) by −∆∂u ∂t and find, owing to (2.2), ‖∇ ∂u ∂t ‖2 + ‖∆ ∂u ∂t ‖2 + ((divϕ1(∇u), ∆ ∂u ∂t )) − ((ϕ2(∇u)f(u), ∆ ∂u ∂t )) +((divϕ3(∇u), ∆ ∂u ∂t )) + ((div(F(u)ϕ4(∇u)), ∆ ∂u ∂t )) −((divϕ5(∇u), ∆ ∂u ∂t )) − ((div(F(u)ϕ6(∇u)), ∆ ∂u ∂t )) CUBO 17, 2 (2015) On an anisotropic Allen-Cahn system 83 −((ωf′(u), ∆ ∂u ∂t )) + ((∆f(u), ∆ ∂u ∂t )) + 1 2 d dt ‖∇∆u‖2 = 0, which yields, proceeding as above, d dt ‖∇∆u‖2 + c‖ ∂u ∂t ‖2H2(Ω) ≤ Q(ǫ −1, ‖u‖H2(Ω)), c > 0. (3.18) 4 Existence and uniqueness of solutions We have the Theorem 1. We assume that u0 ∈ H 2(Ω) ∩ H10(Ω). Then, (2.1)-(2.4) possesses a unique solution u such that u ∈ L∞(0, T ; H2(Ω) ∩ H10(Ω)) ∩ L 2(0, T ; H3(Ω)) and ∂u ∂t ∈ L2(0, T ; H10(Ω)), ∀T > 0. Furthermore, if u0 ∈ H 3(Ω)∩H10(Ω), then u ∈ L ∞(0, T ; H3(Ω)∩H10(Ω)) and ∂u ∂t ∈ L2(0, T ; H2(Ω)∩ H10(Ω)), ∀T > 0. Proof. a) Existence: The proof of existence is based on the a priori estimates derived in the previous section and, e.g., a classical Galerkin scheme. In particular, we first deduce from (3.10) that u ∈ L∞(0, T ; H10(Ω)) ∩ L 2(0, T ; H2(Ω)), ∀T > 0. Having this, it follows from (3.15) that u ∈ L∞(0, T ; H2(Ω)) and ∂u ∂t ∈ L2(0, T ; H10(Ω)). The only difficulty here is to pass to the limit in the nonlinear terms when considering Galerkin approximations. More precisely, we have, owing to classical Aubin-Lions compactness results, a sequence um of solutions to approximated problems such that um → u in L ∞ (0, T ; H2(Ω)) weak star, L2(0, T ; H1(Ω)) and a.e.. We consider, for instance, the passage to the limit in the term F(um)ϕ4(∇um) (the other terms can be handled similarly or are simpler to treat). We have |F(um)ϕ4(∇um) − F(u)ϕ4(∇u)| ≤ |F(um)(ϕ4(∇um) − ϕ4(∇u))| + |(F(um) − F(u))ϕ4(∇u)|, so that, proceeding as in the previous section (using, in particular, the fact that ϕ′4 is bounded), ‖F(um)ϕ4(∇um) − F(u)ϕ4(∇u)‖ ≤ Q(ǫ −1, ‖um‖H2(Ω), ‖u‖H2(Ω))‖um − u‖H1(Ω), 84 Alain Miranville CUBO 17, 2 (2015) hence the convergence in L2(0, T ; L2(Ω)). Furthermore, noting that ωm (defined as in (2.2)) converges to ω in L∞(0, T ; L2(Ω)) weak star and f′(um) converges to f ′(u) in L2(0, T ; H1(Ω)), we easily see that ωmf ′(um) converges to ωf ′(u) in L 3 2 (0, T ; L 3 2 (Ω)) weak. b) Uniqueness: Let u1 and u2 be two solutions to (2.1)-(2.3) (ω1 and ω2 being defined as in (2.2)) with initial data u0,1 and u0,2, respectively. We set u = u1 − u2, ω = ω1 − ω2, u0 = u0,1 − u0,2 and have ∂u ∂t − ∂∆u ∂t − div(ϕ1(∇u1) − ϕ1(∇u2)) + ϕ2(∇u1)f(u1) − ϕ2(∇u2)f(u2) (4.1) −div(ϕ3(∇u1) − ϕ3(∇u2)) − div(F(u1)ϕ4(∇u1) − F(u2)ϕ4(∇u2)) +div(ϕ5(∇u1) − ϕ5(∇u2)) + div(F(u1)ϕ6(∇u1) − F(u2)ϕ6(∇u2)) +ω1f ′(u1) − ω2f ′(u2) − ∆ω = 0, ω = −∆u + f(u1) − f(u2), (4.2) u = ω = 0 on Γ, (4.3) u|t=0 = u0. (4.4) We multiply (4.1) by u and obtain, owing to (4.2), 1 2 d dt (‖u‖2 + ‖∇u‖2) + ((ϕ1(∇u1) − ϕ1(∇u2), ∇u)) (4.5) +((ϕ2(∇u1)f(u1) − ϕ2(∇u2)f(u2), u)) + ((ϕ3(∇u1) − ϕ3(∇u2), ∇u)) +((F(u1)ϕ4(∇u1) − F(u2)ϕ4(∇u2)), ∇u)) − ((ϕ5(∇u1) − ϕ5(∇u2), ∇u)) −((F(u1)ϕ6(∇u1) − F(u2)ϕ6(∇u2)), ∇u)) + ((ω1f ′ (u1) − ω2f ′ (u2), u)) CUBO 17, 2 (2015) On an anisotropic Allen-Cahn system 85 +((f(u1) − f(u2), u)) + ‖∆u‖ 2 = 0. We have, for instance (again, the other terms can be handled similarly or are easier to treat) and proceeding as in the previous section, |((F(u1)ϕ4(∇u1) − F(u2)ϕ4(∇u2)), ∇u))| ≤ |((F(u1)(ϕ4(∇u1) − ϕ4(∇u2)), ∇u))| +|(((F(u1) − F(u2))ϕ4(∇u2), ∇u))| ≤ Q(ǫ −1, T, ‖u0,1‖H2(Ω), ‖u0,2‖H2(Ω))‖∇u‖ 2. Furthermore, owing to (4.2) and a proper interpolation inequality, |((ω1f ′(u1) − ω2f ′(u2), u))| ≤ |((ωf ′(u1), u))| + |((ω2(f ′u1) − f ′(u2)), u))| ≤ Q(T, ‖u0,1‖H2(Ω), ‖u0,2‖H2(Ω))‖∆u‖‖u‖. We thus find an inequality of the form d dt (‖u‖2 + ‖∇u‖2) ≤ Q(ǫ−1, T, ‖u0,1‖H2(Ω), ‖u0,2‖H2(Ω))‖∇u‖ 2, hence the uniqueness, as well as the continuous dependence with respect to the initial data in the H1-norm. Remark 4.1. The viscous Cahn-Hilliard system associated with the free energy (1.6) reads ∂u ∂t − α ∂∆u ∂t − ∆[−div(δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u) + δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u) (4.6) − 1 2 div( |∇u|2 (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) − div( F(u) (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) + 1 2 div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u |∇u|2∇u (ǫ + |∇u| 2 ) 3 2 ) + div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u F(u)∇u (ǫ + |∇u| 2 ) 3 2 ) +βωf′(u) − β∆ω] = 0, α > 0, 86 Alain Miranville CUBO 17, 2 (2015) ω = −∆u + f(u). (4.7) Taking, for simplicity, Dirichlet boundary conditions, u = ∆u = ω = ∆ω = 0 on Γ, we can rewrite (4.6) equivalently as (−∆)−1 ∂u ∂t + α ∂u ∂t − div(δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u) + δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u) (4.8) − 1 2 div( |∇u|2 (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) − div( F(u) (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) + 1 2 div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u |∇u|2∇u (ǫ + |∇u| 2 ) 3 2 ) + div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u F(u)∇u (ǫ + |∇u| 2 ) 3 2 ) +βωf′(u) − β∆ω = 0. Even though (4.8) bears some resemblance with (2.1), we have less regularity on ∂u ∂t and thus cannot proceed as above to prove the existence and uniqueness of solutions. We can however prove the well-posedness in one space dimension (see [5]). Received: May 2013. Accepted: May 2013. References [1] S.M. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. 27 (1979), 1085–1095. [2] F. Chen and J. Shen, Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn-Hilliard systems, Commun. Comput. Phys., to appear. [3] M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D 92 (1996), 178–192. [4] A. Miranville, Asymptotic behavior of a sixth-order Cahn-Hilliard system, submitted. [5] A. Miranville, Existence of solutions for a one-dimensional Allen-Cahn equation, submitted. CUBO 17, 2 (2015) On an anisotropic Allen-Cahn system 87 [6] A. Miranville and R. Quintanilla, A generalization of the Allen-Cahn equation, submitted. [7] A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in Material instabilities in continuum and related problems, J.M. Ball ed., Oxford University Press, Oxford, 329–342, 1988. [8] K. Promislow and H. Zhang, Critical points of functionalized Lagrangians, Discrete Cont. Dyn. System 33 (2013), 1231–1246. [9] J.E. Taylor and J.W. Cahn, Diffuse interfaces with sharp corners and facets: phase-field models with strongly anisotropic surfaces, Phys. D 112 (1998), 381–411. [10] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Second edition, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997. [11] S. Torabi, J. Lowengrub, A. Voigt and S. Wise, A new phase-field model for strongly anisotropic systems, Proc. R. Soc. A 465 (2009), 1337–1359. [12] S.M. Wise, C. Wang and J.S. Lowengrub, Solving the regularized, strongly anisotropic Cahn- Hilliard equation by an adaptative nonlinear multigrid method, J. Comput. Phys. 226 (2007), 414–446. Introduction Setting of the problem A priori estimates Existence and uniqueness of solutions