Phase-field systems with nonlinear coupling and dynamic boundary - - PowerPoint PPT Presentation

Phase-field systems with nonlinear coupling and dynamic boundary conditions

Cecilia Cavaterra

Dipartimento di Matematica “F . Enriques” Università degli Studi di Milano

cecilia.cavaterra@unimi.it

VIII Workshop

• n Partial Differential Equations

Rio de Janeiro, August 25-28, 2009 Joint work with C.G.Gal, M.Grasselli, A.Miranville

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Phase-field system

A well-known system of partial differential equations which describes the behavior of a two-phase material in presence of temperature variations, and neglecting mechanical stresses, is the so-called Caginalp phase-field system δψt − D∆ψ + f(ψ) − λ′(ψ)θ = 0, Ω × (0, ∞) (εθ + λ(ψ))t − k∆θ = 0, Ω × (0, ∞) Ω ⊂ R3, bounded domain with smooth boundary Γ ψ order parameter (or phase-field) θ (relative) temperature δ, D, ε, k positive coefficients λ function related to the latent heat f = F ′, F nonconvex potential (e.g., double well potential)

2 / 46

A naive derivation of the system

bulk free energy functional EΩ(ψ, θ) =

• Ω

D 2 |∇ψ|2 + F(ψ) − λ(ψ)θ − ε 2θ2

• dx

∂θEΩ(ψ, θ), ∂ψEΩ(ψ, θ): Frechét derivatives of EΩ(ψ, θ) equation for the temperature (−∂θEΩ(ψ, θ))t + ∇ · q = 0, q = −k∇θ equation for the order parameter ψt = −∂ψEΩ(ψ, θ)

3 / 46

Motivation for studying the case λ nonlinear

Assuming λ linear is satisfactory for solid-liquid phase transitions However, when one deals, for instance, with phase transitions in ferromagnetic materials, where ψ represents the fraction of lattice sites at which the spins are pointing “up”, then a quadratic λ is a more appropriate choice (see M.Brokate & J.Sprekels (1996))

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Standard boundary conditions for ψ and θ

ψ: Neumann boundary condition ∂nψ = 0 n: outward normal to Γ θ: Dirichlet, Neumann, Robin boundary conditions b∂nθ + cθ = 0 (D): b = 0, c > 0 (N): b > 0, c = 0 (R): b > 0, c > 0

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Dynamic boundary condition for ψ

In order to account for possible interaction with the boundary Γ, we can consider dynamic boundary conditions for ψ surface free energy functional EΓ(ψ) =

• Γ

α 2 |∇Γψ|2 + β 2ψ2 + G(ψ)

• dS

∇Γ tangential gradient operator α, β > 0, G nonconvex boundary potential ∂ψEΓ(ψ): Frechét derivative of EΓ(ψ) ψt = −∂ψEΓ(ψ, θ) − ∂nψ ψt − α∆Γψ + ∂nψ + βψ + g(ψ) = 0 ∆Γ Laplace-Beltrami operator , g = G′

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(IBVP) for the phase-field system

Without loss of generality we take D = k = δ = ε = 1 evolution equations in Ω × (0, ∞)    ψt − ∆ψ + f(ψ) − λ′(ψ)θ = 0 (θ + λ(ψ))t − ∆θ = 0 boundary conditions on Γ × (0, ∞)    ψt − α∆Γψ + ∂nψ + βψ + g(ψ) = 0 b∂nθ + cθ = 0 initial conditions on Ω θ(0) = θ0, ψ(0) = ψ0

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Main goals

Well posedness Existence of the global attractor Existence of an exponential attractor Convergence of solutions to single equilibria

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Definitions of global attractor and exponential attractor

Dynamical system: (X, S(t)) X: complete metric sp. S(t) : X → X semigroup of op. Hausdorff semidistance: distX (W, Z) = sup

w∈W

inf

z∈Z d(w, z)

The global attractor A ⊂ X is a compact set in X : A is fully invariant (S(t)A = A, ∀ t ≥ 0) A is an attracting set w.r.t. H-semidistance: ∀ bdd B ⇒ lim

t→∞ distX (S(t)B, A) = 0

An exponential attractor E ⊂ X is a compact set in X : E is invariant (S(t)E ⊆ E, ∀ t ≥ 0) E has finite fractal dimension E is an exponential attracting set w.r.t. H-semidistance: ∀ bdd B ⇒ ∃ CB > 0, ω > 0 s.t. distX (S(t)B, E) ≤ CBe−ωt, ∀ t ≥ 0

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Literature: λ linear + standard b.c. + smooth potentials

C.M.Elliott & S.Zheng (1990); A.Damlamian, N.Kenmochi & N. Sato (1994); G.Schimperna (2000) well-posedness V.K.Kalantarov (1991); P .W.Bates & S.Zheng (1992); D.Brochet, X.Chen & D.Hilhorst (1993); O.V.Kapustyan (2000); A.Jiménez-Casas & A.Rodríguez-Bernal (2002) longtime behavior of solutions existence and smoothness of global attractors existence of exponential attractors Z.Zhang (2005) asymptotic behavior of single solutions

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Literature: λ nonlinear + standard b.c. + smooth potentials

Ph.Laurençot (1996); M.Grasselli & V.Pata (2004) well-posedness, global and exponential attractors M.Grasselli, H.Wu & S.Zheng (2008) global and exponential attractors, asymptotic behavior of single solutions (nonhomogenous b.c.)

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Literature: λ nonlinear + standard b.c. + singular potentials

M.Grasselli, H.Petzeltová & G.Schimperna (2006) well-posedness and asymptotic behavior

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Literature: λ linear + dynamic b.c. + smooth potentials

R.Chill, E.Fašangová & J.Prüss (2006) F with polynomial controlled growth of degree 6, G ≡ 0 well-posedness convergence to single equilibria via Łojasiewicz-Simon inequality (when F is also real analytic) S.Gatti & A.Miranville (2006) construction of a s-continuous dissipative semigroup ∃ global attractor Aε upper semicontinuous at ε = 0 ∃ exponential attractors Eε

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Literature: λ linear + dynamic b.c. + smooth potentials

C.G.Gal, M.Grasselli & A.Miranville (2007) ∃ family of exponential attractors {Eε} stable as ε ց 0 when ∂nθ = 0 C.G.Gal & M.Grasselli (2008, 2009) F and G smooth potentials (more general than S.Gatti & A.Miranville) (possibly) dynamic boundary condition for θ (a, b, c ≥ 0) aθt + b∂nθ + cθ = 0 construction of a dissipative semigroup (larger phase spaces w.r.t. S.Gatti & A.Miranville) ∃ global attractor, ∃ exponential attractors

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Literature: λ linear + dynamic b.c. + singular potential

L.Cherfils & A.Miranville (2007) F singular potential defined on (−1, 1) G smooth potential (sign restrictions) construction of a s-continuous dissipative semigroup ∃ global attractor of finite fractal dimension convergence to single equilibria via Ł-S method (when F is real analytic and G ≡ 0) S.Gatti, L.Cherfils & A.Miranville (2007, 2008) F (strongly) singular potential defined on (−1, 1) G smooth potential (sign restrictions are removed) separation property and existence of global solutions existence of global and exponential attractors

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Notations

·p norm on Lp (Ω) ·p,Γ norm on Lp (Γ) ·, ·2 usual scalar product inducing the norm on L2(Ω) (even for vector-valued functions) ·, ·2,Γ usual scalar product inducing the norm on L2(Γ) (even for vector-valued functions) ·Hs(Ω) norm on Hs (Ω), for any s > 0 ·Hs(Γ) norm on Hs (Γ), for any s > 0

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The operators AK

In order to account different cases of boundary conditions, we introduce the linear operators AK = −∆ : D(AK) → L2 (Ω) D(AK) = H1

0(Ω) ∩ H2(Ω),

if K = D D(AK) = {θ ∈ H2(Ω) : b∂nθ + cθ = 0}, if K = N, R D, N, R stand for Dirichlet, Neumann, or Robin bdry conds AK generates an analytic semigroup e−AK t on L2(Ω) AK is nonnegative and self-adjoint on L2(Ω)

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The functional spaces Z1

K

Z1

D = H1 0(Ω)

Z1

K = H1(Ω), if K ∈ {N, R}

θ2

Z1

K =

             ∇θ2

2 ,

if K = D, ∇θ2

2 + c b

• θ|Γ
• 2

2,Γ ,

if K = R, ∇θ2

2 + θ2 Ω ,

if K = N, where we have set vΩ := |Ω|−1

• Ω

v (x) dx the norm in Z1

K is equivalent to the standard H1-norm

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The function spaces Vs

Vs = Cs Ω ·Vs , s > 0 ψVs =

• ψ2

Hs(Ω) +

• ψ|Γ
• 2

Hs(Γ)

1/2 Vs = Hs (Ω) ⊕ Hs (Γ) V0 = L2 (Ω) ⊕ L2 (Γ) Vs is compactly embedded in Vs−1, ∀ s ≥ 1 H1 (Ω) ֒ → L6 (Ω) H1/2 (Γ) ֒ → L4 (Γ) H1 (Γ) ֒ → Ls (Γ) , ∀ s ≥ 1

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Enthalpy conservation

In the case K = N we define the enthalpy IN (ψ(t), θ(t)) := λ (ψ(t)) + θ(t)Ω and this quantity is conserved in time for any given solution

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The weak formulation

Problem Pw

K

∀ (ψ0, θ0) ∈ V1 × L2 (Ω) find (ψ, θ) ∈ C

• [0, +∞); V1 × L2 (Ω)
• :

ψt ∈ L2 ([0, +∞); V0) , ∇θ ∈ L2 [0, +∞); (L2(Ω))3 ψt, u2 + ∇ψ, ∇u2 + f (ψ) − λ′ (ψ) θ, u2 + ψt, u2,Γ +α∇Γψ, ∇Γu2,Γ + βψ + g (ψ) , u2,Γ = 0, ∀ u ∈ V1, a.e. in (0, ∞) (θ + λ (ψ))t, v2 + ∇θ, ∇v2 + dθ, v2,Γ = 0, ∀ v ∈ Z1

K, a.e. in (0, ∞)

θ(0) = θ0, ψ(0) = ψ0 and, if K = N, IN (ψ(t), θ(t)) = IN (ψ0, θ0) , ∀ t ≥ 0 Here d = c

b if K = R, d = 0 otherwise

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The strong formulation

Problem Ps

K

∀ (ψ0, θ0) ∈ V2 × Z1

K

find (ψ, θ) ∈ C

• [0, +∞); V2 × Z1

K

• :

(ψt, θt) ∈ L2 [0, +∞); V1 × L2 (Ω)

• AKθ ∈ L2 ([0, +∞) × Ω)

ψt − ∆ψ + f(χ) − λ′(ψ)θ = 0, a.e. in Ω × (0, +∞) (θ + λ(ψ))t − ∆θ = 0, a.e. in Ω × (0, +∞) ψt − α∆Γψ + ∂nψ + βψ + g(ψ) = 0, a.e. in Γ × (0, +∞) b∂nθ + cθ = 0, a.e. in Γ × (0, +∞) θ(0) = θ0, ψ(0) = ψ0 and, if K = N, IN (ψ(t), θ(t)) = IN (ψ0, θ0) , ∀ t ≥ 0

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Assumptions on the nonlinearities f, g and λ

(H1) f, g ∈ C1(R) satisfy lim

|y|→+∞ inf f ′ (y) > 0,

lim

|y|→+∞ inf g′ (y) > 0

(H2) ∃ cf, cg > 0, q ∈ [1, +∞) :

• f ′ (y)
• ≤ cf
• 1 + |y|2

,

• g′ (y)
• ≤ cg
• 1 + |y|q

, ∀y ∈ R (H3) λ ∈ C2 (R) :

• λ′′ (y)
• ≤ cλ,

∀y ∈ R, where cλ > 0 (H4) ∃ a > 0 and γ ∈ C2 (R) ∩ W 1,∞ (R) : λ (y) = γ (y) − ay2, ∀y ∈ R (H5) ∃ η1 > 0 and η2 ≥ 0 : f (y) y ≥ η1 |y|4 − η2, ∀y ∈ R

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Remarks

Remark The growth restriction (H2) is only needed to analyze Pw

K .

Remark Assumption (H4) is only needed to handle the case K = N. Note that (H4) is justified from a physical viewpoint (see, e.g., M.Brokate & J.Sprekels (1996)).

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The global existence result for Pw

K

Theorem Let f, g, λ satisfy assumptions (H1), (H2) and (H3). Then, for each K ∈ {D, N, R} , problem Pw

K admits a global

weak solution.

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Sketch of the proof

Proof. suitable Faedo-Galerkin approximation scheme based on the characterization of a suitable self-adjoint positive

• perator B on V0 satisfying Wentzell b.c. (cf. C.G.Gal,

G.Goldstein, J.Goldstein, S.Romanelli & M.Warma (2009)) system of nonlinear ODEs for the approximating solutions a priori estimates for the approximating solutions global existence (in time) + strong convergence passage to the limit in the nonlinear terms convergence of the approximating solutions to the solution

• f Pw

K

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The global existence result for Ps

K

Theorem Let f, g, λ satisfy (H1) and (H3). Then, for each K ∈ {D, N, R} , problem Ps

K admits a global

strong solution. Proof. global existence result for Pw

K

higher order estimates

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Pw

K: continuous dependence estimate

Lemma Let f, g, λ satisfy assumptions (H1), (H2) and (H3). Let (ψwi, θwi) be two global solutions to Pw

K corresponding to the

initial data (ψ0i, θ0i) ∈ V1 × L2 (Ω) , i = 1, 2. Then, for any t ≥ 0, the following estimate holds: (ψw1 − ψw2) (t)2

V1 + (θw1 − θw2) (t)2 2

+

t

• (ψw1 − ψw2)t (s)2

V0 + (θw1 − θw2) (s)2 Z1

K

• ds

≤ CweLwt ψ01 − ψ022

V1 + θ01 − θ022 2

• Here Cw and Lw are positive constants depending on the

norms of the initial data in V1 × L2 (Ω), on Ω and on the parameters of the problem, but are both independent of time.

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Ps

K: continuous dependence estimate

Lemma Let f, g, λ satisfy (H1) and (H3). Let (ψsi, θsi) two global solutions to Ps

K corresponding to the

initial data (ψ0i, θ0i) ∈ V2 × Z1

K, i = 1, 2.

Then, for any t ≥ 0, the following estimate holds: (ψs1 − ψs2) (t)2

V1 + (θs1 − θs2) (t)2 2

+

t

• (ψs1 − ψs2)t (s)2

V0 + (θs1 − θs2) (s)2 Z1

K

• ds

≤ CseLst ψ01 − ψ022

V1 + θ01 − θ022 2

• Here Cs and Ls are positive constants depending on the norms
• f the initial data in V2 × Z1

K, on Ω and on the parameters of the

problem, but are both independent of time.

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The dynamical system generated by Pw

K

Corollary Let f, g, λ satisfy assumptions (H1), (H2) and (H3). Then, for each K ∈ {D, N, R} , we can define a strongly continuous semigroup Sw

K (t) : V1 × L2 (Ω) → V1 × L2 (Ω)

by setting, for all t ≥ 0, Sw

K (t) (ψ0, θ0) = (ψw (t) , θw (t))

where (ψw, θw) is the unique solution to Pw

K.

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The dynamical system generated by Ps

K

Corollary Let f, g, λ satisfy (H1) and (H3). Then, for each K ∈ {D, N, R} , we can define a semigroup Ss

K (t) : V2 × Z1 K → V2 × Z1 K

by setting, for all t ≥ 0, Ss

K (t) (ψ0, θ0) = (ψs (t) , θs (t))

where (ψs, θs) is the unique solution to Ps

K.

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The phase spaces for Sw

K and Ss K

Due to the enthalpy conservation, in the case K = N we define the complete metric spaces w.r.t. the metrics induced by the norms where the constraint M ≥ 0 is fixed

• V1 × L2(Ω)

M :=

• (u, v) ∈ V1 × L2(Ω) : |IN (u, v)| ≤ M
• V2 × Z1

N

M :=

• (u, v) ∈ V2 × Z1

N : |IN (u, v)| ≤ M

• Phase-space for Sw

K (t)

Y0,K =

• V1 × L2(Ω),

if K ∈ {D, R}

• V1 × L2(Ω)

M , if K = N Phase-space for Ss

K(t)

Y1,K =

• V2 × Z1

K,

if K ∈ {D, R}

• V2 × Z1

N

M , if K = N

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A dissipative estimate for Sw

K and Ss K

Lemma Let f, g satisfy assumptions (H1) and (H5). Let λ satisfies either (H3), if K ∈ {D, R} , or (H4), if K = N. Then, ∀ (ψ0, θ0) ∈ Y0,K, the following estimate holds (ψ (t) , θ (t))2

Y0,K

+ t+1

t

• ψt (s)2

V0 + θ (s)2 Z1

K + ψ (s)4

L4(Ω)

• ds

≤ CK

• (ψ0, θ0)2

Y0,K + F (ψ0) , 12 + G (ψ0) , 12,Γ + 1

• e−ρt

+ C∗

K,

∀ t ≥ 0 where F (y) = y

0 f (r) dr,

G (y) = y

0 g (r) dr,

∀y ∈ R. Here ρ, CK, C∗

K are independent of t and of the initial data.

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Existence of a compact absorbing set

Lemma Let f, g, λ satisfy (H1), (H2), (H3) and (H5). If K = N, assume also λ fulfilling (H4). Then, ∀ R0 > 0, ∃ a positive nondecreasing monotone function Q and t0 = t0(R0) > 0 : Sw

K (t) (ψ0, θ0)V2×H2(Ω) ≤ Q(R0),

∀ t ≥ t0 ∀ (ψ0, θ0) ∈ B(R0) ⊂ Y0,K, where B(R0) is a ball of radius R0. Lemma Let f, g, λ satisfy (H1), (H3) and (H5). If K = N, assume also λ fulfilling (H4). Then, ∀ R1 > 0, ∃ a positive nondecreasing monotone function Q and t1 = t1(R1) > 0 : Ss

K (t) (ψ0, θ0)V3×H3(Ω) ≤ Q(R1),

∀ t ≥ t1 ∀ (ψ0, θ0) ∈ B(R1) ⊂ Y1,K, where B(R1) is a ball of radius R1.

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Existence of the global attractor

Theorem Let f, g, λ satisfy assumptions (H1), (H2), (H3) and (H5). If K = N, assume also λ fulfilling (H4). Then, Sw

K (t) possesses the connected global attractor

Aw

K ⊂ Y0,K, which is bounded in V2 × H2 (Ω).

Theorem Let f, g, λ satisfy assumptions (H1), (H3) and (H5). If K = N, assume also λ fulfilling (H4). Then, Ss

K (t) possesses the connected global attractor

As

K⊂ Y1,K, which is bounded in V3 × H3 (Ω).

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Existence of exponential attractors for Sw

K (t)

Theorem Let f, g ∈ C2(R) and λ ∈ C3(R) satisfy (H1), (H2), (H3) and (H5). If K = N, assume also λ fulfilling (H4). Then, Sw

K (t) has

an exponential attractor Mw

K , bounded in V2 × H2 (Ω) , namely,

(I) Mw

K is compact and positively invariant w.r.t Sw K (t) , i.e.,

Sw

K (t) (Mw K ) ⊂ Mw K ,

∀ t ≥ 0. (II) The fractal dimension of Mw

K w.r.t. Y0,K-metric is finite.

(III) There exist a positive nondecreasing monotone function Qw and a constant ρw > 0 such that distY0,K (Sw

K (t) B, Mw K ) ≤ Qw(BY0,K )e−ρwt,

∀ t ≥ 0, where B is any bounded set of initial data in Y0,K. Here distY0,K denotes the non-symmetric Hausdorff distance in Y0,K and BY0,K stands for the size of B in Y0,K.

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Existence of exponential attractors for Ss

K(t)

Theorem Let f, g ∈ C2(R) and λ ∈ C3(R) satisfy (H1), (H3) and (H5). If K = N, assume also λ fulfilling (H4). Then, Ss

K(t) has an

exponential attractor Ms

K, bounded in V3 × H3 (Ω) , namely,

(I) Ms

K is compact and positively invariant w.r.t. Ss K (t) , i.e.,

Ss

K (t) (Ms K) ⊂ Ms K,

∀ t ≥ 0. (II) The fractal dimension of Ms

K w.r.t. Y1,K-metric is finite.

(III) There exist a positive nondecreasing monotone function Qs and a constant ρs > 0 such that distY1,K (Ss

K (t) B, Ms K) ≤ Qs(BY1,K )e−ρst,

∀ t ≥ 0, where B is any bounded set of initial data in Y1,K. Here distY1,K denotes the non-symmetric Hausdorff distance in Y1,K and BY1,K stands for the size of B in Y1,K.

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Finite fractal dimension of the global attractor

Remark Thanks to the existence of exponential attractors for Sw

K (t) and

Ss

K(t) we deduce that the global attractors Aw K and As K have

finite fractal dimension.

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Lyapunov functional (strong solutions)

Proposition Let f, g, λ satisfy (H1) and (H3). Then Ss

K(t) has a (strict)

Lyapunov functional defined by the free energy, namely, LK(ψ0, θ0) := 1 2

• ∇ψ02

2 + α ∇Γψ02 2,Γ + β ψ02 2,Γ + θ02 2

• +
• Ω

F (ψ0(x)) dx +

• Γ

G (ψ0(S)) dS, K ∈ {D, N, R} where F (y) = y

0 f (r) dr,

G (y) = y

0 g (r) dr,

∀y ∈ R. In particular, for all t > 0, we have d dt LK(Ss

K(t)(ψ0, θ0))

= − ψt (t)2

2 − ψt (t)2 2,Γ − ∇θ (t)2 2 − d

• θ|Γ (t)
• 2

2,Γ

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Equilibrium points (strong solutions)

(ψ∞, θ∞) ∈ Y1,K is an equilibrium for Ps

K if and only if it is a

solution to the boundary value problem − ∆ψ∞ + f (ψ∞) − λ′(ψ∞)θ∞ = 0, in Ω − α∆Γψ∞ + ∂nψ∞ + βψ∞ + g (ψ∞) = 0,

• n Γ

− ∆θ∞ = 0, in Ω, b∂nθ∞ + cθ∞ = 0,

• n Γ

if K ∈ {D, R}, then θ∞ ≡ 0 if K = N, then θ∞ = IN (ψ0, θ0) − λ(ψ∞)Ω If K = N, then ψ∞ is solution of a nonlocal boundary value problem (⇒ a special version of the Łojasiewicz-Simon inequality is needed)

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K = N: convergence to equilibrium (strong solutions)

Theorem Let f, g, λ satisfy assumptions (H1), (H3), (H4) and (H5). Suppose, in addition, that the nonlinearities F, G, λ are analytic. Then, ∀ (ψ0, θ0) ∈ Y1,N, the solution (ψ (t) , θ (t)) = Ss

N(t)(ψ0, θ0) converges to a single equilibrium

(ψ∞, θ∞) in the topology of V2 × Z1

N, that is,

lim

t→+∞

• ψ(t) − ψ∞V2

2 + θ(t) − θ∞Z1 N

• = 0

Moreover, ∃ C > 0 and ξ ∈ (0, 1/2) depending on (ψ∞, θ∞) : ψ(t) − ψ∞V2

2 + θ(t) − θ∞Z1 N + ψt (t) V0 2 ≤ C(1 + t)−ξ/(1−2ξ)

for all t ≥ 0.

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Remarks

Remark As

K coincides with the unstable manifold of the set of equilibria.

Remark Since we have gradient systems with a set of equilibria bounded in the phase-space, we could avoid to prove the existence of a bounded absorbing set and we could directly show the existence of the global attractor. However, the existence of a uniform dissipative estimate can be easily adapted to some nonautonomous systems.

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Further issue: coupled dynamic boundary conditions

surface free energy functional EΓ(ψ, θ) =

• Γ

α 2 |∇Γψ|2 + β 2ψ2 + G(ψ) − ℓ(ψ)θ − a 2θ2

• dS

∂ψEΓ(ψ, θ), ∂θEΓ(ψ, θ): Frechét derivatives of EΓ(ψ, θ) ψt = −∂ψEΓ(ψ, θ) − ∂nψ −(∂θEΓ(ψ, θ))t = −b∂nθ − cθ    ψt − α∆Γψ + ∂nψ + βψ + g(ψ) − ℓ′(ψ)θ = 0 (aθ + ℓ(ψ))t + b∂nθ + cθ = 0 a > 0 , g = G′ ℓ ∈ C2 (R) :

• ℓ′′ (y)
• ≤ cℓ,

∀y ∈ R, where cℓ > 0

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Open issue: singular potentials

bdry coupling with singular F (and, possibly, G) of the form F(s) = γ1[(1 + s) ln(1 + s) + (1 − s) ln(1 − s)] − γ2s2

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Open issue: memory effects

                               ψt + ∞ h1(s)(−∆ψ + f(ψ) − λ′(ψ)θ)(t − s)ds = 0 (θ + λ(ψ))t − ∞ h2(s)∆θ(t − s)ds = 0 ψt + ∞ h3(s)(−∆Γψ + ψ + g(ψ) + ∂nψ)(t − s)ds = 0 b∂nθ + cθ = 0 θ(s) = ˜ θ0(−s), ψ(s) = ˜ ψ0(−s) in Ω, s ≥ 0 h1, h2, h3 ≥ 0 smooth exp. decreasing relaxation kernels

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Thanks for your attention!

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