Slow Motion in One-Dimensional Cahn-Morral Systems
In this paper we study one-dimensional Cahn-Morral systems, which are the multicomponent analogues of the Cahn-Hilliard model for phase separation and coarsening in binary mixtures. In particular, we examine solutions that start with initial data close to the preferred phases except at finitely many transition points where the data has sharp transition layers, and we show that such solutions may evolve exponentially slowly; i.e., if ε is the interaction length then there exists a constant C such that in exp(C/ε) units of time the change in such a solution is o(1). This corresponds to extremely slow coarsening of a multicomponent mixture after it has undergone fine-grained decomposition.
SIAM J. Math. Anal. Vol. 26, No. 1, pp. 21{34, January 1995 SLOW MOTION IN ONE-DIMENSIONAL CAHN-MORRAL SYSTEMS CHRISTOPHER P. GRANT Abstract. In this paper we study one-dimensional Cahn-Morral systems, which are the multicomponent analogues of the Cahn-Hilliard model for phase separation and coarsening in binary mixtures. In particular, we examine solutions that start with initial data close to the preferred phases except at nitely many transition points where the data has sharp transition layers, and we show that such solutions may evolve exponentially slowly; i.e., if " is the interaction length then there exists a constant C such that in exp(C=") units of time the change in such a solution is o(1). This corresponds to extremely slow coarsening of a multicomponent mixture after it has undergone ne-grained decomposition. Key words. Cahn-Hilliard equation, phase separation, transition layers, metastability AMS subject classications. 35B30, 35B25, 35K55 1. Introduction. One of the leading continuum models for the dynamics of phase separation and coarsening in a binary mixture is the Cahn-Hilliard equation, which in the one-dimensional case can be written as ut = ( "2uxx +W0(u))xx; x2 (0; 1) ux = uxxx = 0; x2 f0; 1g: (1.1) Here W represents the bulk free energy density as a function of the concentration u of one of the two components of the mixture. (If, as is typically assumed, the total concentration of the mixture is a constant then the concentration of the second is determined by the concentration of the rst.) The parameter " represents an interaction length and is assumed to be a small positive constant. This equation was derived in [8] based on the free energy functional of van der Waals [29] E"[u] Z 1 0 W(u) + "2 2 juxj2 (1.2) dx: We will usually work with the scaled energy E"[u] " 1E"[u]. Also, we will write E"[u; a; b] when the integral is over the interval [a; b] instead of [0; 1]. In the early 1970s, Cahn and Morral [24] and DeFontaine [13] [14] initiated the study of systems of partial di erential equations that model the phase separation of mixtures of three or more components in essentially the same way that the Cahn- Hilliard equation models the separation of binary mixtures. (See Eyre [20] for a comprehensive survey of these systems.) If the domain is again taken to be [0; 1], then, after a change of variables, such systems can be written in the form ut = ( "2uxx +DW(u))xx; x2 (0; 1) ux = uxxx = 0; x2 f0; 1g; (1.3) where u is now an n-vector (for a mixture with n + 1 components), and W maps D(W) Rn into R. Again, E" dened by (1.2) represents the total free energy of Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, Atlanta, Georgia 30332. Present address, Department of Mathematics, Brigham Young University, Provo, Utah 84602. 21 22 CHRISTOPHER P. GRANT the mixture, and it is easy to check that it provides a Lyapunov functional for (1.3). Note, also, that the mass R 1 0 udx of a solution is conserved. We will make the following assumptions on W. D(W) is open, convex, and connected; W 0 throughout its domain, and W has only nitely many zeros, call them fz1; z2; : : : ; zmg, (corresponding to the preferred homogeneous states, or phases, of the system); W is C3 on D(W) and has a continuous extension to its closure D(W); The Hessian D2W is positive denite at each zero of W, and W is bounded away from 0 outside of each neighborhood of these points. Additionally, we need to require that W increases as the boundary @D(W) of the domain is approached. The precise assumption we shall make is the following: For each point u in @D(W), there is a closed, convex set S D(W) n u such that 1. W is nonzero on , the connected component of D(W) n S containing u; 2. the function ' that maps each point of Rn to its nearest point in S satises W('(u)) W(u) for all u 2 . This assumption is trivially satised when D(W) = Rn. It also holds whenever W is C1 on D(W), @D(W) is a locally compact, oriented hypersurface of class C2, and the (exterior) normal derivative of W is positive. (See, e.g., [21].) However, we state the assumption in this general way because some of the most important examples of D(W) do not have smooth boundaries. For example, Eyre [20] and Elliott and Luckhaus [18] study situations where D(W) is a convex polytope and W satises the assumptions given above. Note that any constant is an equilibrium solution to (1.3). A linear analysis of the equation about an unstable constant equilibrium suggests that typical solutions that start near such a constant undergo ne-grained decomposition with a characteristic length scale that is O("). (See [22] for a precise mathematical formulation and rigorous verication of this heuristic concept in the two-component case.) This ne-grained decomposition of initially homogeneous mixtures has also been frequently observed in physical experiments [7], [9]. In this paper we investigate the way solutions evolve after this initial stage of decomposition. We, therefore, conne our attention to solutions to (1.3) with initial data u(x; 0) = u0(x) close to the zeros of W through most of the domain, with sharp transition layers, or interfaces, separating the intervals where u is nearly constant. Consider when n = 1 (i.e., the original Cahn-Hilliard equation (1.1)), the case for which the most work has been done. Carr, Gurtin, and Slemrod [10] showed that all of the local minimizers of E" with any specied mass are monotone, so, in general, we would expect that the ne-grained structure of u would coarsen as t!1. Numerical work by Elliott and French [17] indicates that this evolution occurs very slowly. (Such slowly evolving states are sometimes said to be dynamically metastable.) Bronsard and Hilhorst [5] have shown that, in a certain space, this evolution occurs at a rate that is O("k) for any power k. Using completely di erent techniques, Alikakos, Bates, and Fusco [1] constructed a portion of the unstable manifold of a two-layer equilibrium that intersects a small neighborhood of a monotone equilibrium and showed that the speed of the flow along this connecting orbit, measured in the H 1 norm, is O(exp( C=")) for some constant C. Recently, Bates and Xun [4] have found exponentially slow motion for the multi-layer states of (1.1) by combining the methods of [1] with those used by Carr and Pego [11] to study reaction-di usion equations. SLOW MOTION IN ONE-DIMENSIONAL CAHN-MORRAL SYSTEMS 23 The results that we present here are similar to those of Bates and Xun in that we also obtain exponentially slow motion, but the methods we use are much simpler, and they are valid not only for the two-component Cahn-Hilliard equation (1.1) but for the multi-component Cahn-Morral system (1.3), as well. It should be mentioned, however, that our results for the two-component two-layer case are weaker than those of Alikakos, Bates, and Fusco, in the sense that we do not explicitly construct heteroclinic orbits. We deal only with the speed of motion and say nothing about the geometric structure of the attractor. In this paper, we apply the elementary, yet powerful, approach introduced by Bronsard and Kohn [6] in their study of slow motion for reaction-di usion equations. The improvement from superpolynomial to exponential speed is made possible by incorporating some ideas of Alikakos andMcKinney [2] about the prole of constrained minimizers of (1.2). Use is also made of techniques of Sternberg [27] for describing the nature of globally stable steady-state solutions of (1.3) in the limit as " ! 0. In Section 2 we present a lower bound on the energy of any function that is su ciently close to a given simple function whose range is a subset of W 1(f0g). This result amounts to an error estimate for a convergence result of Baldo [3]. In Section 3 we show how this estimate yields our main result on slow evolution of solutions with transition layers. As in [6], the only information used about the timedependent partial di erential equation is the time rate of change of the energy along a solution path in phase space. Finally, in Section 4 we consider what the main result implies about the motion of the transition layers themselves. The questions of existence and regularity of solutions for (1.1) and (1.3) have been extensively studied, and di erent authors have obtained various conditions on W that ensure global existence of solutions [15], [16], [18], [19], [20], [25], [26], [28], [30]. Rather than restricting ourselves to one particular set of such conditions, we shall simply assume that W is such that for su ciently smooth initial data with range in D(W) there exists a global solution that is in C(R+;H2(0; 1)) \ L2(0; T ;H4(0; 1)). Given that global solutions exist, our goal is to provide some information about how some of them evolve. 2. Error Estimates. Fix v : [0; 1] ! W 1(f0g) having (exactly) N jumps located at fx1; x2; : : : ; xNg (0; 1). Fix r so small that B(xk; r) [0; 1] for each k, and B(xk; r) \ B(x ; r) = ; whenever k 6= . (Here and below, B(x; r) represents the open ball of radius r centered at x in the relevant space.) Let j be the minimum of the eigenvalues of D2W(zj ), and let = minf j : zj 2 W 1(f0g)g: For any function z on [0; 1] we write z(x) R x 0 z(s) ds. We are interested in solutions corresponding to initial data u(x; 0) = u0(x) such that u0 is close to v in the L1 norm. To the discontinuous function v we assign an asymptotic energy E0[v] XN k=1 (v(xk r); v(xk + r)); where ( 1; 2) def = inf fJ[z] : z 2 AC([0; 1];D(W)); z(0) = 1; z(1) = 2g ; 24 CHRISTOPHER P. GRANT and J[z] def = p2 Z 1 0 p W(z(s))jz0(s)j ds: It is easy to check that is a metric on the domain of W. Also, note that Young's inequality and a change of variable imply that E"[z; a; b] (z(a); z(b)): Lemma 2.1. Let C be any positive constant less than rp2 . Then there are constants C1; > 0 (depending only on W, v and C) such that, for " su ciently small, Z 1 0 j u(x) v(x)j dx ) E"[u] E0[v] C1 exp( C="): Proof. Let K be a compact set in the domain of W containing W 1(f0g) in its interior, and set = supfkD3W( )k : 2 Kg. Choose ^r > 0 and 1 so small that Copyright (r ^r)p2 n 1 and that B(zj; 1) is contained in K for each zj 2 W 1(f0g). Choose 2 so small that inf ( 1; 2) : zj 2 W 1(f0g); 1 62 B(zj; 1); 2 2 B(zj; 2) > sup (zj; 2) : zj 2 W 1(f0g); 2 2 B(zj; 2) ; and jzj z j > 2 2 if zj and z are di erent zeros of W. Let F( 2) = inff ( 1; 2) :zj1; zj2 2 W 1(f0g); zj1 6= zj2 ; 1 2 B(zj1; 2); j( 2 zj2 ) (zj2 (2.1) zj1 )j 2jzj2 zj1 jg: By our assumptions about W, F( 2) > 0, so there exists M 2 N such that MF( 2) > E0[v]. Pick such an M, and set = ^r2 2=(5M2). Now assume that R 1 0 j u(x) v(x)j dx , and let us focus our attention on B(xk; r), a neighborhood of one of the transition points of v. For convenience, let v+ = v(xk + r) and v = v(xk r). Suppose ju vj 2 throughout (xk; xk + ^r), and let IM be an open subinterval of (xk; xk+^r) of width ^r=M. If we assume without loss of generality that E"[u] E0[v] then for " su ciently small there must be some ^x 2 IM such that u(^x) 2 B(zj1; 2) for some zj1 2 W 1(f0g). (Otherwise the rescaled bulk free energy would be too high.) If (u v) zj1 v+ jzj1 v+j 2 throughout IM then it is not hard to check that we would have Z IM j u(x) v(x)j dx Z IM ( u(x) v(x)) zj1 v+ jzj1 v+j dx > ; which is a contradiction. Hence, (u v) zj1 v+ jzj1 v+j < 2 SLOW MOTION IN ONE-DIMENSIONAL CAHN-MORRAL SYSTEMS 25 somewhere on IM. But then the rescaled energy on IM must be no less than F( 2). Partitioning (xk; xk +^r) into M equal intervals of width ^r=M and using the preceding result, we have E"[u; xk; xk + ^r] MF( 2) > E0[v], contrary to assumption. Hence, there is some r+ 2 (0; ^r) such that ju(xk + r+) v+j < 2: Similarly, there is some r 2 (0; ^r) such that ju(xk r ) v j < 2: Next, consider the unique minimizer z : [xk + r+; xk + r] ! Rn of the functional E"[z; xk + r+; xk + r] subject to the boundary condition z(xk + r+) = u(xk + r+): If the range of z is not contained in B(v+; 1) then E"[z; xk + r+; xk + r] inff (z(xk + r+); ) : 62 B(v+; 1)g (2.2) (z(xk + r+); v+); by the choice of 2 and the choice of r+. Suppose, on the other hand, that the range of z is contained in B(v+; 1). Then the Euler-Lagrange equation for z is z00(x) = " 2DW(z(x)); x2 (xk + r+; xk + r) z(x) = u(xk + r+); x= xk + r+ z0(x) = 0; x= xk + r: (2.3) If we dene (x) jz(x) v+j2 then 0 = 2(z v+) z0 and 00 = 2(jz0j2 + (z v+) z00) 2 "2 (2.4) (z v+) DW(z): Now Taylor's theorem and the choice of 1 imply that (2.5) DW(z) = D2W(v+)(z v+) + R; where jRj n jz v+j2=2. Substituting (2.5) into (2.4) gives 00 2 "2 (z v+) D2W(v+)(z v+) n "2 jz v+j3 2 "2 jz v+j2 n 1 "2 jz v+j2 2 "2 jz v+j2 = 2 "2 ; where = C=(r ^r). Thus, satises 00(x) ( =")2 (x) 0; x2 (xk + r+; xk + r) (x) = ju(xk + r+) v+j2; x= xk + r+ 0(x) = 0; x= xk + r: 26 CHRISTOPHER P. GRANT Following Alikakos and McKinney [2], we compare to the solution ^ of ^ 00(x) ( =")2 ^ (x) = 0; x2 (xk + r+; xk + r) ^ (x) = ju(xk + r+) v+j2; x= xk + r+ ^ 0(x) = 0; x= xk + r; which can be explicitly calculated to be ^ (x) = ju(xk + r+) v+j2 cosh [( =")(r r+)] cosh h " (x (xk + r)) i : By the maximum principle, (x) ^ (x), so, in particular, (xk + r) ju(xk + r+) v+j2 cosh [( =")(r r+)] 2 22 exp Copyright " : Consequently, jz(xk (2.6) + r) v+j 2p2 exp( C=(2")): Because W is quadratic at v+, (2.6) implies that, for some constant C1, E"[z; xk + r+; xk + r] (z(xk + r+); z(xk + r)) (z(xk + r+); v+) (v+; z(xk + r)) (2.7) (z(xk + r+); v+) (C1=(2N)) exp( C="): Combining (2.2) and (2.7), we see that the constrained minimizer of the proposed variational problem satises E"[z; xk + r+; xk + r] (z(xk + r+); v+) (C1=(2N)) exp( C="): But the restriction of u to [xk+r+; xk+r] is an admissable function, so it must satisfy the same estimate E"[u; xk + r+; xk + r] (u(xk + r+); v+) (C1=(2N)) exp( C="): A similar estimate holds for the energy of u on the interval [xk r; xk r ]. Hence, E"[u; xk r; xk + r] = E"[u; xk r; xk r ] + E"[u; xk r ; xk + r+] + E"[u; xk + r+; xk + r] (v ; u(xk r )) (C1=(2N)) exp( C=") + (u(xk r ); u(xk + r+)) + (u(xk + r+); v+) (C1=(2N)) exp( C=") (v(xk r); v(xk + r)) (C1=N) exp( C="): Assembling all of our estimates, E"[u] XN k=1 E"[u; xk r; xk + r] E0[v] C1 exp( C="); as was claimed. SLOW MOTION IN ONE-DIMENSIONAL CAHN-MORRAL SYSTEMS 27 3. Slow Evolution. In this section we will consider a family of solutions u"(x; t) to (1.3), parametrized by the corresponding interaction length ". Lemma 3.1. Suppose that C < rp2 and the initial data u" 0 satises Z 1 0 j u" 0(x) v(x)j dx 2 and E"[u" 0] E0[v] + 1 g(") for some function g and for all " small, where is as in Lemma 2.1. Then lim "!0 ( sup 0 t minfg(");exp(C=")g Z 1 0 j u"(x; t) u" 0(x)j dx ) (3.1) = 0: Proof. First note that the scaled total energy E"[u"( ; t)] of the solution of a Cahn-Morral system is nonincreasing in t, since d dt E"[u"( ; t)] = " 1 Z 1 0 DW(u") u"t + "2u" x u" xt dx = " 1 Z 1 0 (DW(u") "2u" xx) u"t dx = " 1 Z 1 0 j u"t j2 dx: Integrating this equation over t 2 (0; T) gives E"[u" 0] E"[u"( ; T )] = " 1 Z T 0 Z 1 0 j u"t j2 (3.2) dx dt: Next, assume that u" 0 satises the conditions of the lemma and that T is small enough that Z T 0 Z 1 0 j u"t j dx dt =2: Then Z 1 0 j u" 0(x) u"(x; T )j dx =2; so by the triangle inequality, Z 1 0 j u"(x; T ) v(x)j dx : Applying, Lemma 2.1 to u"( ; T) gives E"[u"( ; T )] E0[v] C1 exp( C="). In combination with (3.2), this yields Z T 0 Z 1 0 j u"t j2 dx dt = "(E"[u" 0] E"[u"( ; T )]) C1" 1 g(") + exp( C=") (3.3) ; 28 CHRISTOPHER P. GRANT assuming, without loss of generality, that C1 1. Using H older's inequality and (3.3) we have Z T 0 Z 1 0 j u"t j dx dt !2 Z T 0 Z 1 0 1 dx dt ! Z T 0 Z 1 0 j u"t j2 dx dt ! C1T" 1 g(") + exp( C=") : Hence, T 1 C1" 1 g(") + exp( C=") 1 Z T 0 Z 1 0 j u"t j dx dt !2 (3.4) : Now suppose that Z 1 0 Z 1 0 j u"t j dx dt =2: Then we can choose T such that R T 0 R 1 0 j u"t j dx dt = =2. For this choice of T , (3.4) yields T 2 4C1" h 1 g(") + exp( C=") i 2 8C1" min fg("); exp(C=")g : Then (3.3) implies that Z 2 minfg(");exp(C=")g=(8C1") 0 Z 1 0 j u"t j2 dx dt C1" 1 g(") + exp( C=") (3.5) : If, on the other hand, R 1 0 R 1 0 j u"t j dx dt < =2, then (3.3) must hold for every T ; therefore, (3.5) is also true for this case. Using H older's inequality and (3.5) we see that for " < 2=(8C1) sup 0 t minfg(");exp(C=")g Z 1 0 j u"(x; t) u" 0(x)j dx Z minfg(");exp(C=")g 0 Z 1 0 j u"t j dx dt min fg("); exp(C=")g Z minfg(");exp(C=")g 0 Z 1 0 j u"t j2 dx dt !1=2 min fg("); exp(C=")gC1" 1 g(") + exp( C=") 1=2 p 2C1": Letting " ! 0 we get (3.1). The strength of estimate (3.1) in Lemma 3.1 depends on the e ciency of the transition layers in the initial data. In Theorem 3.3 below, we show that, in a neighborhood of the step function v, there exist initial data that smooth out the discontinuities of v SLOW MOTION IN ONE-DIMENSIONAL CAHN-MORRAL SYSTEMS 29 in an e cient enough manner that the corresponding solutions of (1.3) evolve exponentially slowly. Before we present this theorem, we shall state and prove a technical lemma about the existence and regularity of minimizing geodesics for the degenerate Riemannian metric . Lemma 3.2. 1. For any two zeros zi and zj of W, there is a Lipschitz continuous path ij from zi to zj , parametrized by a multiple of Euclidean arclength, that realizes the distance (zi; zj ); i.e., (zi; zj) = J[ ij ]. 2. There exists a positive constant C2 such that j ij(y) zij C2y for y su - ciently small, and j ij(y) zjj C2(1 y) for y su ciently near 1. Proof. Recall that outside of a neighborhood of its zeros W is bounded away from 0; therefore, it is possible to nd a bounded set B D(W) such that if (0) = zi, (1) = zj, and J[ ] (zi; zj) + 1 then the image of is contained in B. Extend W continuously to B, and consider the problem of minimizing J[ ] over all satisfying these boundary conditions and having images contained in B. Now, J[ ] is a parametric integral, and it is known that this new minimization problem has an AC global minimizer ij [12]. The parameter of this minimizer can be chosen to be proportional to arclength, and then ij will be Lipschitz continuous. We claim that ij([0; 1]) is contained in D(W). Suppose it is not. Then there exists some y 2 (0; 1) such that ij(y) 2 @D(W). By the assumptions on W, there is a closed, convex set S D(W) n ij(y) such that W is nonzero on the connected component of D(W) nS containing ij (y), and the function ' that maps each point of Rn to its nearest point in S satises W('(u)) W(u) for all u in . Consider the modied path ij from zi to zj dened by ij (y) = '( ij (y)); if ij (y) 2 ij (y); otherwise : Note that ' is Lipschitz continuous with Lipschitz constant 1. Because of this and the fact that S separates from the rest of D(W), ij is Lipschitz continuous. It is also easy to check that J[ ij ] < J[ ij ]. This contradicts the optimality of ij ; hence, the claim holds. This veries that (zi; zj) = J[ ij ]. We now prove the estimate on ij near zi; the estimate near zj can be derived similarly. Again, we consider a modication of ij , this time the path ij dened by ij(y) = zi + (y= )( ij( ) zi); if 0 y ij(y); otherwise : The optimality of ij implies that p2 Z 0 q W( ij(s))j 0ij (s)j ds p2 Z 0 q W( ij(s)) j ij ( ) zij (3.6) ds: Because D2W(zi) is positive denite, there are positive constants M1 and M2 such that M1ju zij p W(u) M2ju zij in a small neighborhood of zi. Using this in (3.6), we nd that j ij( ) zij2 M3 Z 0 j ij(s) zij ds 30 CHRISTOPHER P. GRANT for some constant M3. Applying a variant of Gronwall's inequality [23] we obtain the desired estimate. Theorem 3.3. Given > 0, there exist constants C; ^" > 0 and a family of initial conditions fu" 0 : 0 " ^"g of (1.3) satisfying homogeneous Neumann boundary conditions and the estimate Z 1 0 j u" 0(x) v(x)j dx 2 such that the corresponding solutions u" of (1.3) satisfy lim "!0 ( sup 0 t exp(C=") Z 1 0 j u"(x; t) u" 0(x)j dx ) = 0: Proof. Lemma 3.2 shows that to each discontinuity xk of v there corresponds an optimal path connecting v(xk r) to v(xk + r). Note that it su ces to prove the present theorem under the assumption that none of these optimal paths passes through any zero of W (except at the endpoints of the path), since if the assumption is not satised then v can be perturbed slightly to create a new step function that does satisfy the assumption. Given ", set u" 0 = v outside of [mj =1B(xk; r). For xed xk, we shall again use the notation v for v(xk r) and will show that for " su ciently small we can dene u" 0 inside B(xk; r) in such a way that u" 0 is very close to v (in the L1 sense) on B(xk; r), E"[u" 0; xk r; xk + r] (v ; v+) + C3 exp( C=") for some C and C3, and u" 0 is continuous at the endpoints of B(xk; r). By taking C slightly smaller and applying Lemma 3.1, the proof of the theorem will then be complete. Let : [0; 1] ! Rn be an optimal path from v to v+ as described in Lemma 3.2. Let be the Euclidean arclength of . Let y : R ! [0; 1] be the solution of dy d = 1 p (3.7) 2W( (y( ))) satisfying y(0) = 1=2. (Since pW and are Lipschitz continuous, a unique C1 solution is guaranteed to exist.) Note that lim !1 y( ) = 1 and lim ! 1 y( ) = 0. Dene u" 0 inside B(xk; r) by u" 0(x) = 8>>>< >>>: v + ( (y (1 r=")) v )(x xk + r)="; xk r < x < xk r + " (y ((x xk)=")); xk r + " x xk + r " v+ + (v+ (y (r=" 1)))(x xk r)="; xk + r " < x < xk + r: It is easy to see that u" 0 is continuous and, for " su ciently small, will satisfy the L1 requirement; therefore, we only need to check the energy requirement. Note that E"[u" 0; xk r; xk + r] = Z r+" r 1 " W(u" 0(x + xk)) + " 2 ju"0 0 (x + xk)j2 dx + Z r " r+" 1 " W(u" 0(x + xk)) + " 2ju"0 0 (x + xk)j2 dx + Z r r " 1 " W(u" 0(x + xk)) + " 2ju"0 0 (x + xk)j2 dx def (3.8) = I1 + I2 + I3: SLOW MOTION IN ONE-DIMENSIONAL CAHN-MORRAL SYSTEMS 31 Now, using (3.7) and the denition of we have I2 = Z r " r+" 1 " W y x " + 1 2" 0 y x " y0 x " 2 dx = Z r=" 1 1 r=" W( (y( ))) + 1 2 j 0(y( ))y0( )j2 d = Z r=" 1 1 r=" p 2W( (y( )))j 0(y( ))jy0( ) d = Z y(r=" 1) y(1 r=") p 2W( (y))j 0(y)j dy (3.9) (v ; v+): Next, we estimate I1 (letting C3 represent a constant whose value may change from line to line): I1 = 1 " Z r+" r W v + (y (1 r=")) v " (x + r) dx + 1 2 y 1 r " v 2 C3 y 1 r " v 2 C3 y 1 r " 2 (3.10) : Now, Lemma 3.2 implies that there exists a constant C > 0 such that, for 0, y0( ) = 1 p 2W( (y( )) Copyright 2rC2 j (y( )) v j Copyright 2r (3.11) (y( )): Applying a simple comparison argument to (3.11) yields y( ) C3 exp Copyright 2r ; for 0. Substituting this into (3.10) we have (3.12) I1 C3 exp( C="): Similarly, (3.13) I3 C3 exp( C="): By substituting (3.9), (3.12), and (3.13) into (3.8), we see that u" 0 satises the energy requirement, so we are done. Remark. For the standard two-component case with W having two minima, the maximum principle can be used more directly in the proof of Lemma 2.1 (see [2]) 32 CHRISTOPHER P. GRANT and an explicit value of C can be obtained in Theorem 3.3. This C agrees with that obtained in [1] and [4]. Remark. The initial data u" 0 just constructed are in W1;1(0; 1). Since E" is continuous on this space and elements of this space can be approximated arbitrarily closely by Cp functions (for arbitrarily large p), the initial data in Theorem 3.3 can be assumed to be arbitrarily smooth. 4. Motion of Transition Layers. From Theorem 3.3, which establishes slow evolution in a certain abstract space, it is natural to infer that the transition layers themselves move extremely slowly. This concept can be made precise in a number of ways, one of which we present here. Fix some closed subset K of D(W) nW 1(f0g), and dene the interface I[u] of a function u by I[u] def = u 1(K): This terminology is natural, since the set K is bounded away from the phases of W, where the bulk energy is low. By analyzing how rapidly I[u] changes, we obtain information on how fast the transition layers move. Let d(A;B) denote the Hausdor distance between the sets A and B, i.e. d(A;B) = max sup a2A d(a;B); sup b2B d(b;A) : We shall show that d(I[u"( ; t)]; I[u" 0]) grows very slowly in t. Theorem 4.1. Fix ^r > 0 and ^ > 0. Then there exist constants C; ^" > 0 and a family of initial conditions fu" 0 : 0 " ^"g of (1.3) satisfying homogeneous Neumann boundary conditions and the estimate Z 1 0 j u" 0(x) v(x)j dx ^ such that the time T (^r) necessary for d(I[u"( ; T (^r))]; I[u" 0]) to exceed ^r satises (4.1) T (r^) exp(C="): Proof. Assume, without loss of generality, that ^r r. Choose ^ small enough that inf ( 1; 2) : zj 2 W 1(f0g); 1 2 K; 2 2 B(zj ; ^ ) > 4N sup (zj; 2) : zj 2 W 1(f0g); 2 2 B(zj ; ^ ) : Choose M 2 N so large that MF(^ ) > E0[v], where F is dened as in (2.1). We claim that there exists "0 > 0 such that for all " "0 and for all functions z : [0; 1] ! Rn satisfying Z 1 0 j z(x) v(x)j dx ^ ^r2 17M2 (4.2) and E"[z] E0[v] + 2N sup (zj; 2) : zj 2 W 1(f0g); 2 2 B(zj ; ^ ) (4.3) ; SLOW MOTION IN ONE-DIMENSIONAL CAHN-MORRAL SYSTEMS 33 we have d I[z]; fxkgN k=1 < ^r 2: Verication of Claim: Note rst that if " is su ciently small then for each k there exist xk 2 (xk ^r=2; xk) and xk+ 2 (xk+; xk + ^r=2) such that jz(xk ) v(xk )j < ^ . This follows as in the proof of Lemma 2.1. Now, suppose the claim is violated. Then, reasoning as before E"[z] XN k=1 E"[z; xk ; xk+] + inf ( 1; 2) : zj 2 W 1(f0g); 1 2 K; 2 2 B(zj ; ^ ) E0[v] 2N sup (zj; 2) : zj 2 W 1(f0g); 2 2 B(zj ; ^ ) + inf ( 1; 2) : zj 2 W 1(f0g); 1 62 K; 2 2 B(zj ; ^ ) > E0[v] + 2N sup (zj; 2) : zj 2 W 1(f0g); 2 2 B(zj ; ^ ) E"[z]; which is a contradiction. Thus, the claim is true. Apply Theorem 3.3 with = minf^ ; ^ ^r2=(17M2) to obtain a parametrized set of initial conditions fu" 0 : 0 " ^"g. Note that z = u" 0 satises (4.2) and, by the construction in the proof of Theorem 3.3, satises (4.3) for " su ciently small. Applying the claim we get d I[u" 0]; fxkgN k=1 < ^r 2 ; for " su ciently small. By Theorem 3.3, the triangle inequality, and the fact that E"[u"( ; t)] is decreasing in t, we see that there is a constant C > 0 such that for " su ciently small, z = u"( ; T ) satises (4.2) and (4.3) if T exp(C="). 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