W, W, W(r) = (1/4)(r 2 1), W (r) = r 3 r

Transcript

1 2018 : msjmeeting-2018sep-09i004 ( ), Cahn Hilliard..,.,. 1.,,,. Dirichlet( ), Neumann( ), Robin(, ), 1, 2, 3., dynamic( ).,.,, system( ) transmission problem( ). 2. Cahn Hilliard 0 < T < +, R d (d = 2, 3) :=. Cahn Hilliard( ) [7] 2 (, Milanville[43] ): u µ = 0 in Q := (0, T ), (1) t µ = u + W (u) f in Q, (2) W, W, W(r) = (1/4)(r 2 1), W (r) = r 3 r., u := u(t, x) 1, u = 1 A, u = 1 B, u = 0 Supported by JSPS KAKENHI Grant Number 17K fukao@kyokyo-u.ac.jp

2 ., ν ν, µ := µ(t, x) Neumann ν µ = 0 on Σ := (0, T ) Σ u, : u(t)dx = u(0)dx for all t [0, T ]. u Neumann., { } 1 E CH (z) := 2 z 2 dx + W(z) dx, d dt E ( ) CH u(t) + µ(t) 2 dx = 0 for all t [0, T ].., Gal[28] Goldstein Miranville Schimperna[31], Cahn Hilliard : u t + νµ µ = 0 on Σ, (3) µ = ν u u + W (u) f on Σ. (4) (3) (1), (2) (1) (4) Cahn Hilliard., (1) (3) u(t)dx + u(t)d = u(0)dx + u(0)d for all t [0, T ].,. (1) (4) { } { } 1 1 E GMS (z) := 2 z 2 dx + W(z) dx + 2 z 2 d + W (z) d, d dt E ( ) GMS u(t) + µ(t) 2 dx + µ(t) 2 d = 0 for all t [0, T ]..

3 3., (1) (4),., (GMS). u, µ : Q R u, µ : Σ R, : u µ = 0 t a.e. in Q, (5) µ = u + ξ + π(u) f, ξ β(u) a.e. in Q, (6) u = u, µ = µ, u t + νµ µ = 0 a.e. on Σ, (7) µ = ν u u + ξ + π (u ) f, ξ β (u ) a.e. on Σ, (8) u(0) = u 0 a.e. in, u (0) = u 0 a.e. on, (9), u u, Laplace Beltrami (, Grigor yan[32] )., W := β + π, β ˆβ W, π ˆπ W, W = ˆβ + ˆπ, β, β R 2 R,. β(r) = r 3, π(r) = r, D(β) = R, W W(r) = 1 4 (r2 1) 2 ; β(r) = ln((1 + r)/(1 r)), π(r) = 2cr, D(β) = ( 1, 1), W log W(r) = ( (1 + r) ln(1 + r) + (1 r) ln(1 r) ) cr 2,, c > 0 ( ) ; β(r) = I [ 1,1] (r), π(r) = r, D(β) = [ 1, 1], W W(r) = I [ 1,1] (r) 1 2 r2,, I [ 1,1] [ 1, 1], I [ 1,1] (r) = { 0 if r [ 1, 1], + otherwise., I [ 1,1] I [ 1,1] (, Barbu[6] )., : H := L 2 (), V := H 1 (), W := H 2 () H := L 2 (), V := H 1 (), W := H 2 (), H := H H, V := { z := (z, z ) V V : z = z a.e. on }, W := (W W ) V., : H 0 := {z H : m(z) = 0},, m : H R m(z) := 1 + { zdx + } z d for all z H

4 , m 0 := m(u 0 )., u 0 := (u 0, u 0 )., V 0 := V H 0., V 0, Poincaré Wirtinger z V 0 := a(z, z) 1/2 (z V 0 )., a(, ) : V V R a(z, z) := z zdx + z z d for all z, z V,., V 0 V 0 F F z, z V 0,V 0 := a(z, z) for all z, z V 0 F : V 0 V 0, C P > 0 C P z 2 V F z, z V 0,V 0 = z 2 V 0 for all z V 0., V 0 H 0 V 0 (Colli F[11, Appendix] )., Kenmochi Niezgódka Paw low[38] (, [37, 39] )., : v := (v, v ) := (u m 0, u m 0 ), v 0 := (v 0, v 0 ) := (u 0 m 0, u 0 m 0 )., : β(z) := (β(z), β (z )), π(z) := (π(z), π (z ))., : (A1) β, β : R 2 R, ˆβ, ˆβ : R [0, + ] ˆβ(0) = ˆβ (0) = 0 β = ˆβ, β = ˆβ ; (A2) π, π : R R Lipshitz ; (A3) D(β ) D(β), ϱ, c 0 > 0 β (r) ϱ β (r) + c0 for all r D(β ),, β (r) := {r β(r) : r = min s β(r) s }; (A4) m 0 intd(β ) ˆβ(u 0 ) L 1 (), ˆβ(u 0 ) L 1 () ; (A5) f := (f, f ) W 1,1 (0, T ; H) f L 2 (0, T ; V ), u 0 V. β β = β. β = β (A3)., (A3), β (A3) β (Calatroni Colli[8])., :

5 2.1.[11, Theorem 2.2] (A1) (A5), (v, µ, ξ) ( µ(t), z ) v H 1 (0, T ; V 0) L (0, T ; V 0 ) L 2 (0, T ; W ), µ L 2 (0, T ; V ), ξ L 2 (0, T ; H), ξ β(v + m 0 ) a.e. in Q, ξ β (v + m 0 ) a.e. on Σ, v (t), z V 0,V 0 + a( µ(t), z ) = 0 for all z V 0, H = a( v(t), z ) + ( ξ(t) + π(v(t) + m 0 1) f(t), z ) H for all z V, for a.a. t (0, T ), v(0) = v 0 in H 0. : (A5) f H 1 (0, T ; H), f(0) V, u 0 V (H 3 () H 3 ()), β (u 0 ) V. 2.2.[11, Theorem 4.2] (A1) (A4), (A5), (u, µ, ξ) u W 1, (0, T ; V ) H 1 (0, T ; V ) L (0, T ; W ), µ L (0, T ; V ) L 2 (0, T ; W ), ξ L (0, T ; H), ξ β(u) a.e. in Q, ξ β (u ) a.e. on Σ, (5) (9)., : F 1 v (t) + ϕ ( v(t) ) = P ( ξ(t) π ( v(t) + m 0 1 ) + f(t) ) in H 0, ξ(t) β ( v(t) + m 0 1 ) in H, for a.a. t (0, T ), v(0) = v 0 in H 0., ϕ : H 0 [0, + ] 1 z 2 dx + 1 z 2 d if z V 0, ϕ(z) := if z H 0 \ V 0., P : H H 0 P z := z m(z)1, 1 := (1, 1)., z D( ϕ) = W V 0, ϕ(z) = ( z, ν z z ) (Colli F[11, Appendix], [13] )., µ = ϕ(v)+ξ+π(v +m 0 1) f, V µ, µ m(µ) Poincaré Wirtinger Kenmochi Niezgódka Paw low[38] 1.

6 GMS, [11, Theorem 2.1]., f F Yamazaki[25], Kajiwara[35]., Cherfils Petcu Pierre[10], Cherfils Petcu[9] Furihata Matsuo[27] F Yoshikawa Wada[26]. Liu Wu [42]. 4. GMS u, ξ : Q R, u, ξ : Σ R u ξ = g, ξ β(u) in Q, (10) t u ξ = ξ, ξ β(u ), t + νξ ξ = g on Σ, (11) u(0) = u 0 in, u (0) = u 0 on. (12) β(= β ) : R 2 R (10) (12). k s r r < 0, β(r) := 0 0 r L, β(r) := r q 1 r (q > 0), β(r) := I [0,1] (r) (13) k l (r L) r > L,, (10) (12) Stefan (k l, k s, L ), (q > 1), fast diffusion (0 < q < 1), Hele-Shaw. Stefan Aiki[1, 2, 3] 1990, Laplace Beltrami, Laplace Beltrami,., Igbida, [5, 34]. Hele-Shaw Igbida[33]., GMS : ε (0, 1] u ε t µ ε = 0 in Q, (14) µ ε = ε u ε + ξ ε + π ε (u ε ) f, ξ ε β(u ε ) in Q, (15) u,ε u,ε = (u ε ), µ,ε = (µ ε ), + ν µ ε µ,ε = 0 a.e. on Σ, (16) t µ,ε = ε ν u ε ε u,ε + ξ,ε + π ε (u,ε ) f, ξ,ε β(u,ε ) a.e. on Σ, (17) u ε (0) = u 0,ε a.e. in, u,ε (0) = u 0,ε a.e. on, (18), π ε : R R, β (13) W := ˆβ + ˆπ ε (14) (18) Cahn Hilliard., β(r) = r 3, π ε (r) = εr., :

7 (A6) g L 2 (0, T ; H 0 ), u 0 H; (A7) ε (0, 1] π ε : R R Lipschitz., c 1 > 0 σ : [0, 1] [0, 1] σ(0) = 0, σ(1) = 1 πε (0) + π ε L (R) c 1 σ(ε)., f L 2 (0, T ; V 0 ) ϕ(f) = g in H 0 a.e. in (0, T ), v 0,ε V 0 v 0,ε + ε ϕ(v 0,ε ) = v 0 := u 0 m 0 1 in H 0, C > 0 v 0,ε 2 H 0 C, ε v 0,ε 2 V 0 C, ˆβ(v 0,ε + m 0 )dx C, ˆβ(v 0,ε + m 0 )d C for all ε (0, 1], v 0,ε v 0 in H 0, u 0,ε := v 0,ε + m 0 1 u 0 in H as ε 0. (10) (12) (14) (18) : 3.1.[21, Theorem 2.1] [22, Theorem 2.1] (A1), (A4), (A6), (A7), (u, ξ) u H 1 (0, T ; V ) L (0, T ; H), ξ L 2 (0, T ; V ), ξ β(u) a.e. in Q, ξ β(u ) a.e. on Σ, u (t), z V,V + u (t), z = V,V + g(t)zdx + ξ(t) zdx + g (t)z d for all z V, for a.a. t (0, T ), u(0) = u 0 in H. ξ (t) z d [21, Theorem 2.2]. Cahn Hilliard., Neumann Colli-F[12] u ε u 2 C([0,T ];V ) + T 0 ( ξε (s) ξ(s), u ε (s) u(s) ) H ds Const.ε 1 2, Robin F Motoda[24] Neumann., [12], Cahn Hilliard β., H 1 () ˆβ, H 1 (),. Neumann, Kenmochi[36], Akagi[4], log Kubo Lu[40]., Robin

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