Approximation of dynamic boundary condition: The Allen Cahn equation

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1 1 Approximation of dynamic boundary condition: The Allen Cahn equation Matthias Liero I.M.A.T.I. Pavia and Humboldt-Universität zu Berlin 6 th Singular Days 2010 Berlin

2 Introduction Interfacial dynamics in semiconductor models Schematic cross-section of thin film a-si:h photovoltaic cell Interaction with domain walls in spinodal decomposition Cahn Hilliard equation 2

3 3 AC Equation with Dynamic Boundary Condition Sprekels and Wu [2010]: Ω R d, d 2, with C 2 boundary t u u + F b (u) = 0, in Ω [0, T ] t φ φ + ν u + φ + F s(φ) = 0, on Ω [0, T ] u = φ, on Ω [0, T ] u(0, ) = u 0, φ(0, ) = φ 0 Laplace Beltrami operator on Ω Cahn Hilliard equation (R. Kenzler et al. [2001]) Caginalp system (R. Chill, E. Fasangova and J. Pruess [2006]) Nonisothermal Allen Cahn equation (C. Gal and M. Grasselli, A. Miranville [2008])

4 Layer Approximation Ω r b t u a b u + F b (u) = 0, in Ω [0, T ]

5 4 Layer Approximation Ω ε def = {x R d : (y, η) Ω (0, ε), x = y + ην(y)} Ω ε Ω ε r b t u a b u + F b (u) = 0, in Ω [0, T ]

6 4 Layer Approximation Ω ε def = {x R d : (y, η) Ω (0, ε), x = y + ην(y)} Ω ε Ω ε r b t u a b u + F b (u) = 0, in Ω [0, T ] r s ε δ t φ a s ε α φ + ε β F s(φ) = 0, in Ω ε [0, T ] u = φ on Ω [0, T ] a b ν u a s ε α ν φ = 0, on Ω [0, T ] a s ε α ν φ = 0, on Ω ε \ Ω [0, T ]

7 Layer Approximation Ω ε def = {x R d : (y, η) Ω (0, ε), x = y + ην(y)} Ω ε Ω ε r b t u a b u + F b (u) = 0, in Ω [0, T ] r s ε 1 t φ a s ε α φ + ε 1 F s(φ) = 0, in Ω ε [0, T ] u = φ on Ω [0, T ] a b ν u a s ε α ν φ = 0, on Ω [0, T ] a s ε α ν φ = 0, on Ω ε \ Ω [0, T ] Today: δ = 1, β = 1, α (, 1)

8 Setting r b t u a b u + F b r (u) = 0, s ε t φ a s ε α φ + 1 ε F s(φ) = 0 Spaces Ṽ ε = { (u, φ) H 1 (Ω) H 1 (Ω ε ) : u Ω = φ Ω } H ε = L 2 (Ω) L 2 (Ω ε ) Energy functionals a b a s ε Ẽ ε (u, φ) = 2 u 2 +F b (u)dx+ Ωε α 2 φ ε F s(φ)dx Ω Dissipation functionals R ε ( u, φ) = Ω r b 2 u 2 dx + 1 ε Ω ε r s 2 φ 2 dx 5

9 6 Setting In this setting r b t u a b u + F b r (u) = 0, s ε t φ a s ε α φ + 1 ε F s(φ) = 0 is formally equivalent to 0 = D R ε ( u, φ) + D Ẽ ε (u, φ) (Force balance) Basic existence theory yields solutions (u ε, φ ε ) L (0, T ; Ṽε) H 1 (0, T ; H ε )

10 Transformation Ω Ω Ω ε Change of coordinates x = y + ην(y) y ην(y) x x (y, η) Ω (0, ε) Scaling η = εη, η (0, 1) Mapping L 2 (Ω ε ) L 2 ( Ω (0, 1)) Spaces Ṽ ε V = {(u, φ) H 1 (Ω) H 1 ( Ω (0, 1)) : u Ω = φ η=0 } H ε H = L 2 (Ω) L 2 ( Ω (0, 1))

11 8 Transformation Energy functionals a b E ε (u, φ) = Ω 2 u 2 + F b (u)dx 1 [ ] + asε α+1 2 ( B ε φ ε 2 η φ 2 ) + F s (φ) J ε dηda Ω 0 Dissipation functionals R ε ( u, φ) = Ω r b2 u 2 dx + Ω 1 0 r s 2 φ 2 J ε dηda Transformed solutions solve DE ε (u ε, φ ε ) + DR ε ( u ε, φ ε ) = 0

12 8 Transformation Energy functionals E ε (u, φ) = + Ω 1 0 E b (u) [ ] asε α+1 2 ( B ε φ ε 2 η φ 2 ) + F s (φ) J ε dηda Dissipation functionals R ε ( u, φ) = Ω r b2 u 2 dx + Ω 1 0 r s 2 φ 2 J ε dηda Transformed solutions solve DE ε (u ε, φ ε ) + DR ε ( u ε, φ ε ) = 0

13 9 A priori estimates E ε (u ε (t), φ ε (t)) + t 0 R ε( u ε, φ ε )dτ E ε (u ε (0), φ ε (0)) Case α 1: Case α ( 1, 1): (u ε, φ ε ) L (0,T ;V ) C (u ε, φ ε ) L (0,T ;V ) C with V := {(u, φ) H 1 (Ω) L 2 ( Ω (0, 1)) : η φ L 2 ( Ω (0, 1)), u Ω = φ η=0 } Bounded dissipation functional ( u ε, φ ε ) L 2 (0,T ;H) C

14 10 Convergent subsequence We have (u ε, φ ε ) (u, φ) in L (0, T ; V ) (resp. in L (0, T ; V )) (u ε, φ ε ) (u, φ) in H 1 (0, T ; H) In particular = t [0, T ] : z ε (t) z(t) in H Moreover, z ε C w ([0, T ], V ) = t [0, T ] : z ε (t) z(t) in V with V = V for α 1 or Ṽ = V for α ( 1, 1)

15 with G ε = DR ε 11 λ-convexity F b and F s are λ-convex, i.e., λ R u F b (u) + λ u 2, φ F s (φ) + λ φ 2 are convex, e.g. C 1,1 -perturbation of a convex function, double well potential W(u) = 1 4 (1 u2 ) 2 F b (u v) F b (u) F b (u)v λ v 2, u, v R Consequence: For z ε = (u ε, φ ε ) E ε (z ε ẑ ε ) + λ ẑ ε 2 H E ε (z ε ) DE ε (z ε ), ẑ ε, ẑ ε V Hence force balance equivalent to E ε (z ε (t) ẑ ε ) + λ ẑ ε 2 H E ε (z ε (t)) + G ε ż ε (t), ẑ ε, ẑ ε V

16 12 Γ-convergence Definition: (Attouch [1984], Braides [2002], Dal Maso [1993]) I 0 : Y R Γ-limit of sequence (I ε ) ε>0 w.r.t. topology τ iff (i) liminf estimate: τ y ε y = I0 (y) lim inf I ε (y ε ) ε 0 (ii) limsup estimate (existence of recovery sequence): τ y Y (ŷ ε ) ε>0 : y ε y and I0 (y) lim sup I ε (ŷ ε ). ε 0 Theorem: (I ε ) ε>0 equi-coercive ( ε>0 {I ε E} precompact) and Γ I ε I 0 ; then min I 0 = lim min I Y ε 0 ε. Y (y ε ) ε>0 minimizers of I ε = cluster points y are minimizers of I 0

17 13 Γ-convergence Minimization problems: Homogenization, Two-scale convergence Singular limits (Cahn Hilliard Sharp interface) Young measure relaxation, penalization Rate independent systems (Mielke, Roubíček, Stefanelli [2008]) Two-scale Homogenization (Mielke & Timofte [2007], Hanke [2009]) Damage to Delamination (Mielke, Thomas, Roubíček in preparation) Dimension reduction in linearized elastoplasticity (Mielke & L. in preparation) Hamiltonian systems (Mielke [2008]) Gradient flows (Sandier & Serfaty [2004])

18 Γ-convergence of the Energies E ε (u, φ) = E b (u)+ Ω 1 0 [ ] asε α+1 2 ( B ε φ ε 2 η φ 2 ) + F s (φ) J ε dηda (i) Case α = 1 V { R, E 0 : E b (u) + a s (u, φ) Ω 2 φ 2 +F s (φ)da η φ = 0, otherwise (ii) Case α < 1 V R {, E 0 : E b (u) + H d 1 ( Ω)F s (φ)da (u, φ) φ constant, otherwise

19 Γ-convergence of the Energies E ε (u, φ) = E b (u)+ Ω 1 0 [ ] asε α+1 2 ( B ε φ ε 2 η φ 2 ) + F s (φ) J ε dηda (i) Case α = 1 V { R, E 0 : E b (u) + a s (u, φ) Ω 2 φ 2 +F s (φ)da (u, φ) V, otherwise (ii) Case α < 1 V R {, E 0 : E b (u) + H d 1 ( Ω)F s (φ)da (u, φ) φ constant, otherwise

20 Γ-convergence of the Energies E ε (u, φ) = E b (u)+ Ω 1 0 [ ] asε α+1 2 ( B ε φ ε 2 η φ 2 ) + F s (φ) J ε dηda (i) Case α = 1 V { R, E 0 : E b (u) + a s (u, φ) Ω 2 φ 2 +F s (φ)da (u, φ) V, otherwise (ii) Case α < 1 V R {, E 0 : E b (u) + H d 1 ( Ω)F s (φ)da (u, φ) V c, (u, φ) otherwise

21 15 Γ-convergence of the Energies (iii) Case α ( 1, 1) V := {(u, φ) H 1 (Ω) L 2 ( Ω (0, 1)) : η φ L 2 ( Ω (0, 1)), u Ω = φ η=0 } V R {, E 0 : E b (u) + (u, φ) Ω F s (φ)da η φ = 0, otherwise In the following: F s convex

22 15 Γ-convergence of the Energies (iii) Case α ( 1, 1) V := {(u, φ) H 1 (Ω) L 2 ( Ω (0, 1)) : η φ L 2 ( Ω (0, 1)), u Ω = φ η=0 } V R {, E 0 : E b (u) + (u, φ) Ω F s (φ)da (u, φ) V, otherwise In the following: F s convex

23 16 Main Result Theorem Let z ε = (u ε, φ ε ) be solutions w.r.t. (E ε, R ε ) with (u ε, φ ε ) (u, φ) in L (0, T ; V ) (resp. in L (0, T ; V )) (u ε, φ ε ) (u, φ) in H 1 (0, T ; H) then z = (u, φ) is a solution w.r.t. (E 0, R 0 ), i.e. E 0 (z(t) ẑ)+λ ẑ 2 H E 0 (z(t))+ G 0 ż(t), ẑ, ẑ ε V (resp. V ) with G 0 = DR 0. In particular, z solves G 0 (ż) + DE 0 (u, φ) = 0

24 17 The limit equations Formally equivalent to r b t u a b u + F b (u) = 0, in Ω [0, T ] Case α = 1 r s t φ a b φ + F s(φ) + a b ν u = 0, on Ω [0, T ] Case α < 1 r s t φ + a b [ ν u] + F s(φ) = 0, in [0, T ] u = φ = const, on Ω [0, T ] α ( 1, 1) r s t φ + a b ν u + F s(φ) = 0, on Ω [0, T ]

25 18 Sketch of proof Set ẑ ε = z ε z ε, with z ε (t) recovery sequence for z(t) z(t). T 0 E ε(z ε ) + λ z ε z ε 2 H dt T 0 E ε(z ε ) + G ε ż ε, z ε z ε dt 2. Integration by parts T 0 G εż ε, z ε z ε dt = T 0 G εz ε, z ε dt + R ε (z ε (T )) R ε (z ε (0)) 3. Passing to the limit + Integration by parts T 0 E 0(z) + λ z z 2 H dt T 0 E 0(z) G 0 ż, z z dt 4. Sequence z δ z ẑ in L 2 (0, T ; V 0 ). By arbitrariness of ẑ E 0 (z(t) ẑ) + λ ẑ 2 H E 0 (z(t)) + G 0 ż(t), ẑ, ẑ V/V

26 19 Outlook Different scalings in dissipation potential More complex dissipation potentials (Heat equation with entropy as driving potential) Interfaces

27 19 Outlook Different scalings in dissipation potential More complex dissipation potentials (Heat equation with entropy as driving potential) Interfaces Thank You