NUMERICAL ANALYSIS FOR A NONLOCAL PHASE FIELD SYSTEM. J(x y)dy u(x) f(u)+lθ,

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1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING, SERIES B Volume, Number, Pages 9 c Institute for Scientific Computing and Information NUMERICAL ANALYSIS FOR A NONLOCAL PHASE FIELD SYSTEM SETH ARMSTRONG, SARAH BROWN AND JIANLONG HAN Abstract. In this paper, we propose a stable, convergent finite difference scheme to solve numerically a nonlocal phase field system which may model a variety of nonisothermal phase separations in pure materials which can assume two different phases, say solid and liquid, with properties varying in space. The scheme inherits the characteristic property of conservation of internal energy. We also prove that the scheme is uniquely solvable and the numerical solution will approach the true solution in the L - norm. Key words. Finite difference scheme; Nonisothermal, Long-range interaction.. Introduction In this wor, we consider the problem (.) u t = J(x y)u(y)dy (.) Ω (θ +lu) t = θ Ω J(x y)dy u(x) f(u)+lθ, in (,T) Ω, with initial and Neumann boundary conditions (.3) (.4) u(,x) = u (x), θ(,x) = θ (x), θ n =, Ω where T > and Ω R n is a bounded domain. Here θ represents temperature, u is an order parameter often used to represent various material phases, l is a latent heat coefficient, the interaction ernel satisfies J( x) = J(x), and f is bistable. In order to derive equations (.)-(.), we begin with the free energy (.5) E = J(x y)[u(x) u(y)] dxdy + [F(u(x))+ ] 4 θ dx, where F is a double well function. We consider the gradient flow associated with (.5) relative to the order parameter u in L and the internal energy e in H (Ω), where by H we mean the dual space of H with mean value zero. This is done because the total internal energy I, with density denoted by e = θ +lu, should be conserved. We have (.6) (.7) where E(u,e) u u t = E(u,e), u e t = E(u,e), e is a linear functional on L and E(u,e) e is a linear functional on H. Received by the editors October 8, 9 and, in revised form, September,. Mathematics Subject Classification. 35K57, 34A34, 65L, 65N6.

2 S. ARMSTRONG, S.BROWN, AND J. HAN If we write F = f, the representative of E(u,e) u in L is E(u, e) = J(x y)u(y)dy + J(x y)dyu(x) (.8) u Ω Ω +f(u) l(e lu), and the representative of E(u,e) e in H is E(u, e) (.9) = (e lu). e The more familiar Ginzburg-Landau free energy [ d u +F(u)+ l ] θ dx used in [7], [8], and in higher order versions by [6] in deriving phase-field systems, is obtained by approximating the interaction term through a truncated Taylor series. For example, when J is fairly localized, one may hope that J(x y)(u(x) u(y)) dxdy is well-approximated by ( ) d u dx, where d = J(y)yi dy is assumed to be independent of coordinate, i. Such an approximation was introduced by Van der Waals in [] in 893, and has been adopted ever since for ease of analysis. If Ω J(x y)u(y)dy J(x y)dyu(x) isreplacedby u, thesystem(.)-(.4) Ω becomes (.) (.) u t = u f(u)+lθ, (θ +lu) t = θ in (,T) Ω, with initial and Neumann boundary conditions (.) (.3) u(,x) = u (x), θ(,x) = θ (x), and θ n =. Ω The system (.)-(.4) and the system (.)-(.3) were proposed as models for nonisothermal phase separation in pure materials which can assume two different phases, say solid and liquid. The function θ represents the temperature field within the material. The function u is an order parameter that describes the phase of the material, where with appropriate scaling u = represents the solid phase and u = representsthe liquidphase; valuesofuwith < u < representamixture of the two phases. The results about existence, uniqueness, and the structure of solutions for both systems can be found in references [,3-5,9-3,5-9]. Tothebestofournowledge,thereareveryfewresultsonthenumericalsolutions to the system (.)-(.4) and the system (.)-(.3). When temperature is fixed in both systems, (.)-(.4) is the nonlocal Allen- Cahn equation (.4) u t = J(x y)u(y)dy J(x y)dyu(x) f(u) Ω Ω

3 NONLOCAL PHASE FIELD SYSTEM 3 and the system (.)-(.3) is the Allen-Cahn equation (.5) u t = u(x) f(u). Numerical analysis related to the nonlocal Allen-Cahn (.4) and Allen-Cahn (.5) equations can be found in [], [4] and the references therein. In this paper, we develop a finite difference scheme for the system(.)-(.4). The scheme inherits the characteristic property of conservation of internal energy. We also prove that the difference scheme is stable and that the numerical approximation converges to the solution of (.)-(.4).. Analysis of the Proposed Scheme In this section, we consider finite difference approximations of the system (.)- (.4) for n = and n =. We will use f(u) = u 3 u, but the analysis applies to a general smooth bistable function if care is taen in the choice of linearization (see Lemma.4). For n = we use the following notation. Setting Ω = ( L,L), we define Ω x = {x i x i = L+i x, i M} and Ω t = {t t =, K}, where x = L/M and = T/K. Then our choice of difference scheme for the system (.)-(.4) for n = is (.) (.) Then (.3) and (.4) u i = u (x i ) for i M,and θ i = θ (x i ) for i M. δ t u i = (J u ) i (J ) i u i +ψ(u i,u + i )+lθi lδ t u i +δ t θ i = δ xθ i, both for i M and K. The Neumann boundary condition is expressed by (.5) with θ θ x =, and θ M+ θ M x δ t u i = u+ i u i = for K,, δ xθ i = θ i+ θ i +θ i x and [ ] M (J u ) i = x J(x x i )u + J(x m x i )u m + J(x M x i )u M. Here we use m= ψ(u i,u+ i ) = u i (u i ) u + i for the discretization of f. It will be shown later that this choice of ψ in place of ψ(u i ) = u i (u i )3 is necessary to prove both the convergence and the stability of of the numerical scheme.

4 4 S. ARMSTRONG, S.BROWN, AND J. HAN Now for n =, meaning a rectangular domain ( L,L) ( W,W) R, we have and Ω x,y = {(x i,y j ) x i = L+i x, y j = W +j y, i M, j N} where x = L/M and y = W/N. Our difference scheme in this case is (.6) (.7) with (.8) Ω t = {t t =, t K}, u = u (x i,y j ) for i M, j N,and θ = θ (x i,y j ) for i M, j N, δ t u = (J u ) (J ) u +ψ(u,u+ )+lθ for i M, j N, K, and (.9) lδ t u +δ t θ = δ xθ +δ yθ for i M, j N, K. The Neumann boundary condition yields (.) In (.8)-(.9), θ,j θ,j x θ i, θ i, y (J u ) = x y + + =, θ M+,j θ M,j x =, θ i,n+ θ i,n y δ t u = u+ = for j N and = for i M. u δ xθ = θ i+,j θ +θ i,j x, δ yθ = θ + θ +θ y, [ M N m= n= M m= N n= J(x m x i,y n y j )u m,n ( J(xm x i,y y j )u m, +J(x m x i,y N y j )u m,n) ( J(x x i,y n y j )u,n +J(x M x i,y n y j )u M,n) + ( J(x x i,y y j )u, 4 +J(x M x i,y y j )u M, +J(x x i,y N y j )u,n +J(x M x i,y N y j )u )] M,N and, as in the n = case, we need choose for the discretization of f ψ(u,u + ) = u (u ) u +.,

5 NONLOCAL PHASE FIELD SYSTEM 5 From (.6)-(.), we have [ +(u ) ] u + = (.) [ (J u ) (J ) u +u +lθ ] +u. If we define r x = x and r y = y, then (.9)-(.) yield (.) θ + = r x (θ i+,j +θ i,j)+( r x r y )θ +r y (θ + +θ ) l(u+ u ) for i M and j N. The boundary conditions give rise to (.3) (.4) (.5) θ +, = r xθ, +( r x r y )θ, +r y θ, l(u +, u,), θ +,N = r xθ,n +( r x r y )θ,n +r y θ,n l(u +,N u,n), θ + M, = r xθ M, +( r x r y )θ M, +r y θm, l(u+ M, u M, ), and (.6) θ + M,N = r xθ M,N +( r x r y )θ M,N +r y θ M,N l(u+ M,N u M,N ), each for K. For j N, (.7) (.8) while for i M, we have (.9) and (.) θ +,j = r x θ,j +( r x r y )θ,j +r y (θ,j+ +θ,j ) l(u+ M,N u M,N ) θ + M,j = r xθ M,j +( r x r y )θ M,j +r y (θ M,j+ +θ M,j ) l(u + M,j u M,j); θ + i, = r x (θ i+, +θ i, )+( r x r y )θ i, +r y θ i, l(u + i, u i,) θ + i,n = r x(θ i+,n +θ i,n)+( r x r y )θ i,n +r y θ i,n l(u+ i,n u i,n ), where for each equation, K. For the sae of conciseness, it will be helpful in the following theorem to mae some preliminary definitions. We note that each Λ i will exist under the assumption of boundedness of u(,x,y) and θ(,x,y). To this end, define Λ = sup u(,x,y), Λ = sup θ(,x,y), Λ 3 = sup Ω J(x y), C = lλ +Λ +(Λ 3 +)+l and C = C e CT, where e. is the standard exponential function.

6 6 S. ARMSTRONG, S.BROWN, AND J. HAN Theorem.. If u(,x,y),θ(,x,y) L (Ω), there exists a unique solution to the system (.6)-(.), and the internal energy is conserved under this scheme. Furthermore, if r x +r y < / and < (l/c) where C is given above, then (.) lmax u +max θ C i.e., the scheme is stable under the maximum norm. Proof. From equations (.)-(.), it is clear that the scheme is uniquely solvable. The internal energy e = Ω (θ +lu)dx under this scheme can be represented by defining e = e() as (.) e = x y + + [ M m= n= M m= N n= N (lu m,n +θ m,n ) ( lu m, +θ m, +lu m,n +θ m,n) ( lu,n +θ,n +lu M,n +θ M,n) + ( lu 4, +θ, +lu M, +θm, +lu,n +θ,n +lu M,N +θm,n)]. For, using (.9) we have (.3) e + e = x y + + M m= N n= [ M N (δx θ m,n +δ y θ m,n ) m= n= ( δ x θ m, +δ y θ m, +δ x θ m,n +δ y θ m,n) ( δ x θ,n +δ yθ,n +δ xθ M,n +δ yθ M,n) + ( δ 4 x θ, +δyθ, +δxθ M, +δyθ M, +δx θ,n +δ y θ,n +δ x θ M,N +δ y M,N)] θ. Using the definition of the central schems and the summation gives (.4) M N (δxθ m,n +δyθ m,n) m= n= N = r x (θm,n θm,n θ,n +θ,n) n= +r y M (θm,n θ m,n θ m, +θ m, ). m= Invoing the Neumann boundary condition and the central schemes definition gives

7 NONLOCAL PHASE FIELD SYSTEM 7 (.5) M m= ( δ x θ m, +δ y θ m, +δ x θ m,n +δ y θ m,n) M = r y ( θm,n +θ m,n +θ m, θ m, ) m= + r x(θ M, θ M, θ, +θ,) + r x(θ M,N θ M,N θ,n +θ,n ), (.6) N n= ( δ x θ,n +δ y θ,n +δ x θ M,n +δ y θ M,n) N = r x ( θm,n +θ M,n +θ,n θ,n ) n= + r y(θ,n θ,n θ, +θ,) + r y(θ M,N θ M,N θ M, +θ M, ), and (.7) 4 (δ x θ, +δ y θ, +δ x θ M, +δ y θ M, +δ x θ,n +δ y θ,n +δ x θ M,N +δ y θ M,N ) = r x( θm, +θ M, +θ, θ, θ M,N +θ M,N +θ,n θ,n ) + r y( θ,n +θ,n +θ, θ, θ M,N +θ M,N +θ M, θ M, ). Using (.)-(.7), we arrive at (.8) e + = e for. Therefore, (.9) e = e, showing that internal energy is conserved. Finally, we prove statement (.) using induction. For =, using definitions preceding the theorem (.3) max u Λ and (.3) max θ Λ, so that (.3) lmax u +max θ lλ +Λ < C, where C is defined before the statement of the theorem.

8 8 S. ARMSTRONG, S.BROWN, AND J. HAN Assuming (.) is true for some natural number, we consider u + and θ +. Here, (.)- (.) imply that (.33) u + = ( (J u ) (J ) u ) +u +lθ +(u ) + +(u ) u and (.34) θ + = r x (θ i+,j +θ i,j )+( r x r y )θ Plugging (.33) into (.34), we have (.35) θ + +r y (θ + +θ ) l(u + u ). = r x (θi+,j +θi,j)+( r x r y )θ +r y (θ+ +θ ) [ ] ((J l u ) (J ) u +u +lθ ) +(u ) (u ) +(u ) u. If we set u = max u and θ = max θ, and if r x +r y < /, then invoing Λ 3 defined before the theorem, (.33)-(.35) imply (.36) and (.37) u + (Λ 3 +) u +l θ + +(u ) u θ + θ +l(λ 3 +) u +l θ + l(u ) +(u ) u. Since h(x) = x +x is an increasing function for x, we have (u ) +(u ) u u + u u. This inequality together with (.37) imply (.38) θ + θ +l(λ 3 +) u +l θ + l u + u u. Since (.36) is true for all i and j, and because there exist i and j such that u + = u + i,j, we arrive at (.39) l u + = l u + i,j l(λ 3 +) u +l θ l + +(u i,j ) u i,j. We need the following lemma. Lemma.. If b a > and /b, then (.4) +a a+ b +b b b. Proof. Since the proof is straightforward, we omit it here.

9 NONLOCAL PHASE FIELD SYSTEM 9 Adding (.38) and (.39) implies (.4) θ + +l u + θ +l(λ 3 +) u +l θ + l u l + u u + +(u i,j ) u i,j. We have assumed the statement is true for, so l u + θ < C. Therefore u < C/l. Also by the definition of u, we have u u i,j. If < (l/c), then < / u. Using Lemma. we have (.4) l +(u i,j ) u i,j + l u + u u l u. Using (.4) and (.4) together with the definiton of C preceding the theorem, we get (.43) θ + +l u + θ +l u +l(λ 3 +) u +l θ (+C )( θ +l u ) (+C ) + ( θ +l u ) e CT C = C. So the statement is true for +. Therefore, (.44) max, l u +max, θ C, where C is independent of and of the spatial mesh size. Remar.3. A similar result holds for n 3. Next we consider error estimates for the proposed scheme. For the solution of (.)-(.4), we have the following lemma. Lemma.4. Suppose that J and f satisfy the following assumptions: (A ) M sup Ω J(x y) dy < and f C(R). (A ) There exist c >, c >, c 3 >, c 4 > and r > such that f(u)u c u r c u, and f(u) c 3 u r +c 4. If assumptions (A ) (A ) are satisfied, u L (Ω) and θ L H (Ω), then there exists a unique solution (u,θ) C([,T],L (Ω)) to the system (.)-(.4) such that u t L (Q T ), and u tt, θ t, θ L (Q T ), where Q T = [,T] Ω. Proof. TheproofofthelemmaisasimilarargumenttothatintheproofofTheorem.3 in [4]; we omit it here. We note that although Lemma.4 is more general than for f(u) = u 3 u that we consider here, the conditions in the lemma are satisfied for our choice of f for the nonlinear function in (.). Theorem.5. If the system (.)-(.4) has a solution (u,θ) C, ([,T] Ω), then the solution of the difference scheme converges to this solution uniformly, and the convergence rate is O(+ x + y ).

10 S. ARMSTRONG, S.BROWN, AND J. HAN Proof. Let (U(t,x,y), V(t,x,y)) be the solution of the system (.)-(.4). We use the following notation: (.45) U = u (x i,y j ), U = u(t,x i,y j ), V = θ (x i,y j ), V = θ(t,x i,y j ). From (.)-(.4), we have for, (.46) U + U = (J U ) (J ) U +U (U )3 +lv +R (, x, y), where R (, x, y) = O(+ x + y ). Then (.47) l U+ U + V + V = V i+,j V +V i,j x + V + V +V y +R (, x, y), with Neumann boundary condition (.48) V,j = V,j +O( x ), V M+,j = V M,j +O( x ) for j N, V i, = V i, +O( y ), V i,n+ = V i,n +O( y ) for i M, where R (, x, y) = O(+ x + y ). We define error terms (.49) X = U u for =,...,K, i =,...,M, j =,...,N and (.5) Y = V θ for =,...,K, i =,...,M, j =,...,N. From (.6), (.7) and (.45), we have (.5) X =, Y = for i =,,M, j =,,N. Turning now to, (.8) and (.46) yield (.5) X + = [(J X ) (J ) X +X (U 3 (t,x i,y j ) (u ) u + )+X +ly ] +X +R (, x, y). Since (.53) (U ) 3 (u ) u + = (U ) (U U + ) +X (U +u )U+ +(u ) X +,

11 NONLOCAL PHASE FIELD SYSTEM equation (.5) implies [ +(u ) ] X + = (.54) [ (J X ) (J ) X +X (U ) (U U+ ) X ] (U +u )U+ +X +ly + O(+ x + y ). From Lemma.4, u and u t are bounded in Q T, so (.55) (U ) (U U + ) (Λ ) Λ 4, where Λ = max, U, Λ 4 = max QT u t. Now equations (.54) and (.55) and the boundedness of U and u suggest that (.56) max X + C(Λ,Λ,Λ 3 )max +max X X +lmax Y + R, where C(Λ,Λ,Λ 3 ) depends only on Λ = max, u, Λ = max, U and Λ 3 = sup Ω J(x y) dy, and where R = O( + x + y ). Equations (.9) and (.47) together lead to (.57) l X+ X + Y + = Y i+,j Y +Y i,j x with Neumann boundary condition (.58) Y + Y + Y +Y y +R 3 (, x, y), Y,j = Y,j +( x ), Y M+,j = Y M,j +O( x ) for j N, Y i, = Y i, +O( y ), Y i,n+ = Y i,n +O( y ) for i M, where R 3 (, x, y) = O(+ x + y ). Thus (.59) Y + = r x (Y i+,j +Y i,j)+r y (Y + +Y ) +( r x r y )Y + l(x + X )+R 3(, x, y). If r x +r y < /, equation (.59) gives rise to (.6) max Y + max Y +lmax X + X +R 3 (, x, y). Invoing equation (.5) and the boundedness of U and u, we have (.6) lmax X + X lc(λ,λ,λ 3 )max X +l max Y +lr (, x, y),

12 S. ARMSTRONG, S.BROWN, AND J. HAN where C(Λ,Λ,Λ 3 ) and R = O(+ x + y ) are defined as before. Finally, using equations (.56), (.6) and (.6), we arrive at (.6) max X + +max Y + (+C(Λ,Λ,Λ 3,l))(max +lr 4 (, x, y) (+C(Λ,Λ,Λ 3,l)) (max X +lr 4 (, x, y). establishing Theorem.5. (+C(Λ,Λ,Λ 3,l)) + (max +( +)R 4 (, x, y) X +max Y ) +max Y ) X +max Y ) TlR 4 (, x, y) = O(+ x + y ), Remar.6. The scheme for the Neumann boundary condition can also be applied to the problem with the Dirichlet boundary condition. However, for the Dirichlet problem, the internal energy is not conserved. 3. Numerical Results Having established stability and convergence of the difference scheme, in this section we investigate some numerical approximations of the solution to (.)-(.4). In the graph, we use T as the temperature instead of θ. For n =, let Ω = (,), f(u) = u 3 u, J(x) = e x and l =. Figures -3 show the numerical results for the phase field system at t =,, and 4 with initial values u (x) =.8sin(πx), θ (x) =.5+.cos(πx). Figures 4-6 show the numerical results for the phase field system at t =,,, and 4 with initial values u (x) =.8sin(πx), θ (x) =.5cos(πx). Figures 7-8 show the numerical results for the phase field system at t =,,, and 4 with initial values u (x) =.8sin(πx), θ (x) =.5+.cos(πx). For n =, let Ω = (,) (,), J ǫ (x,y) = e (x +y ), =.5, x =.5 and y =.5. Figures - show the numerical results at t =,4 and 5 with initial values u (x,y) =.8cos(πx) cos(πy), θ (x,y) =.5+.cos(πx) cos(πy). Figures 3-5 show the numerical results at t =,4 and 5 with initial values u (x,y) =.8cos(πx) cos(πy), θ (x,y) =.cos(πx) cos(πy). Figures 6-8 show the numerical results at t =,4 and 5 with initial data u (x,y) =.8cos(πx) cos(πy), θ (x,y) =.5+.cos(πx) cos(πy).

13 NONLOCAL PHASE FIELD SYSTEM 3 u(,x) u(,x) u(,x).5.5 u(4,x) Figure. The order parameter u in the phase field system for n = with u (x) =.8sin(πx), θ (x) =.5+.cos(πx), =., x=.. T(,x) T(,x) T(,x).5.5 T(4,x) Figure. The temperature T in the phase field system for n = with u (x) =.8sin(πx), θ (x) =.5 +.cos(πx), =., x=.. The internal energy The internal energy t axis Figure 3. The internal energy in the phase field system for n = with u (x) =.8sin(πx), θ (x) =.5+.cos(πx), =., x=..

14 4 S. ARMSTRONG, S.BROWN, AND J. HAN u(,x) u(,x) u(,x).5.5 u(4,x) Figure 4. The order parameter u in the phase field system for n = with u (x) =.8sin(πx), θ (x) =.5cos(πx), =., x=.. T(,x) T(,x) T(,x).5.5 T(4,x) Figure 5. The temperature T in the phase field system for n =. u(,x) u(,x) u(,x).5.5 u(4,x) Figure 6. Theorderparameteruinthephasefieldsystemforn = with u (x) =.8sin(πx), θ (x) =.5 +.cos(πx), =., x=..

15 NONLOCAL PHASE FIELD SYSTEM 5 T(,x) T(,x) T(,x).5.5 T(4,x) Figure 7. The temperature T in the phase field system for n = with u (x) =.8sin(πx), θ (x) =.5 +.cos(πx), =., x=.. u(,x,y ) u(,x,y ) u(4,x,y ) u(5,x,y ) Figure 8. The order parameter u in the phase field system for n = with u (x,y) =.8cos(πx) cos(πy), θ (x,y) =.5 +.cos(πx) cos(πy), =.5, x=.5, y =.5. T(,x,y ) T(,x,y ) T(4,x,y ) T(5,x,y ) Figure 9. The temperature T in the phase field system for n =.

16 6 S. ARMSTRONG, S.BROWN, AND J. HAN The internal energy 3 The internal energy t axis Figure. The internal energy in the phase field system for n = with u (x,y) =.8cos(πx) cos(πy), θ (x,y) =.5 +.cos(πx) cos(πy), =.5, x=.5, y =.5. u(,x,y ) u(,x,y ) u(4,x,y ) u(5,x,y ) Figure. The order parameter u in the phase field system for n = with u (x,y) =.8cos(πx) cos(πy), θ (x,y) =.cos(πx) cos(πy), =.5, x=.5, y =.5. T(,x,y ) T(,x,y ) T(4,x,y ) T(5,x,y ) Figure. Thetemperature Tin the phasefieldsystem forn = with u (x,y) =.8cos(πx) cos(πy), θ (x,y) =.cos(πx) cos(πy), =.5, x=.5, y =.5.

17 NONLOCAL PHASE FIELD SYSTEM 7 u(,x,y ) u(,x,y ) u(4,x,y ) u(5,x,y ) Figure 3. The order parameter u in the phase field system for n = with u (x,y) =.8cos(πx) cos(πy), θ (x,y) =.5 +.cos(πx) cos(πy), =.5, x=.5, y =.5. T(,x,y ) T(,x,y ) T(4,x,y ) T(5,x,y ) Figure 4. The temperature T in the phase field system for n = with u (x,y) =.8cos(πx) cos(πy), θ (x,y) =.5 +.cos(πx) cos(πy), =.5, x=.5, y =.5. In Figures -3 and Figures -, we see that when the initial temperature is greater than zero, θ will decrease to a negative constant and u will approach a piecewise defined constant function which will approach in the greater portion of the region. During this process, the internal energy is conserved as predicted by Theorem.. Figures 4-5 and Figures - suggest that when the initial temperature oscillates about zero, θ will approach zero while u approaches a piecewise defined constant function. Both are also steady state solutions for the phase-field system. During this process, we have checed that the internal energy is also conserved as expected.

18 8 S. ARMSTRONG, S.BROWN, AND J. HAN The graphs in figures 6-7 and Figures 3-4 demonstrate the reasonable result that over time and with the negative, nonconstant initial temperature, θ will increase to a positive constant on Ω, while u approaches a piecewise-constant function that is represented by state value of in the greater portion of Ω. The theoretical analysis for this steady state solution will be investigated in a future paper. Remar 3.. We also checed the numerical result with discontinuous but bounded functions used as initial conditions functions. The numerical results show behavior similar to that obtained with smooth initial functions agreeing with the finding that they need only be bounded functions. Acnowledgment. The authors would lie to than the referees for their helpful suggestions. References [] P. W. Bates, On some nonlocal evolution equations arising from materials science Fields Inst. Communications, Vol 48(6), 3-5. [] P. W. Bates, S. Brown and J. Han Numerical analysis for a nonlocal Allen-Cahn equation International Journal of Numerical Analysis and Modeling, Vol 6(9), [3] P. W. Bates, F. Chen, and J. Wang, Global existence and uniqueness of solutions to a nonlocal phase-field system, in US-Chinese Conference on Differential Equations and Applications, P. W. Bates, S-N. Chow, K. Lu, X. Pan, Edt., International Press, Cambridge, MA, 997, 4-. [4] P. W. Bates, P. C. Fife, R. A. Gardner, and C. K. R. T. Jones, Phase field models for hypercooled solidification, Physica D. 4 (997), -3. [5] P. Bates, J. Han, and G. Zhao, On a nonlocal phase-field system, Nonlinear Analysis 64 (6), No., [6] P. W. Bates and S. Zheng, Intertial manifolds and inertial sets for the phase-field equations, J. Dynam. Differential Equations 4 (99), [7] G. Caginalp, Analysis of a phase field model of a free boundary, Arch. Rat. Mech. Anal. 9 (986), [8] G. Caginalp and P. C. Fife, Dynamics of layered interfaces arising form phase boundaries, SIAM J. Appl. Math. 48 (988), [9] J. Carr, M. Gurtin and M. Slemrod, Structured phase transitions on a nite interval, Arch. Rational Mech. Anal., 86(984), [] J. Carr, M. Gurtin and M. Slemrod, One-dimensional structured phase transformations under prescribed loads, J. Elasticity, 5(985), [] X.Chen and X.-P. Wang, Phase transition near a liquid-gas coexistence equilibrium, SIAM J. Appl. Math. 6 (), [] C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations, In Free Boundary Problems, K. H. Hoffmann and J. Spreels (eds.), International Series of Numerical Mathematics, Vol. 95, Birhauser Verlag, Basel, (99), [3] E. Feireisl, F. I. Roch, and H. Petzeltova, A non-smooth version of the Lojasiewicz-Simon theorem with applications to nonlocal phase-field systems, J. Diff. Eq. 99 (4), -. [4] X. Feng and A. Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows, Numer. Math. 3 (3), [5] MM. Gander,M. Mei, and G. Schmidt, Phase transition for a relaxtion model of mixed type with periodic boundary condition, Applied mathematics Research Express, 7, -34. [6] D.-Y.Hsieh and X.-P. Wang, Phase transition in van der Waals fluid,on a nite interval, SIAM J. Appl. Math. 57 (997), [7] M. Mei, Y.S. Wong and L. Liu, Stationary solutions of phase transition in a coupled viscoelastic system, Nonlinear Analysis Research Trends, Edited by N.Roux, Nova Science Publishers, Inc. 8, [8] M. Mei, Y.S. Wong and L. Liu, Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition:(i) Existence and uniform boundedness, Discrete and Continuous Dynamical Systems-Series B, 7 (7) [9] M. Mei, Y.S. Wong and L. Liu, Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition:(ii) Convergence, Discrete and Continuous Dynamical Systems- Series B, 7 (7)

19 NONLOCAL PHASE FIELD SYSTEM 9 [] J.D. van der Waals, The thermodynamic theory of capillarity flow under the hypothesis of a continuous variation in density, Verhandelingen der Koninlije Nederlandsche Aademie van Wetenschappen te Amsterdam (893), -56. Department of Mathematics, Southern Utah University, Cedar City, UT 847, USA armstrong@suu.edu and brown s@suu.edu and han@suu.edu