Gradient Systems of Phase Transitions

Transcription

1 Gradient Systems of Phase Transitions Fang Hua Lin Courant Institute, NYU January 2011 Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

2 Early Work Fast Reaction and Slow Diffusion Consistency Conditions Note Proof Remark Example: N = N 1 N2 Dynamical Picture Conclusions Another Example KRS Problem Note Guess Energy Minimization Theorem Another Theorem Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

3 Early Work Early Work In the early 1970s E. De Giorgi proposed the Γ-convergence method for understanding the limits of scalar phase field equations in both the static (1) ε u ε 1 ε F (u ε ) = 0 and dynamic (2) u ε t = ε u ε 1 ε F (u ε ) cases. The first rigorous mathematical work is probably the paper by L. Modica and S. Mortola (1977) on the Γ-convergence of (1). Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

4 Early Work Since the mid-1980s many papers have been written on the gradient theory of phase transitions; e.g., L. Modica (1986), Luckhaus-Modica (1989), Kohn-Sternberg (1989), Fonseca-Tartar (1989) for the elliptic case; and R. Pego (1989), X.F. Chen (1990), M. Soner (1991), T. Ilmanen (1993) (convergence of Allen-Cahn to Brakke flows) for the parabolic case. P. Souganidis et al. studied problems involving statistical effects. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

5 Early Work There was also work on geometric motion by mean curvature (level set methods). In the late 1990s there were works on high dimensional and higher co-dimensional motion by mean curvature using Ginzburg-Landau equations by Ambrosio-Soner (1998), Lin (1999), Lin-Rivière (2001) and Bethuel-Orlandi-Smets (2005 6). Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

6 Fast Reaction and Slow Diffusion Fast Reaction and Slow Diffusion WKB analysis is attributed to Keller-Rubenstein-Sternberg (KRS) (1989) and may be described as follows. Let u(x, t, ε) R m be a solution of the following initial boundary value problem for x in a bounded smooth domain Ω of R n : { u t = ε u 1 (3) ε F (u), (x, t) Ω R +, u(x, 0) = g ε (x), x Ω, with boundary conditions (N) u(x, t) = 0, x Ω, ν ν is the exterior unit normal along Ω; (D) u(x, t) = h ε (x), x Ω. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

7 Consistency Conditions Consistency Conditions In addition, one has consistency conditions: (NC) (DC) ν g ε (x) = 0 g ε (x) = h ε (x) on Ω, on Ω. F(u) is typically a smooth function on R m such that F(u) dist 2 (u, N) whenever dist(u, N) δ N. Here N is a smooth, connected, compact submanifold of R m. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

8 Consistency Conditions Let τ = t ε be the fast and η = εt the slow time variables. Keller-Rubinstein-Sternberg argued formally that Here (a) u(x, t, ε) = u (x, τ, η) + O(ε 2 ). { u τ (x, τ, η) = F (u ), u (x, 0, 0) = g(x). For η = 0 the above ODE can be solved globally in τ [0, ), and (b) F(u (x,, 0)) = 0 = u (x,, 0) N. Let φ(x) = lim τ + u (x, τ, 0) = u (x,, 0), then in the (x, η)-variables (as τ ) one has { u η (c) + u T u (N) u (x,, η) = φ(x) at η = 0. u satisfies the equation of heat flow of harmonic maps into N. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

9 Consistency Conditions Note Note Note KRS showed the above statement rigorously when N = S m 1 and all data are radially symmetric. Indeed, when N = S m 1, then the equation in (x, η) variables becomes (4) u ε η = u ε + u ε ε 2 (1 u ε 2 ). (Here F(u) = 1 4 (1 u 2 ).) Suppose u ε (x, 0) = φ(x) H 1 (Ω, S m 1 ), then under either of conditions (D) or (N) one has u ε (x, η) u(x, η) a solution of heat flow of harmonic maps into S m 1. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

10 Consistency Conditions Proof Proof Sketch of the Proof If then E ε (u ε (t)) = 1 2 Ω E ε (φ) E ε (u ε (t)) = u ε 2 (t)dx + 1 4ε 2 ( u ε 2 1) 2 dx Ω t 0 Ω u ε 2 dx dη. η Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

11 Consistency Conditions Proof We have and as ε 0 (*) u u η u ε u ε η = div(u ε u ε ) = div(u u). Since u = 1, (*) implies that (5) u η = u + u 2 u. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

12 Consistency Conditions Remark Remark Remark [Chen-Struwe, Chen-Lin] (6) { u ε η = uε 1 ε 2 grad F(u ε ) u ε (x, 0) = φ(x) H 1 (Ω, N) with either (D) or (N) one can show u ε u weakly in H 1 such that u η = u + A(u)( u, u). See [Lin-Wang] for a further understanding of this. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

13 Example: N = N 1 N2 Example: N = N 1 N2 Consider 1 E ε,λ (u, v) = Ω 2 ( u 2 + v 2 ) + 1 [ (1 u 2 4ε 2 ) 2 + (1 v 2 ) 2] dx λ + 2 (u v)2 dx. Ω Here u, v are complex-valued functions on Ω R 2, N 1 and N 2 are circles in R 4, Ω a bounded smooth domain. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

14 Example: N = N 1 N2 Again one has boundary conditions (D) or (N). { u (7) t = u + u (1 u 2 ) λ(u v)v, ε 2 v t = v + v (1 v 2 ) λ(u v)u. ε 2 References: Geshkenbein and Larkin, JETP Lett., Sigrist, Rice and Veda, Phys. Rev. Lett., Multiple components of BECs, after λ log 1 ε Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

15 Dynamical Picture Dynamical Picture [Lin-T.C. Lin] We start with the following: a u component has a vortex at a and a v component has a vortex at b, both are of degree 1 2. There is a domain wall Γ 0 connecting vortces a and b so that phases of u and v both have jumps π across Γ. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

16 Dynamical Picture Conclusions Conclusions We may draw the following four conclusions: For 0 t o( λ), u, v simply adjust to become energy minimizing subject to the prescribed domain wall Γ 0 and the location of vortices a and b. Moreover, in this time interval. E ε,λ (u, v) = 2π log 1 ε + c 0l(Γ 0 ) λ + O(1) If o( λ) t O( λ), then Γ = Γ(t) is a motion by curvature while the endpoints a, b are fixed. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

17 Dynamical Picture Conclusions For t = O(log 1 ε / λ), Γ(t) is straight and ( 1 2 degree) vortices at locations a(t), b(t) such that d dt a(t) = ab, d dt b(t) = ab. For t = O(log 1 ε ) it starts with a(0) = b(0) = 1 2 (a 0 + b 0 ). Then a(t) = b(t) motion is given by the ODE d dt a(t) = aw(a). Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

18 Dynamical Picture Another Example Another Example [Lin-Lin-Wei] E,β (u, v) = u 2 + v 2 + ( 1 u 2 v 2) 2 + β u 2 v 2 S 3 2 for u, v H 1 (S 3, C) such that S3 u 2 = c 1 S 3, (, β +.) S 3 v 2 = c 2 S 3, 0 < c 1, c 2 < 1, c 1 + c 2 = 1. One may find linked vortex ring solutions (skyrmions) as critical points of two-component BECs. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

19 KRS Problem KRS Problem When N = N 1 N2, a union of two smooth, compact and connected submanifolds N 1 and N 2 such that N 1 N 2 =. KRS (1989) asked the behavior of solutions of (3) with suitably prepared initial and boundary conditions. That is to understand (8) u ε t = ε uε 1 ε grad F(uε ) with u ε (x, 0) = g ε (x) and (D) or (N) as ε 0 +. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

20 KRS Problem Note Note Note g ε (x) and h ε (x) are suitably prepared. One may assume F(u) = d 2 N (u). where { dist(u, N) if dist(u, N) δ N d N (u) = smooth and c N > 0 if dist(u, N) > δ N. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

21 KRS Problem Guess Guess Guess Fast time variable dynamics are similar; i.e., u τ = grad F(u). It will bring the data, after O(ε) time, the initial data g ε (x) for (8) becomes well prepared as in the diagram, to For example, εe ε (g ε ) A, g ε (Ω 1 ) N 1, g ε (Ω 2 ) N 2. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

22 KRS Problem Guess In O(1) time scale, the shape interface Γ will evolve according to the motion by mean curvature. Formal analysis by KRS showed that if N 1 = {P}, N 2 = {q}, then Γ(t) is indeed a motion by mean curvature. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

23 Energy Minimization Energy Minimization such that u ε = g ε on Ω. [ 1 min Ω 2 u ε ] ε 2 F(uε ) dx = min E ε (u ε ) Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

24 Energy Minimization Theorem Theorem Theorem [Lin-Pan-Wang] If g ε is admissible and u ε is a minimizer, then (9) εe ε (u ε ) = C F 0 (N)Hn 1 (Γ) + o ε (1) u ε (Ω ε 1 ) N 1 u ε (Ω ε 2 ) N 2 Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

25 Energy Minimization Theorem Γ ε Γ is an area-minimizing hypersurface and C F 0 (N) is the energy of the minimimal connecting orbit between N 1 and N 2. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

26 Energy Minimization Theorem For p N 1, q N 2, one can define C ε (p, q) to be the value + [ 1 min du ε ] 2 ds ε 2 F(u ε) ds such that u ε ( ) = p, u ε (+ ) = q. C F 0 (N) = ε min{c ε(p, q) : p N 1, q N 2 }. Let Σ 1 = {p N 1 : q N 2 s. t. εc ε (p, q) = C F 0 (N)} Σ 2 = {q N 2 : p N 1 s. t. εc ε (p, q) = C F 0 (N)} Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

27 Energy Minimization Another Theorem Another Theorem Theorem (Lin-Pan-Wang) If, in addition, Γ is smooth and stable, then (10) εe ε (u ε ) = C F 0 (N)Hn 1 (Γ) + εd + o(ε) whenever D <. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

28 Energy Minimization Another Theorem Here such that u = g on Ω and { } 1 D = min Ω 2 u 2 : u(ω) N 1 N 2 u(x) + Σ 1, u(x) Σ 2, ε C ε (u(x) +, u(x) ) = C F 0 (N) x Γ. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28

29 Energy Minimization Another Theorem The parabolic case is under investigation. Fang Hua Lin (CIMS) Gradient Systems of Phase Transitions January / 28