Stability Analysis of Stationary Solutions for the Cahn Hilliard Equation

Transcription

1 Stability Analysis of Stationary Solutions for the Cahn Hilliard Equation Peter Howard, Texas A&M University University of Louisville, Oct. 19, 2007 References d = 1: Commun. Math. Phys. 269 (2007) d 2: Physica D 229 (2007)

2 The Cahn Hilliard Equation We consider the Cahn Hilliard equation u t = {M(u) (F (u) κ u)}; x R d, d 1 (CH) where M C 2 (R), F C 4 (R), and we make the standard phenomenological assumptions M(u) M 0 > 0, and that F (u) has a double well form. F(u) u 1

3 The Physical Setting u t = {M(u) (F (u) κ u)} One setting in which the Cahn Hilliard equation arises is in the modeling of spinodal decomposition, a phenomenon in which the rapid cooling of a homogeneously mixed binary alloy causes separation to occur, resolving the mixture into regions in which one component or the other is dominant, with these regions separated by steep transition layers. More precisely, this phase separation is typically considered to take place in two steps: First, a relatively fast process occurs in which the homogeneous mixture quickly begins to separate (this is spinodal decomposition), followed by a second coarsening process in which the regions continue to grow on a relatively slower timescale. In this context, u denotes the concentration of one component of the alloy (or a convenient affine transformation of this concentration), F denotes the bulk free energy density of the alloy, M denotes the molecular mobility, and κ is a measure of interfacial energy. 2

4 Miller et al. Experiment with Fe Cr,

5 Phenomenological Derivation Since u is a conserved density it satisfies a conservation law u t + J = 0. If we let E(u) denote the total free energy for a system in configuration u, then a standard phenomenological assumption is J = M(u) δe δu. In 1958, Cahn and Hilliard recognized that an appropriate energy is E(u) = Ω F (u) + κ 2 u 2 dx. 4

6 Form of the Free Energy Density For Helmholtz free energy F = U T S, F T = S. F(u) F(u) Temperature Drops u u l u r u 5

7 (CH): u t = {M(u) (F (u) κ u)} For any line G(u) = Au + B, we can replace F with F G, and we can also shift F with a change of variable w = u c. Without loss of generality, we can put F in the (not necessarily even) standard form: F(u) u u 1 2 u The values u 1 and u 2 are called the binodal values. This transformation places the generic homogeneous configuration at u h = 0. 6

8 Stability of Stationary Solutions Generally, we say that a stationary solution ū(x) is stable if for u(t, x) = ū(x) + v(t, x), with v(0, x) small in some appropriate sense, v(t, x) remains small as time evolves. Upon substitution of u into (CH) we obtain an equation for v: v t = Lv + Nv, where Lv = {M(ū) (F (ū)v κ v)}. 7

9 The Stationary Solutions for d = 1 In d = 1, u t = {M(u)(F (u) κu xx ) x } x. In this case, there are only three types of non-constant bounded stationary solutions ū(x) to (CH). Periodic solutions Pulse type (reversal) solutions ū(± ) = u ±, u = u + Transition waves ū(± ) = u ±, u u + The homogeneous state corresponds with a constant solution to u(t, x) u h, which is typically in the spinodal region {u : F (u) < 0}. 8

10 1 Periodic Solutions and Limiting Transition Wave F (u) = 1 8 u4 1 4 u2 M(u) 1, κ = 1 Equilibrium Solutions x Leading eigenvalues as a function of amplitude 0.07 u=0 corresponds with constant state 0.06 Leading Eigenvalue u=1 corresponds with transition wave Amplitude 9

11 Multiple Dimensions u t = {M(u) (F (u) κ u)} For d 2, we have a broader cast of stationary solutions. For example, in the case d = 2, some of the players are: 1. Three types of planar solutions. 2. Radial solutions ū(r) that satisfy ū (0) = 0, ū (r) < 0 for r > 0, and lim r ū(r) = u. 3. Saddle solutions, which have infemum u 1 and supremum u 2 (the binodal values), and which have the same sign as xy. 4. Doubly periodic solutions. Focus on the transition front (or kink) solution. 10

12 Existence and Structure of the Kink Solution Given any F in double well form, there exist precisely two kink solutions to (CH), ū(x 1 ) and ū( x 1 ), both of which are strictly monotonic, and both of which approach the binodal values u 1 < u 2 : lim ū(x 1) = u 1 ; x 1 lim ū(x 1) = u 2. x 1 + F(u) u u 1 2 u 11

13 The Result of Jasnow and Zia Upon linearization of (CH) about the kink solution ū(x 1 ), we obtain a perturbation equation with linear part v t = Lv := {M(ū) (F (ū)v κ v)}. The relevant eigenvalue problem is Lφ = λφ. We take the Fourier transform in the transverse variable x = (x 2, x 3,..., x d ) ( x ξ): ˆL ξ ˆφ = λ ˆφ. In the paper Crossover of Interfacial Dynamics, [Physical Review A 36 (1987) ], David Jasnow and R. K. P. Zia showed that for M = M 0 = constant and a double Gaussian F, the leading eigenvalue λ (ξ) satisfies λ (ξ) = c ξ 3 + o( ξ 3 ). 12

14 The Result of Korvola, Kupiainen, and Taskinen Theorem. For the case M 1, κ = 1, and F (u) = 1 8 u4 1 4 u2, the kink solution ū(x 1 ) = tanh(x 1 /2) is stable for d 3 as follows: if ( u(0, x) ū(x 1 ) (1 + x ) r) δ, sup x R d for some δ sufficiently small, then u(t, x) = ū(x 1 ) + Aū (x 1 )φ(t, x) + v(t, x), where Here A = 1 2 R d u(0, x) ū(x 1 )dx; sup v(t, x) Ct 1 12 d 1 3. x R d φ(t, x) = (2πt 1/3 ) 1 d e iξ x t 1/3 e 1 3 ξ 3 dk. R d 1 13

15 The KKT Methodology The result of Korvola, Kupiainen, and Taskinen appears as Anomalous Scaling for Three-Dimensional Cahn Hilliard Fronts, Comm. Pure Appl. Math. 58 (2005) The method is based on a renormalization group approach. The main difference between the approach of [KKT] and the current approach is the manner in which λ (ξ) is treated. In [KKT], a correction associated with λ (ξ) is added as a first order term in a timeasymptotic expansion. In the current approach, λ (ξ) is accomodated by a consideration of its effect in shifting the front location. 14

16 d=1 u(0,x) _ u(x) x δ(t) u(t,x) _ u(x) x d=2 Shock Layer x 2 δ (t,x ) 2 x 1 δ (t,x ) 2 15

17 Local Tracking Implementation General perturbations of ū(x 1 ) lead to transient shifts along the transition front, and we build a tracking method into our linearization. (Follows Howard Zumbrun ARMA 2000, Hoff-Zumbrun IUMJ 2000, Howard-Hu ARMA 2006.) Setting u(t, x) = ū(x 1 δ(t, x)) + v(t, x), where x = (x 2, x 3,..., x d ) denotes the transverse variable, we find ( t L)v = ( t L)(δ(t, x)ū x1 (x 1 )) + Q, where Lv = {M(ū) (F (ū)v κ v)} Q = Q(ū, δ, v). 16

18 The Spectral Analysis The relevant eigenvalue problem is Lφ = λφ. We take the Fourier transform in the transverse variable x = (x 2, x 3,..., x d ) ( x ξ): where For ξ 0, we set ϕ = D 1/2 ξ D ξ H ξ ˆφ = λ ˆφ, D ξ := x1 M(ū) x1 + ξ 2 M(ū) H ξ := κ x1 x 1 + (F (ū) + κ ξ 2 ). ˆφ, so that L ξ := D 1/2 ξ H ξ D 1/2 ξ ϕ = λϕ. The advantage of this formulation is that L ξ is self-adjoint (on D(L ξ )). 17

19 The Spectrum for ξ 0 1. The essential spectrum of L ξ lies entirely on the real axis and is bounded to the left of both M(u ± )F (u ± ) ξ 2 κm(u ± ) ξ The point spectrum of L ξ lies entirely on the real axis and is bounded to the left of κm 0 ξ 4 (recall: M(u) M 0 > 0). 3. For ξ sufficiently small, the leading eigenvalue of L ξ satisfies λ ( ξ ) c 1 ξ 3, and the remainder of the point spectrum lies to the left of c 2 ξ 2. In order to carry out the nonlinear analysis, we require a very precise understanding of the leading eigenvalue λ ( ξ ). For this, we use the Evans function. 18

20 The Evans Function In this setting, the Evans function is simply D(λ, ξ) = det φ 1 φ 2 φ + 1 φ + 2 φ 1 φ 1 φ 1 φ 2 φ 2 φ 2 φ + 1 φ + 1 φ + 1 φ + 2 φ + 2 φ + 2 x1 =0, where the φ ± k denote solutions of L ξφ = λφ that decay at ±. The Evans function can be analytically defined in terms of the three variables r := ξ 2 ρ ± := λ + b± r + c ± r 2 ; b ± := M(u ± )F (u ± ); c ± := κm(u ± ). b ± + 2c ± r 19

21 The Leading Eigenvalue For ξ sufficiently small, the leading eigenvalue of L ξ satisfies λ ( ξ ) = λ 3 ξ 3 + O( ξ 4 ), where λ 3 = 2κ(M(u ) + M(u + )) [u] 2 max{u,u + } min{u,u + } F (x) F (u )dx. 20

22 The Nonlinear Analysis Return now to ( t L)v = ( t L)(δ(t, x)ū x1 (x 1 ))+ Q. Let G(t, x; y) denote the Green s function associated with t L. Then v(t, x) = G(t, x; y)v 0 (y)dy + δ(t, x)ū x1 (x 1 ) R d t + G(t s, x; y) Qdyds. 0 R d There exists a splitting G = G(t, x; y) + ū x1 (x 1 )E(t, x; y) so that G L p (R d ) t d 2 (1 1 p ) ū x1 E L p (R d ) t d 1 3 (1 1 p ). 21

23 Choosing δ(t, x) We write t v(t, x) = G(t, x; y)v0 (y)dy + G(t s, x; y) Qdyds R d 0 R d + δ(t, x)ū x1 (x 1 ) + ū x1 (x 1 ) E(t, x; y)v 0 (y)dy R d t + ū x1 (x 1 ) E(t s, x; y) Qdyds. R d 0 We choose δ(t, x) = E(t, x; y)v 0 (y)dy R d t 0 R d E(t s, x; y) Qdyds. 22

24 Nonlinear Stability of Planar Transition Waves Theorem. [H. Physica D, 2007] Under the assumptions of the talk, ū is nonlinearly stable as follows: for Hölder continuous initial perturbations (u(0, x) ū(x)) C 0+γ (R d ), γ > 0, with u(0, x) ū(x 1 ) L 1 x E 0 (1 + x 1 ) 3/2, for some E 0 sufficiently small, there holds, u(t, x) ū(x 1 δ(t, x)) L p (R d ) C [ (1+t) d 2 (1 1 p ) +(1+t) (d )(1 1 p ) 1 6 +σ], for all 1 < p, and where δ(t, x) L p (R d 1 ) C(1 + t) d 1 3 (1 1 p )+σ. 23

25 Further Work One direction of further study regards the consideration of additional stationary solutions. (Classified here by nodal structure.) 1. Periodic bands x x

26 Further Work 2. Dang, Fife, and Peletier have established the existence of saddle solutions to (CH) in d = 2 (ZAMP, 1992). Also, Junping Shi (CPAM, 2002)

27 Further Work 3. Holzmann and Kielhöfer have established the existence of doubly periodic solutions with rectangular nodal domains (Math. Annalen 1994)

28 Further Work 4. Fife, Kielhöfer, Maier Paape, and Wanner have established the existence of doubly periodic solutions with nodal domains that are not rectangular (Physica D, 1997)

29 The case d = 1 F (u) = 1 8 u4 1 4 u2 M(u) 1, κ = 1 Equilibrium Solutions Periodic Solutions and Limiting Transition Wave x Leading eigenvalues as a function of amplitude 0.07 u=0 corresponds with constant state 0.06 Leading Eigenvalue u=1 corresponds with transition wave Amplitude

30 The Case d = 1: u t = (M(u)(F (u) κu xx ) x ) x Let u 1 and u 2 denote the binodal values, and let u 3 and u 4 satisfy u 1 u 3 < u 4 u 2. Then there exists a bounded stationary solution to (CH) satisfying inf ū(x) = u 3; sup ū(x) = u 4 x R x R if and only if there holds F (u 4 ) [F ] [u] F (u 3 ), where [u] = (u 4 u 3 ) and [F ] = F (u 4 ) F (u 3 ). Two strict inequalities periodic solutions (including constants) One equality reversal solutions ū(± ) = u ±, u = u + Two equalities transition waves ū(± ) = u ±, u u + 29

31 Stability Periodic solutions are spectrally unstable, and the leading eigenvalues decrease as the amplitudes increase. Reversal solutions are spectrally unstable in the following weak sense: the eigenvalue λ = 0 has multiplicity 2. Transition front solutions are orbitally stable. 30

32 Coarsening Rates The curve λ max (P ) can be approximated by Langer s large P limiting equation λ max (P ) = c e P P, and the leading eigenvalue associated with u h = 0 can be used to show c = πm(0) F (0) 3 4κ e2π q κ F (0). considered the case, and they approximated the double-well potential with a double Gaussian F (u) = ν2 2 { (u K) 2 u > 0 (u + K) 2 u < 0. 31

33 They showed that the leading eigenvalue of ˆL ξ satisfies λ (ξ) = M 0σ 2K 2 ξ 3, where σ = + ū (x 1 ) 2 dx 1. 32

34 The Dispersion Relation The essential spectrum of ˆL ξ satisfies: σ ess (ˆL ξ ) (, M(u ± )( ξ 2 F (u ± ) + κ ξ 4 ], with σ ess (ˆL 0 ) (, 0]. The eigenvalue λ (0) lies in the essential spectrum. Since the linear perturbation equation asymptotically has second order diffusion, we would like to have a quadradic scaling for λ ( ξ ). 33

35 Numerical Study of de Mello and Filho, 2005 u x (0) = u x (1) = 0 u xxx (0) = u xxx (1) = 0 1 U(t) (a=1, b=1, ε 2 =0.001, t=1/100 x=1/50) 0.5 u(x,t) X t=0 t=5 t t=10 t t=50 t t=100 t t=500 t Fig. 2. The time evolution of the profile of U n i for different times starting at the initial condition (t ¼ 0) till the total spinodal decomposition is attained at n ¼ 500: The size of the system is L ¼ 1: 34

36 Coarsening Rates In the paper, Theory of Spinodal Decomposition in Alloys [Annals of Physics 65 (1971) 53 86], James Langer suggested the following method for obtaining coarsening rates: at each time t suppose that the solution to (CH), u(t, x), is near a stationary solution ū(x), and that the difference between u and ū is determined primarily by thermal fluctuations in the material. If we adapt Langer s analysis to the current setting, we find that if P (t) denotes the period of a periodic stationary solution at time t then dp dt = 1 2 λ max(p )P, where λ max (P ) denotes the leading eigenvalue associated with a stationary periodic solution with period P. 35

37 The Fundamental Questions Alloys undergoing spinodal decomposition and subsequent coarsening typically exhibit changes in characteristic properities (e.g., hardness typically increases), and one fundamental question in applications regards the time scale on which these changes occur. We would like to take a slight perturbation of the constant homogenously mixed state as an initial condition, and determine the amount of coarsening that has occurred for all subsequent times. 36