HARALD GARCKE AND AMY NOVICK{COHEN modied chemical potentials which are prescribed as functional derivatives of the free energy functional E(u) = Z (u

Transcription

1 A SINGULAR LIMIT FOR A SYSTEM OF DEGENERATE CAHN{HILLIARD EQUATIONS HARALD GARCKE AND AMY NOVICK{COHEN Abstract. A singular limit is considered for a system of Cahn-Hilliard equations with a degenerate mobility matrix near the deep quench limit. Via formal asymptotics, this singular limit is seen to give rise to geometric motion in which the interfaces between the various pure phases move by motion by minus the surface Laplacian of mean curvature. These interfaces may couple at triple junctions whose evolution is prescribed by Young's law, balance of uxes, and continuity of the chemical potentials. Short time existence and uniqueness is proven for this limiting geometric motion in the parabolic Holder space C + 4 ; 4+ t; p ; < <, via parametrization of the interfaces.. Introduction Isothermal phase separation in multicomponent systems can be modeled by a system of Cahn{Hilliard equations. In such a system the diusional mobility matrix in general can be expected to be dependent on the concentration of the components. In regions in which one component is predominant, the mobility tends to be much smaller than in regions in which there is a mixture of dierent components. This implies that diusion in interfacial regions is relatively enhanced. In particular, this phenomenon becomes more pronounced at low temperatures where entropy contributions are smaller. Therefore, it is reasonable to consider systems of Cahn{Hilliard equations in which the mobility matrix depends on the concentrations of the dierent components (cf. [,, 3]). The mass balance for the various components then gives the following evolution equations for the concentrations u i ; i = ; :::; N, u i t = r J i ; J i = j= L ij (u)rw j ; () w i = D i (u) N (D (u) e) ui ; where e = (; :::; ) t and u = (u ; :::; u N ) t. The number N denotes the number of the components and J i is the ux of the i th component, which is taken here to be a linear combination of the thermodynamical forces rw j. The functions w i are 99 Mathematics Subject Classication. 35B5 35K65 35K55 8C4. Key words and phrases. Cahn{Hilliard equations, phase separation, multi{phase systems, surface diusion.

2 HARALD GARCKE AND AMY NOVICK{COHEN modied chemical potentials which are prescribed as functional derivatives of the free energy functional E(u) = Z (u) + jruj where the variational P derivative has been taken over the set of all functions which N satisfy the constraint i= ui =. This constraint reects the fact that the u i are dened here as volume (or molar) fractions of the components. In this paper we assume the homogeneous part of the free energy to be of regular solution type; i.e., (u) = i= u i ln u i u Au; where A is a N N matrix. The gradient energy coecient > that appears in the gradient part of the free energy is a small parameter and we will see that p is directly proportional to the thickness of the interfaces. Furthermore, E is taken to be dened on a bounded domain IR n with a suciently smooth boundary. As boundary conditions for the system (), we assume no{ux conditions and natural boundary conditions for the concentrations; i.e., J i = and ru i = where is the exterior normal Phase separation for multicomponent systems were rst studied by Morral and Cahn [3] and De Fontaine [3]. In particular their linear stability analysis suggested that systems of Cahn{Hilliard equations model spinodal decomposition in multicomponent systems. This was later supported by the numerical simulations of Eyre [] and Blowey, Copetti, and Elliott [5]. We also refer to the work of Barrett and Blowey [3, 4] who analyze a numerical method for a system of Cahn{Hilliard equations with a concentration dependent mobility matrix. They performed numerical simulations with a concentration dependent mobility which is taken to be small in the pure component and as expected they observed that the time scale of the evolution increases as the mobility in the pure component decreases. A derivation of the system () from basic thermodynamical principles was given by Elliott and Luckhaus [8] in the case of a constant mobility matrix. They also proved global existence and uniqueness. Elliott and Garcke [7] studied a system of Cahn{Hilliard equations with a concentration dependent mobility matrix which allowed the mobility to vanish for the pure component. They discussed properties of the mobility matrix and showed existence of a weak solution of the resulting system of fourth order degenerate parabolic equations. We also refer to the work of Tombakoglu and Ziya Akcasu [3] who derived the system () with a concentration dependent mobility matrix from a mean eld theory. The simplest choice of a concentration dependent mobility matrix which is physically relevant, has as its entries L ij (u) = u i ( ij u j ) i; j = ; :::; N () P (cf. [3, 7]). This matrix is symmetric and fullls the P N condition j= Lij = which N is necessary in order to ensure that the constraint i= ui = is preserved during the evolution. Although our results are also true for more general mobility matrices

3 A SINGULAR LIMIT FOR A SYSTEM OF DEGENERATE CAHN{HILLIARD EQUATIONS 3 (cf. [7]), we restrict our discussion to the form () in order to keep the presentation simple. Our goal in this paper is to relate the system () to a sharp interface model in the limit in which the interfacial thickness tends to zero as the time is rescaled appropriately. The rst result in this direction for a single Cahn{Hilliard equation with a constant mobility was obtained by Pego [6]. By the means of formal asymptotic expansions, Pego [6] showed that in the scaling = ", t! "t, as " & solutions of the scalar Cahn{Hilliard equation with constant mobility can be expected to converge to solutions of the Mullins{Sekerka evolution problem. Later Pego's result was justied rigorously in the radially symmetric case by Stoth [3] and by Alikakos, Bates, and Chen [] in the case that the limiting motion has a smooth solution. Cahn, Elliott and Novick{Cohen performed a formal asymptotic analysis for a single Cahn{Hilliard equation with a concentration dependent mobility. With the scaling = ", t! " t, they obtained motion by surface diusion as the singular limit near the deep quench; i.e. as "; &. This evolution law is dened purely in terms of local geometric quantities in contrast to Mullins{Sekerka evolution where the geometric data of the interfaces are coupled with the evolution of bulk elds. We also mention the work by Elliott and Garcke [6] and Escher, Mayer, and Simonett [9] for existence and stability results for diusive surface motion laws. The connection between a multiphase Ginzburg{Landau system and a sharp interface problem was rst demonstrated by Baldo []. He applied {convergence techniques to show that functions that minimize E over a set of functions with prescribed mass converge to solutions of a sharp interface problem which consists of nding a partition of into N phases which fullls a minimal interface condition. Here, each of the phases is constrained to have xed volume. Via formal asymptotic expansions, Bronsard and Reitich [8] related a vector valued Allen{Cahn equation to mean curvature evolution for interfaces which are coupled by an angle condition, known as Young's law [33], at triple junctions. They also showed existence of solutions for the limiting evolution problem. The angle condition at the triple junction can be viewed as a balance of mechanical forces and it arises in the formal asymptotic expansions as a solvability condition for a three layered stationary wave. Recently Bronsard, Gui and Schatzman [6] proved the existence of such a wave in the case of a symmetric three{well potential. Cahn and Novick{Cohen [] and Novick{Cohen [4] studied an Allen{Cahn/Cahn{ Hilliard system modelling simultaneous phase separation and ordering in binary alloys. Assuming a degenerate mobility, in [4] formal asymptotic expansions were applied to show that the singular limit of this system couples motion by mean curvature and quasi{static diusion with surface diusion. Triple junctions have also been derived recently by formal asymptotics in the context of a phase eld model for solidication of a eutectic alloy in the stationary case by Wheeler, McFadden and Boettinger [3]. In parallel with our work, Bronsard, Garcke and Stoth [6] studied an asymptotic limit of a system of Cahn{Hilliard equations with a constant mobility matrix. They obtained an evolution law that couples Laplace's equation in bulk

4 4 HARALD GARCKE AND AMY NOVICK{COHEN regions, the Gibbs{Thomson law on interfaces, Young's law at triple junctions to a dynamic law of Stefan type. For the limiting evolution they demonstrated a conditional existence result using an implicit time discretization and methods from geometric measure theory. In this paper we show that with the scaling = ", t! " t, the singular limit of () in the deep quench limit yields motion by minus the Laplacian of the mean curvature for each of the interfaces. At triple junctions Young's law, a condition that follows from the continuity of chemical potentials and a no{ux condition hold. At the intersection of an interface with an external boundary a Neumann type angle condition and a no{ux condition are obtained. Furthermore, we derive a fourth order parabolic boundary value problem such that solutions of this initial boundary value problem parametrize curves which solve this limiting geometric evolution problem. We prove local in time existence for this initial boundary value problem. Taking into account equivalence classes of solutions obtained by reparametrization of the evolutionary curves and allowing for variability in xing the tangential velocity uniqueness for the geometric problem is also proved.. Formal asymptotic expansions In what follows we consider the scaling t! " t and = " (cf. Cahn, Elliott, Novick{Cohen []) and write () as " 3 u i t = "r J i ; (3) J i = j= L ij (u)rw j ; (4) w i = D i (u) N (D (u) e) " u j ; (5) where is chosen to be of regular solution P type. We assume that A = (a ij ) i;j=;:::;n N is positive denite when restricted to fu j i= ui = g. The coecients a ij are pair interaction coecients of components i and j and since it is reasonable to assume that a phase does not interact with itself, a ii is taken here to be zero (cf. De Fontaine [3]). This implies that the minima of are of equal height in the limit &. The assumptions on A give rise to a N{well structure for when > is suciently small. Moreover, each of the N phases can be identied by the property that one of the N components is predominant. Throughout our discussion, we assume that = O("): The derivation of the outer and inner solutions will only be sketched. For a more detailed analysis in the binary case we refer to Cahn, Elliott, and Novick{Cohen [] and Novick{Cohen [4]... The outer expansion. Let us denote P by a(k), k = ; :::; N, the N local minima N of on the Gibbs simplex G = fu j i= ui = ; u i g. This implies that the a(k)

5 A SINGULAR LIMIT FOR A SYSTEM OF DEGENERATE CAHN{HILLIARD EQUATIONS 5 are solutions of the system of algebraic equations D i (u) (D (u) e) = i = ; :::; N: N Since the a(k) converge at the corners of the Gibbs triangle exponentially fast as " &, we can label the a(k) so that a k (k) C e C " ; a i (k) C e C " for i 6= k; where C ; C are constants which are independent of " for " suciently small and are therefore independent of. Let us consider now a region of the outer solution which is dominated by component k; i.e., u a(k). We make here the self{consistent ansatz that u is of the form u i = a i (k) + "u i + " u i + ::: e C " + H:O:T: for i = ; :::; N; J i = J i + "J i + " J i + ::: e C " + H:O:T: for i = ; :::; N; w i = "w i + " w i + ::: + H:O:T: for i = ; :::; N; where by H.O.T. we denote higher order terms... The inner expansion. We now analyze the behaviour of the solution in an interfacial region separating two phases under the assumption that we are nitely away from both triple junctions and external boundaries. The two phases which bound this interfacial region are denoted here as k and k. We employ the "classical" variables (; s) (cf. Caginalp and Fife [9] and Rubinstein, Sternberg, and Keller [7]) for the scaled distance function from the interface = d=" where d is the signed distance from the interface and s is an arc{length variable along the interface. We use the convention that the unit normal dening the sign of points into the domain denoted as k. Any given function v(x; t) written in terms of the new variables as v(t; ; s) satises rv v rs (6) v t v " t: (7) Here V = d t denotes the velocity of the interface. We now assume that there exist asymptotic expansions for u i ; J i ; w i, i = ; :::; N, given in terms of these variables; i.e., u i = u i + "u i + ::: ; i = ; :::; N; J i = J i + "J i + ::: ; i = ; :::; N; w i = "w i + " w i + ::: ; P P N such that i= ui N = and i= ui k =, k = ; ; :::. i = ; :::; N

6 6 HARALD GARCKE AND AMY NOVICK{COHEN At O(), for i = ; :::; N = n J i n J i = j= ; ; (8) L ij (u )w j ; ; (9) = D i (u ) N (D (u ) e) u i ; () where n denotes the unit normal to the interface and the notation v ; indicates the derivative of a function v with respect to the variable. Integrating (8) yields the existence of a function g i which is independent of such that n J i = g i (s; t); () and matching of the normal components of the uxes with the uxes in the outer region shows that g i (s; t) = T:S:T:; () where T.S.T. denote terms which are transcendentally small in " (cf. [4]). Equation () with boundary conditions dictated by the outer expansions, has a stationary wave connecting the minima of as its solution (cf. Sternberg [9]). We shall assume that the stationary wave solution has only two non{trivial components which we denote by k ; k ; i.e., This implies that Neglecting transcendentally small terms u i = T:S:T: for i 6= k ; k : (3) L ij (u ) = T:S:T: for i; j = fk ; k g: (4) L k k (u ) = L k k (u ) = L k k (u ) = L k k (u ) (5) and these terms are dierent from zero. Thus there are eectively only four nontrivial entries in L ij (u ); i; j = ; :::; N. From here on we employ the notation L ij := L ij (u ); i; j = ; :::; N. From (9), (), (4), and (5) it follows that there exists a function ~g(s; t) which is transcendentally small, such that on the interface between k and k, n J k = L k k w k = L k k ; + L k k w k ; + ~g(s; t) = w k ; w k ; + ~g(s; t) = g k (s; t): (6) From (6), since L k k 6= it follows that w k w k ; = ~g gk L k k : (7) By our assumptions on L k k and on the normal uxes in the outer regions, ~g g k = O() : L k k

7 A SINGULAR LIMIT FOR A SYSTEM OF DEGENERATE CAHN{HILLIARD EQUATIONS 7 Hence, matching the chemical potentials in the inner region with the chemical potentials in the outer region yields and where h(s; t) is not necessarily T.S.T.. At O("): = n J i w i = j= ~g g k = O e C " (8) w k w k = h(s; t) (9) ; + J i D ij u j N ;s ; () u D e u i ; k ;k u i ; ; () where the sign convention is used that the curvature k ;k is computed with respect to the normal pointing into the k phase and k ;k denotes the rst term in a regular perturbation expansion in " for k ;k. Since employing the Frenet formulas s J i = k ;k n J i, we conclude from () that s J i = T:S:T: and therefore () simplies to = n J i + J i ; ;s + T:S:T: : By matching with the outer solution, it follows that n J i = T:S:T:. Multiplying () by u i ;, integrating over (; ) with respect to the variable, and summing over i gives Z Z Z i= w i u i ; = i;j= Z i= D ij u j u i ; u i ; u i ; Z Integrating by parts and employing () yields Z i= w i u i ; = k ;k Z i= i= i= k ;k N D u e u i ; () u i ; : u i ; d + T:S:T:: (3) Now we dene k k to be the surface energy of the interface between the k phase and the k phase; i.e., Z k k := u i ; d i= where u i is the connecting orbit between the minima of corresponding to the k and k phases (cf. Sternberg [9]). From the above discussion, u i = T:S:T: for i 6= k ; k,

8 8 HARALD GARCKE AND AMY NOVICK{COHEN hence: k k := Z n u k ; + u k o ; d + T:S:T:: (4) Since the k and k phases are assumed to be distinct, k k P >. By construction N i= ui =. This implies now that u k + u k = + T:S:T:: (5) Employing (4) and (5) in (3) yields w k Z w k u k ;d + T:S:T: = k ;k k k ; (6) and this implies that to lowest order w k w k u k k k = k ;k k k : (7) Here [v] k k denotes the jump of a function v across the interface between the k and k phases with the sign convention that we subtract the value in the k phase from the value in the k phase. At O(" ), for i = ; :::; N, V u i ; = n J i J i : (8) ; ;s At this level, the equation of conservation of mass alone is needed. Integrating (8) between and, using the fact that ;s J i is T.S.T., and matching with the normal uxes in the outer region Z V u i k k = J i ;s + T:S:T: : (9) Taking i = k (or k ), to lowest order V u k Z k k = L k k and thus Dening Z w k ;ssd + L k k w k V u k k Z k = w k ;ss w k ;ss l k k := Z L k k which is a positive constant, we can rewrite (3) as V = lk k u k k k w k Employing (7) in (3) and using the fact that u k L k k ;ssd + T:S:T:; d : (3) d (3) ;ss w k ;ss : (3) k k = yields V k k = l k k k k k k ;ss : (33)

9 A SINGULAR LIMIT FOR A SYSTEM OF DEGENERATE CAHN{HILLIARD EQUATIONS 9 Here, the upper indices in V k k and k k indicate that the velocity and curvature are evaluated at the interface between k and k. Additionally, the subscript indicating the order of the expansion has been omitted..3. Expansion near the intersection with the external boundary. Let the interface between the k phase and the k phase be denoted by k k. At the intersections of k k with the external boundary three conditions must hold. (i) Attachment: An isolated interface k k does not detach from the The rational for the attachment constraint arises from the physical interpretation of the partitioning of the phases. (ii) A Neumann boundary condition: Let m(t) denote the point of intersection of the interface k k Following the arguments of Owen, Rubinstein, and Sternberg [5], Cahn and Novick{Cohen [] and Novick{Cohen [4] we introduce a rectangle R " whose sides are proportional to ". Let m(t) be located at the midpoint of one of the sides of the rectangle and let that side be tangent at m(t). To obtain the Neumann boundary condition, we introduce the variables = xm(t) = (; n) " where (; n) are orthogonal coordinates and is a variable which is tangent at m(t). We now multiply the equations for the modied chemical potentials by u i ;, integrate over the rectangle R ", and sum over the components i = ; :::; N. This gives to leading order = = Z R " i= ( ZR" u i ; D i (u ) N ((D (u )) e) ( i= u i N (D (u )) e i= u i ; u i P N Since i= ui =, the middle term in the above equation term vanishes. Hence = ( (u )) i= u i ; u i This identical form appeared also in [4] and the same arguments may now be employed to deduce that the contact angle with the external boundary is up to O(") accuracy. ) : ) : (iii) The no ux condition: Integration of the balance of mass equation for component k over the rectangle R " gives Z Z " 3 u k t = " r J k : R " R " It is easy to see that I := " 3 ZR " u k t = O(" vol(r " )) = O(" 3 ) :

10 HARALD GARCKE AND AMY NOVICK{COHEN Dening Z II := " r J k ; R " we can argue as in Novick{Cohen [4] to obtain II = O(" vol(r " )) = O(" ): In particular I = o(ii), and to leading order from the balance of ux condition, Gau' identity, and matching with the outer solution = II = Z = " " J k = L k k w k = "( e n ) w k ;s w k ;s ;s w k ;s Z en d + O(" ) = L k k d + O(" ) ; where denotes the exterior normal and e n denotes the unit vector pointing in direction of the coordinate n. Here we have used the fact that across the interface k k R only quantities involving k and k give contributions to leading order. Since Lk k d 6= and since the contact angle is (which implies that e n 6= ), we conclude that w k ;s w k ;s = : Matching with the inner solution and noting (7) then gives k ;k ;s = :.4. Conditions at the triple junction. Four dierent types of conditions must hold at any triple junction. The rst condition is a persistence condition; i.e., the condition three interfaces which meet at an isolated triple junction remain together. The second condition is Young's law; i.e., a condition which determines the angles between the interfaces which meet at the triple junction which results from a balance of mechanical forces. The remaining two conditions consist of a balance of ux condition and a continuity condition resulting from imposing continuity of the chemical potentials across the triple junction. Since the derivation of the rst two conditions is similar to the analysis which appears in the treatment of other problems which include triple junctions we do not present the details here. Instead we refer to the works of Bronsard and Reitich [8] and Novick{Cohen [4]. Without loss of generality, it is assumed that phases ; and 3 meet at the triple junction under consideration. Let i; j; k f; ; 3g be mutually dierent. Let the interface between i and j be denoted by k. The same convention is employed for k and l k. (i) Persistence: A single isolated triple junction does not pull apart. (ii) Young's law: Following the arguments in Bronsard and Reitich [8] and Novick{ Cohen [4] and our discussion of the Neumann condition, it is easy to demonstrate

11 A SINGULAR LIMIT FOR A SYSTEM OF DEGENERATE CAHN{HILLIARD EQUATIONS that Young's law holds at any triple junction (see also Bronsard, Gui and Schatzman [7]). This condition may be written as: sin = sin = 3 sin 3 where k represents the angle of the k phase at the triple junction and k := ij where ij is as dened in subsection.. (iii) A balance of ux condition: To proceed, let us integrate the mass balance equation over a triangle T " which is constructed so that its midpoint coincides with the triple junction m(t), its sides are proportional to ", and each of its sides is intersected by one of the three interfaces. Employing the stretched variable and dening and = x m(t) ; " I i := Z Z T " " 3 u i t II i := r J i ; T " it is easy to check that I i = O (" 3 ) and II i = O("). More precisely, we may conclude that that I i = o(ii i ) and may therefore be neglected. Thus, using Gau' theorem, matching with the outer and inner solutions, and employing the structure of the inner solutions (cf. [] and [4]), we obtain to leading order: = Z = T " r J i = " 3X k= " Z Z (@T ") k k ( " J i ds (34) L ij w j ;s ) e k ds; where ds refers to integration with respect to arc{length. Here, (@T " ) k, k = ; ; 3, refer to the edge of the triangle T " which intersects the k'th interface, k denotes the exterior normal to T " on (@T " ) k, and e k is the unit tangent along the k'th interface pointing away from the triple junction. By matching with the appropriate inner solutions, we see that along each of the 3 interfaces only two of the N chemical potentials vary to leading order. Hence, in total, at any given triple{junction only three chemical potentials denoted here as w, w and w 3, enter signicantly. Let us choose three mutually distinct numbers p; q; r f; ; 3g. Assuming that we are in the non-degenerate case in which < i <, i = ; ; 3, a triangle T " may be constructed such that p e p =. Since the component p varies only along interfaces q and r, we obtain from (34) taking i = p that to lowest order Z Z = " + " : (35) (@T ") q q e q L pr w r ;s w p ;s (@T ") r (r e r ) L pq w q ;s w p ;s

12 HARALD GARCKE AND AMY NOVICK{COHEN Evaluating the integrals in (35), we conclude from (3) that at the triple junction l pr w r ;s w p ;s q + lpq (w q s w p s) r = : (36) Here the notation (:) i indicates that the expression in brackets is evaluated at the interface i. Matching with the inner solutions and employing equation (7), gives that the above equation may be written as: l q q rp ;s + l r r qp ;s = : Dening := 3, := 3 and 3 :=, and noting that ij = ji, we conclude that l ;s = l ;s = l ;s : (iv) Continuity of the chemical potentials: It follows from (7), using the notation from above that w w 3 = ; w3 w = ; w w 3 = 3 3 : Under the assumption that each of the three chemical potentials are continuous across the triple junction, it follows that w = w 3 ; w = w 3 ; w 3 = w 3 ; and we may sum the above equations to obtain: = : (37) 3. The limiting motion and geometric properties of the evolution Let us formulate the limiting motion for a three phase system. Generalization to more phases may be obtained in a straightforward manner. The interfaces shall be denoted by i (t), i = ; ; 3. The quantities i ; l i and i will be dened as in subsection.4. Furthermore, we set V := V 3 ; V := V 3 ; V 3 := V : We remark that the velocities V ij are antisymmetric in i and j; i.e., V ij = V ji. The free boundary problem for the interfaces may now be stated as (A) Along each interface i, V i = l i i i ;ss, (B) At each intersection of i with the external boundary, the following conditions hold (i) an attachment condition, (ii) the contact angle is, (iii) a no{ux condition ;s =. (C) At each triple junction (i) the triple junction does not pull apart,

13 A SINGULAR LIMIT FOR A SYSTEM OF DEGENERATE CAHN{HILLIARD EQUATIONS 3 (ii) the angles of the phases at the triple junction fulll Young's law [33]; i.e.: sin = sin = 3 sin ; (38) 3 which may also be written as a balance of mechanical forces (cf. []) 3X i= i i = (39) where i denotes a unit tangent to the interface i pointing away from the triple junction. (iii) a balance of uxes condition holds: l ;s = l ;s = l ;s : (iv) the chemical potentials are continuous across a triple junction; this implies that = : (4) Let us now state some geometric properties. It is easy to demonstrate that the total area occupied by each of the dierent phases is preserved. We leave the simple proof to the reader. A further property which we wish to show is that the energy E(t) = 3X i= i L i (t) is non{increasing. Here by L i (t) we denote the length of i (t). In this context we remark that Baldo ([]) has been proven that the weighted sum of lengths E is the {limit as "! of the free energy E. For simplicity we state the property for three curves which each intersect the external boundary at one endpoint and which all meet at a single triple junction at their other endpoint. A generalization to more complex networks of interfaces is straightforward. Claim: The total energy is a non-increasing function of time; moreover, d 3X dt E(t) = i= i l i Z i (t) i ;s ds; where ds denotes integration with respect to arc{length. Proof. By the transport theorem for evolving curves (see for example Gurtin [] Section I.): Z Z d dt Li (t) = i V i ds i _R i (4) i i (t) where i denotes the unit tangent to i (t) pointing into the interfaces at their endpoints and _R i is the velocity of the endpoint. For an endpoint i (t) lying on the exterior we conclude from the angle condition that i _R i =.

14 4 HARALD GARCKE AND AMY NOVICK{COHEN Therefore, only triple points can contribute to the boundary integral. Summing (4) over the three interfaces each weighted with the appropriate interfacial energy, and employing the law (A) of interfacial motion yields d 3X Z 3X dt E(t) = i i l i i i ;ssds i i _m(t) : (4) i= i (t) Here, m(t) denotes the location of the triple junction. By (39) the last term in (4) vanishes. Integration by parts and using the no{ux condition i ;s = at the external boundary gives d 3X Z dt E(t) = ( i ) l i i 3X ;s ds + i l i i i ;s (43) m(t) i= i (t) where the notation (:) m(t) indicates that the term in the bracket has been evaluated at the point m(t). Employing (iii) and (iv) in (C), the last term in (43) can be seen to vanish which proves the claim. i= i= 4. A local existence theorem for the limiting evolution problem In what follows we restrict ourselves to a situation in which three phases are present which meet at exactly one triple junction. The study of more complex geometries can be treated in a similar manner. We describe the three interfaces i, i = ; ; 3, by functions X i : [; T ] [; ]! IR ; (t; p) 7! X i (t; p) such that for t [; T ], X i (t; ) is a parametrization of i (t) whose properties will be specied in the discussion below, and we formulate an initial boundary value problem in terms of the parametrizations (X i ) i=;;3 whose solution will parametrize curves that solve the evolution problem (A){(C) derived in section 3. We then prove local existence within the framework of the parabolic Holder spaces C + 4 ; 4+ t; p ; < < relying on the results of Solonnikov [8]. First we derive a partial dierential equation for X i such that the evolutionary curves parametrized by X i fulll the law of motion which may be written equivalently as V i = l i i i ;ss ; X i t N i = l i i i ;ss : Here, N i is the unit normal to i (t). We use the convention that the sign of N i is chosen so that (T i ; N i ) is positively oriented, where T i = Xi ;p jx i ;pj is a unit tangent. The Frenet formulas imply that X i ;ss = i N i ;

15 A SINGULAR LIMIT FOR A SYSTEM OF DEGENERATE CAHN{HILLIARD EQUATIONS 5 where the index s indicates as before dierentiation with respect to arc{length; i.e., for a given function f(p) dened on the curve, f ;s = f;p : jx ;pj Setting and X i ;pp i := T i jx;pj i i := 3 i i ;s + i ;ss i i ; (44) we obtain, omitting the index i for convenience ;ss N + T = X ;pppp jx ;p j 4 5 X ;ppp jx ;p j 3 ;sn ;s N ;s X ;pp jx ;p j N X ;pp jx ;p j : The last equality implies that a solution of the partial dierential equation i l X i i t = X ;pppp i jx;pj + X i i 4 5i ;ppp jx;pj + i 3 i ;s i N i + i i ;sn i + (45) X i ;pp X i ;pp + i ;s jx;pj i (i ) 3 N i + ( i ) jx;pj i fullls V i = i l i i ;ss ; since (45) may also be written as i l i X i t = i ;ssn i i T i : (46) To keep the presentation simple, we dene a IR {valued function G such that (45) is equivalent to X i t = i l i X i ;pppp jx i ;pj 4 + G(X i ;p; X i ;pp; X i ;ppp) i = ; ; 3: (PDE) We remark that G is smooth as long as its rst variable is bounded away from zero. There is some degree of freedom in specifying the parametrization of the evolutionary curves (see the next section), and we nd it convenient to make the following choice at the endpoints p = ; jx i ;p(t; )j = jx i ;p(t; )j = ; i = ; ; 3 :

16 6 HARALD GARCKE AND AMY NOVICK{COHEN This choice yields the following formulation of the boundary condition at the triple junction (cf. (C) in Section 3): 3X i= X (t; ) = X (t; ) = X 3 (t; ); 3X i= jx i ;p(t; )j = ; i = ; ; 3; (II) i X i ;p(t; ) = ; i X i ;pp(t; ) N i (t; ) = ; l ;s = l ;s = 3 l 3 3 ;s: (I) (III) (IV) Assuming is described as the {level set of a smooth function b : IR! IR such that is a regular value, we obtain as boundary conditions at the external boundary that for i = ; ; 3 (cf. (B) in Section 3) b(x i (t; )) = ; X i ;p b ;x (X i )(t; ) X i ;p b ;x (X i )(t; ) = ; i ;s(t; ) = ; jx i ;p(t; )j = : (V) (VI) (VII) (VIII) Altogether this yields boundary conditions at p = and p = for a system of fourth order parabolic equation which contains six equations. This is precisely the number of conditions which one would expect for this type of system. To show local in time existence we follow the approach of Bronsard and Reitich [8] where a second order system was studied arising in the context of motion by mean curvature. But since our system is of fourth order the analysis here is more complicated. In order to solve the system of equations (PDE) together with the boundary conditions (I){(IX) and given initial conditions, we choose parametrizations X i Cp 4+ ([; ]; IR ), i = ; ; 3, of the initial curve which satisfy (IX) jx i ;p()j = jx i ;p()j = for i = ; ; 3: (47) The existence proof will be carried out by a contraction argument on the set (T; M) = f(x ; X ; X 3 ) C + 4 ;4+ t;p [; T ] [; ]; IR 3 X i (; ) = X i and j X i j + 4 ;4+ M for i = ; ; 3 g: The numbers T and M will be chosen later. Here C + 4 ;4+ t;p dened as a parabolic Holder space (cf. Solonnikov [8], Chapter IV) with the norm j : j + 4 ;4+, and the space (T; M) is taken to inherit this norm. Assume ( X ; X ; X 3 ) (T; M) is given. Then linearizing equation (PDE) gives j [; T ] [; ]; IR is X i t + i l i X i ;pppp jx i ;pj 4 = i l i X i ;pppp jx i ;pj 4 i l i X i ;pppp j X i pj 4 + G( X i ;p; X i ;pp; X i ;ppp) : (P DE L )

17 A SINGULAR LIMIT FOR A SYSTEM OF DEGENERATE CAHN{HILLIARD EQUATIONS 7 This equation is to be solved in conjunction with linearized versions of the boundary conditions (I){(IX) which we shall denote by (I L ){(IX L ). The conditions (I) and (III) are already linear. The linearized boundary conditions (II L ) and (IV L ) are 3X i= X i ;p()x i ;p(t; ) = X i ;p() X i ;p(t; ) i X i ;pp(t; )N i () = 3X i= i X i ;pp(t; )N i () j X i ;p(t; )j ; (II L ) 3X i= i X i ;pp(t; ) N i (t; ) (IV L ) where N i is the unit normal to the parametrization of the initial curve X i and N i is the unit normal to X i. Using the identity i ;s = X i ;ppp jx i ;pj 3 N i 3 i i ; we can linearize the boundary condition (V) as i l i X i ;ppp(t; )N i () j l j X j ;ppp(t; )N j () = i l i X i ;ppp(t; )N i () j l j X j ;ppp(t; )N j () i l i i ;s(t; ) + j l j j ;s(t; ) (V L ) for (i; j) = (; ) and (i; j) = (; 3). The boundary conditions at the point p = can be linearized similarly to give linear conditions (V I L ){(IX L ). We now demonstrate that the boundary conditions (I L ){(IX L ) together with equation (P DE L ) fulll the complementary conditions in the sense of Solonnikov (cf. [8]). This will allow us to apply the C {theory for linear parabolic systems. Introducing the notation j= (u ; :::; u 6 ) := (X ; X ; X ; X ; X 3 ; X 3 ); the boundary conditions (I L ){(IX L ) may be written in the form 6X B qj u j (t; p) = q (t; p) for q = ; :::; ; and p = ; (cf. Solonnikov [8], Bronsard and Reitich [8]). Here, the q are dened to be the right hand sides of equations (I L ){(IX L ). The B qj 's are polynomials with respect and here each qj is homogeneous of a degree that is denoted here by q which does not depend on j. The index indicates that this is the degree at the point p =. We use the notation q to indicate the degree of the polynomial B qj at the point p =. Therefore, using the notation of Solonnikov [8], we obtain that B qj = Bqj where Bqj is the principal part of B qj. Below we discuss only the boundary condition at the point p =, as the analysis for p = is similar but simpler. Dening (X ; X ;Y ; Y ; Z ; Z ) := (X ;p(; ); X ;p(; ); X ;p(; ); X ;p(; ); X 3 ;p(; ); X 3 ;p(; ));

18 8 HARALD GARCKE AND AMY NOVICK{COHEN we get ( Bqj (; i) ) j=;:::;6 = ( ; ; ; ; ; ) for q = ; = ( ; ; ; ; ; ) for q = ; = ( ; ; ; ; ; ) for q = 3; = ( ; ; ; ; ; ) for q = 4; = (X i ; X i ; ; ; ; ) for q = 5; = ( ; ; Y i ; Y i ; ; ) for q = 6; = ( ; ; ; ; Z i ; Z i) for q = 7; = ( i ; ; i ; ; 3 i ; ) for q = 8; = ( ; i ; ; i ; ; 3 i) for q = 9; = ( X ; X ; Y ; Y ; 3 Z ; 3 Z ) for q = ; = ( l X i 3 ; l X i 3 ; l Y i 3 ; l Y i 3 ; ; ) for q = ; = ( ; ; l Y i 3 ; l Y i 3 ; 3 l 3 Z i 3 ; 3 l 3 Z i 3 ) for q = where i = p. The principal part of the equation (P DE L ) is described by the matrix where L @p ) = (l kj) k;j=;:::;6 l kk = i l i jx;pj l kj = if k 6= j ;! k + if i = where [:] is the Gau bracket and indicates the integer part. To formulate the complementary condition we dene! 3Y L := det L (t; p; r; i) = j l j jx;pj 4 + r ; j 4 ^L = ^l kj i.e., for p = ; we get ^l kk = := L L 3Y 3Y j l j 4 + r j= j= j6=i ; j= ^l kj = if k 6= j; k + : j l j 4 + r if i = The complementary condition at the point p = as formulated by Solonnikov [8] requires that the rows of the matrix A(t; p; r; i) := B ^L ;

19 A SINGULAR LIMIT FOR A SYSTEM OF DEGENERATE CAHN{HILLIARD EQUATIONS 9 are linearly independent for all r CI and Re r >, modulo the polynomial M + (r; ) = 3Y j= j (r) j (r) : All functions here are to be interpreted as polynomials in, and by j l (l = ; and j = ; ; 3) we denote those roots of L(t; ; r; i) which have positive imaginary part. To determine the roots j l we must calculate the roots of r + j l j 4. Therefore writing r in polar coordinates; i.e., r = jrje i with ( ; ), we nd that for each j = ; ; 3 there are precisely two roots with positive imaginary part, namely j = 4p r jrj 4 j l ei( + j 4 4 ) ; j = 4p r jrj 4 j l ei( + 3 j 4 4 ) : To decide whether or not the complementary condition is satised, we must determine whether there exists a nontrivial vector (! q ) q=;:::; such that X q=! q B qj()^l jj() mod 3Y l= l l for j = ; :::; 6, or equivalently whether there exists a nontrivial vector (! q ) q=;:::; such that X j + q=! q Bqj() mod k k where k = Thus we must decide whether the set of twelve linear equations X q= X q=! q B qj( k ) = ;! q B qj( k ) = ; and j = ; :::; 6: j+ where k = and j = ; :::; 6, has a nontrivial solution (!q ) q=;:::;. To derive a more practical version of the complementary condition, we calculate the determinant of the matrix which corresponds to the linear system from above and check the conditions at which the determinant vanishes. Using the property that determinants are multilinear, up to a non-zero factor we must calculate the

20 HARALD GARCKE AND AMY NOVICK{COHEN determinant of the matrix D = X X X X X X Y Y Y Y Y Y Y Y 3 Z Z Z 3 3 Z Z Z 3 i X i X i X i X i X i X i Y i Y i Y i Y i Y i Y i Y i Y i 3 Z i Z i Z 3 i 3 Z i Z i Z 3 C A ; where we have employed the notation j = ( j l j ) 4. Assuming that the interfaces are labeled sequentially and counterclockwise, it is easily seen that j= sin = Y Z Y Z ; sin = Z X Z X ; sin 3 = X Y X Y ; where j denotes the angle opposite j, the j'th interface, at the triple junction. Employing the above identities and the symbolic calculation routines of MAPLE, we nd! 3Y 3X 3!! X det D = 8( + i) j 4 3X 3 j j ( j ) 3 j j sin j + j sin j 5 : j= Therefore, the determinant can only vanish if This condition is fullled if j= sin j = for j = ; and 3: < j < for j = ; and 3: (48) Other cases are not possible. If one angle is zero then by Young's law and the positivity of the surface energies j, the two others have to be and this would imply that the determinant is zero. The case in which one angle is can also not occur. This can be seen as follows. Assume without loss of generality that 3 =. It then follows that T = T and therefore the balance of tensions (39) implies that ( )T + 3 T 3 = ; j=

21 A SINGULAR LIMIT FOR A SYSTEM OF DEGENERATE CAHN{HILLIARD EQUATIONS hence, 3 =. This, however, cannot be admitted within the framework of our analysis because it would imply degeneracy of the equation of evolution for X 3. To check the complementary condition at p = is less complicated and no additional conditions arise. Remark. It is easy to check that angles ; ; 3 satisfying Young's law (38) satisfy condition (48) if and only if the positive surface tensions ; ; 3 fulll i < j + k (P artial ) for all permutations (i; j; k) of (; ; 3). This condition is referred to in the literature as a partial wetting condition on the surface tension (cf. de Gennes [4]) and we will henceforth impose this condition on our system. We shall assume, moreover, that X i Cp 4+ ([; ]) fulll the compatibility condition of order for the initial boundary value problem to ((P DE L ), (I L ){(IX L )); this means that the initial condition satisfy (I L ){(IX L ) as well as further compatibility conditions which formally are derived by dierentiating conditions (I L ) and (V I L ) with respect to time (see Solonnikov [8] for details). The compatibility condition for (I L ) implies that at the point p = l X ;pppp + G(X;p; X;pp; X;ppp) = l X ;pppp + G(X;p; X;pp; X;ppp) ; (CI ) and l X ;pppp + G(X ;p; X ;pp; X ;ppp) = 3 l 3 X 3 ;pppp + G(X 3 ;p; X 3 ;pp; X 3 ;ppp) : (CI ) The compatibility condition for (V I L ) at the point p = reduces to i l i X i ;pppp + G(X i ;p; X i ;pp; X i ;ppp) X i ;p = i = ; ; 3: (CV I i ) Finally, we assume that the X i, i = ; ; 3, are proper parametrizations; i.e., that jx i ;p(p)j is uniformly bounded away from zero for p [; ] and i = ; ; 3. If in addition the ( i ) i=;;3 are positive and fulll the partial wetting conditions (P artial ) we obtain Lemma 4.. Let X (T; M); and let X i Cp 4+ ([; ]; IR ), i = ; ; 3, satisfy the compatibility conditions of order. Then there exists a unique solution X i C + 4 ;4+ t;p, i = ; ; 3 of ((P DE L ), (I L ){(IX L )), such that X i () = X. i Moreover there exists a constant C >, such that the inequality 3X j X i j + 4 ;4+ C j ( X ;p; j X ;pp; j X ;ppp; j X ;pppp) j j 4 ; + j X j j 4+ j= X + C j q (:; ) + j j+ q q (:; ) j+ q 4 4 is valid. q=

22 HARALD GARCKE AND AMY NOVICK{COHEN Proof. The lemma follows from Theorem 4.9 of Solonnikov [8]. To prove local existence for the nonlinear problem it suces to demonstrate that the operator which maps ( X ; X ; X 3 ) to a solution (X ; X ; X 3 ) of ((P DE L ), (I L ){ (IX L )) maps the set (T; M) into itself and is a contraction for T suciently small and M suciently large. This however is straightforward employing the a priori estimates given in Lemma 4. (see Bronsard and Reitich [8] for the details in the second order case). Therefore we can conclude that under the assumption that the ( i ) i=;;3 are positive and fulll the partial wetting conditions (P artial ) the following local existence result holds. Theorem 4.. Let X i Cp 4+ ([; ]; IR ), i = ; ; 3, satisfy the compatibility conditions of order. Then there is a time T > and a positive real number M such that for all M > M there exists a unique solution X (T; M) of ((P DE), (I){(IX)). In the next section we see what geometric constraints on the initial curves are implied by the above compatibility conditions. 5. Uniqueness and the geometric evolution problem Although the contraction argument in the preceding section gives existence and uniqueness for the initial boundary value problem ((PDE), (I){(IX)), there remains the question of identifying geometrically admissible initial data (curves) and resolving the possibility of non{uniqueness for the geometric evolution problem. We amplify this issue as follows. A given set of three curves in the plane which we wish to take as initial data or a set of three evolving curves which we envision as a possible solution to the evolution problem (A){(C) formulated in Section 3 may be described by a family of smooth parametrizations. This is due to possible reparametrization of the curves. Thus any such set of curves can be equated with an equivalence class of parametrizations which may be used to describe it. Therefore, the rst question to resolve is to determine which initial data (curves) are geometrically admissible in the sense that they admit a smooth solution in the following sense: Denition 5.. The geometric evolution problem (A){(C) with initial curves parametrized by functions X i Cp 4+ ([; ]; IR ), i = ; ; 3, has a smooth solution on the time interval [; T ] if and only if there exist parametrizations Y i : [; T ] [; ]! IR ; i = ; ; 3; and functions i : [; ]! [; ], i = ; ; 3, such that i) Y i C + 4 ;4+ t;p [; T ] [; ]; IR ; i Cp 4+ ([; ]), ii) Yt i N i = i l i i ;ss, iii) the curves parametrized by Y i (t; ), t T, fulll { the persistence conditions, { the angle conditions, { the balance of uxes

23 A SINGULAR LIMIT FOR A SYSTEM OF DEGENERATE CAHN{HILLIARD EQUATIONS 3 both at the points of intersection with the exterior boundary and at the triple junction as well as the condition of continuity of the chemical potentials at the triple junction; i.e., the conditions (B) and (C) of Section 3 are satised. iv) Y i (; (p)) i = X(p); i v) () i =, () i =, and d dp (i ) (p) > for all p [; ]. We shall see that the initial curves need to fulll in addition to (B) and (C) only a single geometric compatibility condition in order to guarantee that there exists parametrizations of these initial curves which fulll the compatibility conditions of order which are needed to apply the existence theorem of Section 4. Another issue we want to address is to verify uniqueness of the ow within the equivalence class of possible reparametrizations. In the discussion which follows, we take into account the variability in the prescription of the tangential velocity along the interfaces { to be more precise: any velocity is permissible which guarantees that the resulting evolution satises the geometric conditions (A){(C) as well as the regularity requirements of Denition 5.. We remark that the geometric conditions (A){(C) do not allow for an arbitrary prescription of the tangential velocities, as the attachment condition and the persistence condition partially determine the tangential velocity at the external boundary and at the triple junction (see (5) and (5) below). Throughout the remainder of this section we assume that the surface tensions ( i ) i=;;3 are positive and fulll the partial wetting condition (P artial ). In addition, we use a subscript to indicate that a term is evaluated at time zero. The following theorem makes precise our claim that it is possible to obtain a smooth solution of the geometric evolution problem (A){(C) provided the initial data fulll a geometric compatibility condition. Theorem 5.. Suppose the initial curves are parametrized by functions X i Cp 4+ ([; ]; IR ), i = ; ; 3, and assume that the initial curves fulll (B), (C) and the geometric compatibility condition ( ) l ;ss + ( ) l ;ss + ( 3 ) l 3 3 ;ss = ; (G eom ) where i indicates the curvatures of the initial curves. Then there exists a time T > such that the geometric evolution problem (A){(C) with initial data parametrized by X i, i = ; ; 3, has a solution on the time interval [; T ] in the sense of Denition 5.. Remark. i) The geometric compatibility condition (G eom ) formally follows from the fact that if the evolution is smooth the velocity _m(t) of the triple junction m(t) has to satisfy _m() N i = i l i i ;ss ; i = ; ; 3: This, however, is only possible if (G eom ) is fullled. ii) A similar such theorem may be stated for triple junction motion within the context of motion by mean curvature (see Bronsard and Reitich [8]). The analogous geometric

24 4 HARALD GARCKE AND AMY NOVICK{COHEN compatibility condition would be then = ; and a proof of the suciency of such a condition can be given following closely the arguments of the proof of Theorem 5. given below. Proof of Theorem 5.. To apply the local existence theorem of the preceding section, the parametrization of the initial curves need to fulll the compatibility conditions of order of the initial boundary value problem ((P DE L ), (I L ){(IX L )). Since the initial curves fulll (B) and (C) the remaining compatibility conditions are (47), (CI ), (CI ) and (CV I i ), i = ; ; 3. In the terminology of (44){(46), the compatibility conditions (CI ) and (CI ) at the triple junction can be stated as l ;ssn + T = l ;ssn + T = 3 l 3 3 ;ssn T 3 ; (49) where the subscript has been used to indicate that the quantities have been evaluated at time t =. Using the geometric compatibility condition (G eom ), it is possible to check that in order to fulll (CI ) and (CI ) the, i i = ; ; 3, must be given by i = "( j ) l j j T i T k T i ( i ) l i ;ss + ( k ) l k k T j N i T k ;ss N i T j for permutations (i; j; k) of (; ; 3). We remark that the right hand side of (5) is invariant under reparametrization. Similarly at points of intersection of an interface with the external boundary the compatibility conditions (CV I i ), i = ; ; 3 may be written as i.e., we must require that # (5) i l i i ;ss N i + i T i T i = ; (5) i = ; i = ; ; 3; (5) at points of intersection of an initial interface with the external boundary. Let us now choose a suciently smooth reparametrization ~X i (p) = X i ( i (p)) of the initial curves in such a way that the ~X i, i = ; ; 3, fulll (47), (5) and (5) at the end points. Noting the denition (44) of i, it is easy to see that a prescription of i at p = or p = via (5) or (5) and the fulllment of condition (47) can be achieved by solving a fourth order ODE for i locally near the ends of [; ]. In this manner, the initial parametrization may be redened to yield a parametrization fullling the compatibility conditions of order. The statement of the theorem now follows from Theorem 4.. In terms of uniqueness for the geometric problem, our results may be stated as follows

25 A SINGULAR LIMIT FOR A SYSTEM OF DEGENERATE CAHN{HILLIARD EQUATIONS 5 Theorem 5.3. Assume C 4+ {initial curves which fulll the boundary conditions (B), (C), and the geometric compatibility condition (G eom ) are given. Let X i Cp 4+ ([; ]; IR ), i = ; ; 3, be parametrizations of the initial curves which fulll the compatibility conditions of order and let (X ; X ; X 3 ) (T; M) be the solution of ((PDE), (I){(IX)) for the initial data (X) i i=;;3 which was obtained in Theorem 4.. Furthermore, assume that (Y ; Y ; Y 3 ) is a solution of the geometric evolution problem (for the same initial curves) in the sense of Denition 5.. Then there exists a time T > and functions i C + 4 ;4+ t;p ([; T ] [; ]; [; ]) which satisfy i (t; ) =, i (t; ) (i ) (t; p) > for all p [; ]; t [; T ] such that X i (t; p) = Y i (t; i (t; p)): Remark. i) In the proof of Theorem 5. we demonstrated that C 4+ {initial curves which satisfy (B), (C) and (G eom ) can be parametrized by functions X i Cp 4+ ([; ]; IR ), i = ; ; 3, such that the compatibility conditions of order zero are satised. ii) Theorem 5.3 shows that the geometric evolution problem dened by (A){(C) in Section 3 has only one geometrically distinct smooth solution in the sense that two smooth parametrizations of evolving curves which fulll (A){(C) and which parametrize the same curves at the initial time, parametrize the same curves at later times throughout their common interval of existence. The maximal interval of existence may depend on the parametrization, though it is possible to formulate a maximal \geometric" interval of existence if one allows for reparametrizations. Proof of Theorem 5.3. Our approach is to derive an initial boundary value problem for reparametrizations ( i ) i=;;3 such that Z i (t; p) := Y i (t; i (t; p)) (53) is a solution of ((PDE), (I){(IX)) whenever i is a suciently smooth solution of this initial boundary value problem. The uniqueness statement in Theorem 4. then implies that X i = Z i, i = ; ; 3, and proves the theorem. Let us formally derive an initial boundary value problem which should be satised by the functions i. Our goal is to nd functions i such that Z i, i = ; ; 3, satises Z i t = i l i i ;ssn i + i T i ; (54) where i is dened as in (44) in terms of the parametrization Z i. Noting (44) through (46), we observe that equation (54) is equivalent to the equation (PDE). For Z i, the