Phase-Field Force Convergence

Transcript

1 Phase-Field Force Convergence Bo Li Department of Mathematics and Quantitative Biology Graduate Program UC San Diego Collaborators: Shibin Dai and Jianfeng Lu Funding: NSF School of Mathematical Sciences Shanghai Jiao Tong University November 14, / 21

2 Interface motion Let x = x(t) be a point on a moving interface Γ = Γ(t). Velocity: ẋ(t) Unit normal: n = n(x) Normal velocity: v n = ẋ n(x) x n Mean curvature: H = H(x) Γ + Curvature driven interface motion Mean-curvature flow: v n = H. Gradient flow: v n = δ Γ E[Γ] with E[Γ] = area (Γ). Surface diffusion: v n = Γ H. Willmore flow: v n = Γ H + 2H(H 2 K). Gradient flow: v n = δ Γ E[Γ] with E[Γ] = Γ H 2 ds. 2 / 21

3 Interface motion driven by curvature and field Two-phase flows ρ ± (u t + (u )u) = µ ± u p + f in ± (t), u = 0 in ± (t), u n = 0 and 2µn D(u)n p = γh on Γ(t), v n = u n on Γ(t). The Mullins Sekerka equations for solidification u = 0 in (t) + (t), u = H on Γ(t), v n = n u on Γ(t). Relaxation of a dielectric boundary ε Γ(t) u = ρ in, v n = γh + 1 ( 1 1 ) εγ(t) n u 2 2 ε + ε (ε ε + ) (I n n) u 2 on Γ(t). 3 / 21

4 Numerical methods for tracking interface motion Front Tracking Method: Level-Set Method x(t + t) x(t) + t v(x(t)). A level-set representation: Γ(t) = {x : φ(x, t) = 0}. By the Chain Rule: t φ + φ ẋ = 0. Use n = φ/ φ to get the level-set equation t φ + v n φ = 0. Phase-Field Method Diffuse interface described by a phase-field function φ : R: φ 1 inside, φ 0 outside. Approximating sharp interface: Γ = {x : φ(x) = 1/2}. Question: Phase-field surface area? χ A / 21

5 The van der Waals Cahn Hilliard Functional [ ε E ε [φ] = 2 φ ] ε W (φ) dx W (φ) = 18φ 2 (φ 1) 2 Under some constraints, e.g., φ 0 in and 0 < φ dx <, 0 1 small E ε [φ]: φ 0 or φ 1, and E ε [φ] area ({φ = 1/2}). Mean-Curvature Flow Sharp Interface Diffuse [ Interface ε Energy E[Γ] = area (Γ) E ε [φ] = 2 φ ] ε W (φ) dx First Variation δ Γ E[Γ] = H δe ε [φ] = ε φ + 1 ε W (φ) Gradient Flow v n = H t φ = ε φ 1 ε W (φ) 5 / 21

6 Energy Convergence (Modica-Mortola 1977, Modica 1987, & Sternberg 1988) Let R d (d 2) be a bounded domain and ε k 0. Recall the perimeter of G in {: } P (G) = sup χ G g dx : g Cc 1 (, R d ), g 1 in. Define E 0 : L 1 () [0, ] by { P (G) if φ = χ G for some G, E 0 [φ] = + otherwise. Theorem {E εk } Γ-converges to E 0 w.r.t. the L 1 -convergence, i.e., (1) If φ k φ in L 1 (), then φ = χ G for some G, and lim inf E ε k [φ k ] E 0 [φ]; (2) φ L 1 (), there exist φ k L 1 () s.t. φ k φ in L 1 () and lim sup E εk [φ k ] E 0 [φ]. 6 / 21

7 Consequences. Consider the set A of all φ H 1 () subject to some constraints, e.g., φ = φ 0 on for a given φ 0, and/or φ dx = θ for some constant θ (0, ). (1) If ε k 0 and φ k = arg min {E εk [φ] : φ A}, then (up to a subsequence) φ k arg min E 0 and min E εk min E 0. (2) If φ 0 is an isoloted local minimizer of E 0, then there exsit φ k, local minimizer of E εk, such that φ k φ 0 in L 1 (). 7 / 21

8 Sketch of Proof. (1) lim inf E ξk [φ k ] E 0 [φ], if φ k φ in L 1 (). Assume {E εk [φ k ]} converges and φ k φ a.e. in. Then, φ k φ in L q () for any q [1, 4). Define η k (x) := φk (x) 0 2W (s) ds, x, k = 1, 2,... Then, {η k } is bounded in L 4/3 (). Since η k = 2W (φ k ) φ k, [ εk sup η k L 1 () sup k 1 k 1 2 φ k ] W (φ k ) dx <. ε k The compact embedding W 1,1 () L 1 () implies that, up to a subsequence, η k η in L 1 (). Let η correspond to some φ 0. So, φ k φ 0 a.e.. Hence φ = φ 0 a.e.. But W (φ k ) L 1 () 0. Hence, φ = χ G, and η k χ G a.e. (this is why 18). Finally, E 0 [φ] = P (G) lim inf η k dx lim inf E ε k [φ k ]. 8 / 21

9 (2) For any φ L 1 (), there exist φ k L 1 () s.t. φ k φ in L 1 () and lim sup E εk [φ k ] E 0 [φ]. Assume E 0 [φ] <. So, φ = χ G with G and P (G) <. Construction of φ k in two steps. Step 1. Assume G is smooth and construct φ k such that 0 φ k χ G in, φ k = 1 in G k := {x G : dist(x, G) ε k }, φ k = 0 in \ G, φ k χ G strongly in L 1 () and a.e. in, [ εk lim sup 2 φ k ] W (φ k ) dx P (G). ε k 9 / 21

10 Details. Write ε for ε k and define q ε : [0, 1] R by t ε q ε (t) = ds. 2ε + 2W (s) 0 Denote λ ε = q ε (1) (0, ε/2) and let p ε : [0, λ ε ] [0, 1] be the inverse of q ε : [0, 1] [0, λ ε ]. Then εp ε(s) = 2ε + 2W (p ε (s)) s [0, λ ε ]. Extend p ε : p ε (s) = 0 if s < 0 and p ε (s) = 1 if s > λ ε. Define φ ε (x) = p ε (d s (x)) x, where d s (x) = dist (x, G) if x G and dist (x, G) if x G c. Note that ε φ ε = 2ε + 2W (φ ε ). Step 2. Assume φ = χ G with 0 < G < and P (G) <. Approximate G by smooth sets. Q.E.D. 10 / 21

11 Force Convergence Let R d be a bounded domain. Let G be open and smooth with ν the unit outer normal of G. Define the boundary force f 0 [ G] := δ G P (G) = (d 1)Hν. Lemma. We have for any V Cc 1 (, R d ) that f 0 [ G] V ds = (I ν ν) : V ds. G G 11 / 21

12 Phase-field force and stress [ f ε (φ) := δ φ E ε [φ]( φ) = ε φ + 1 ] ε W (φ) φ, [ ε T ε (φ) = 2 φ ] ε W (φ) I ε φ φ. Lemma. We have for any φ H 2 () tht f ε (φ) = T ε (φ) a.e., f ε (φ) V dx = T ε (φ) : V dx V Cc 1 (, R d ). 12 / 21

13 Theorem (Force convergence). Assume G, P (G) <, φ k χ G a.e., E εk [φ k ] P (G), and Ψ C c (, R d d ). Then T εk (φ k ) : Ψ dx = (I ν ν) : Ψ dh d 1. (1) lim G If, in addition, all φ k W 2,2 (), G is open, and G is of C 2, then [ lim ε k φ k + 1 ] W (φ k ) φ k V dx ε k = (d 1) Hν V ds V Cc 1 (, R d ). Remarks. G The set G is the reduced boundary of G (De Giorgi 1955). With Ψ = V, we see that f εk (φ k ) f 0 ( G) weakly. Energy convergence is necessary. 13 / 21

14 Definition. A point x G, if P B(x,r) (E) = Dχ E > 0 for all r > 0; B(x,r) B(x,r) The limit ν(x) = lim Dχ E r 0 B(x,r) Dχ E ν(x) = 1. exists; and Structure Theorem. Assume G and P (G) <. Then G = j=1 K j Q, where G (Q) = 0, and each K j is a compact subset of a C 1 -hypersurface S j with ν G is normal to S j, and all K j are disjoint. Moreover, j=1 Hd 1 (K j ) = H d 1 ( G) = G () = P (G). 14 / 21

15 Example. Let a > 0 and W a (s) = W (s)/a. Define φ k w.r.t. W a ε k φ k = 2ε k + 2W a (φ k ). Then, φ k χ G in L 1 () and a.e. in. We have [ ] εk lim dx = lim = lim = P (G) 1 2 φ k 2 + W (φ k) ε k ( Wa (φ k ) + ε k H n 1 ({φ k (x) = s}) aw a (φ k ) 2 [Wa (φ k ) + ε k ] ( Wa (s) + ε k 2 + ) φ k dx 1 + a 2 Wa (s) ds = 1 + a 2 a P (G) > P (G). ) aw a (s) ds 2 [Wa (s) + ε k ] Let Ψ = η d s d s (η: a cut-off function). Then (1) fails. 15 / 21

16 Lemma (Asymptotic equipartition of energy). We have 2 lim εk 2 φ W (φ k ) k ε k dx = 0, k ε 2 φ k 2 1 W (φ k ) ε k dx = 0. lim Proof. We have 2 0 lim sup εk 2 φ W (φ k ) k ε k dx [ εk = lim sup 2 φ k W (φ k ) ] 2W (φ k ) φ k dx ε k = P (G) lim inf η k dx 0. k ξ 2 φ k 2 1 W (φ k ) ξ k = ( ) ( ) ( ) + ( ). Q.E.D. 16 / 21

17 Proof of Theorem on Force Convergence. Need only to prove that for any Ψ C c (, R d d ) T εk (φ k ) : Ψ dx = (I ν ν) : Ψ dh d 1. lim G If suffices to prove that for any Ψ C c (, R d d ) ε k φ k φ k : Ψ dx = ν ν : Ψ dh d 1. lim G Assume this is true. With (I : Ψ)I replacing Ψ, we have [ εk lim 2 φ k ] W (φ k ) I : Ψ dx ε k = lim ε k φ k 2 I : Ψ dx = lim ε k φ k φ k : (I : Ψ)I dx = ν ν : (I : Ψ)I dh d 1 = I : Ψ dh d 1. G G 17 / 21

18 Fix Ψ C c (, R d d ) and prove now ε k φ k φ k : Ψ dx = lim G ν ν : Ψ dh d 1. Let σ > 0. Recall that G = ( j=1 K j) Q, where K j s are disjoint compact sets, each being a subset of a C 1 -hypersurface S j, and Q G with G (Q) = 0. Moreover, j=1 Hd 1 (K j ) = H d 1 ( G) = G () = P (G) <. Choose J so that j=j+1 Hd 1 (K j ) < σ. Choose open U j such that K j U j U j (j = 1,..., J). For each j, we define d (j) s : U j R to be the signed distance to S j, and zero-extend d (j) s to \ U j. Choose ζ j Cc 1 () be such that 0 ζ j 1 on, ζ j = 1 in a neighborhood of K j, supp (ζ j ) U j, and ζ j d s (j) C c (, R d ). Set ν J = J j=1 ζ j d s (j) C c (, R d ) and note that ν j 1 on and ν j = ν on each K j (1 j J). 18 / 21

19 We rewrite ε k φ k φ k as ε k φ k φ k = ( ε k φ k + ε k φ k ν J ) ε k φ k + 2W (φ k ) ε k φ k ν J ε k φ k ε k ν J 2W (φ k ) φ k. Finally, lim sup ε k φ k φ k : Ψ dx ν ν : Ψ dh n 1 G ( ) 4σ sup ε k φ k L 2 () k 1 Ψ L () + 2σ Ψ L (). This completes the proof. Q.E.D. 19 / 21

20 Phase-Field Free-Energy Functional for Molecular Solvation [ ξ F ξ [φ] = P 0 φ 2 dx + γ 0 2 φ ] ξ W (φ) dx + ρ 0 (φ 1) 2 U vdw dx + F ele [φ] [ F ele [φ] = ε(φ) ] 2 ψ φ 2 + ρψ φ (φ 1) 2 B(ψ φ ) dx ε(φ) ψ φ (φ 1) 2 B (ψ φ ) = ρ ψ φ = ψ on Main results: Free-energy convergence and force convergence. in dielectric boundary w n p Γ ε p=1 ε w=80 xi Q i 20 / 21

21 Thank you! 21 / 21