Chaos, Solitons an Fractals (7 64 73 Contents lists available at ScienceDirect Chaos, Solitons an Fractals onlinear Science, an onequilibrium an Complex Phenomena journal homepage: www.elsevier.com/locate/chaos Frontiers Well-poseness of the Cahn Hilliar equation with fractional free energy an its Fourier Galerkin approximation Mark Ainsworth, Zhiping Mao Division of Applie Mathematics, Brown University, 8 George St, Provience, RI 9, USA a r t i c l e i n f o a b s t r a c t Article history: Receive 3 January 7 Revise 5 May 7 Accepte 6 May 7 Available online 6 May 7 MSC: 65 653 655 Keywors: Fractional Cahn Hilliar equation on-local energy Well-poseness Fourier spectral metho Error estimates We stuy the well-poseness of the Cahn Hilliar equation in which the graient term in the free energy is replace by a fractional erivative. We begin by establishing the existence an uniqueness of a Fourier-Galerkin approximation an then erive a number of priori estimates for the Fourier-Galerkin scheme. Compactness arguments are then use to euce the existence an uniqueness of the solution to the fractional Cahn Hilliar equation. An estimate for the rate of convergence of the Fourier-Galerkin approximation is then obtaine. Finally, we present some numerical illustrations of typical solutions to the fractional Cahn Hilliar equation an how they vary with the fractional orer β an the parameter ε. In particular, we show how the with τ of the iffuse interface epens on ε an β, an erive the scaling law τ = O (ε /β which is verifie numerically. 7 Elsevier Lt. All rights reserve.. Introuction The classical Cahn Hilliar equation is often use as a iffuse interface moel for the phase separation of a binary alloy [4,5] in which the free energy functional takes the form ( ε u + F (u, ( where R n an F (s = 4 s 4 s. Despite the wiesprea aoption of the moel, there are goo reasons [8,9] for preferring a moel in which the free energy is non-local an takes the form ε (u(x u (y x y n + β + F (u, ( where / < β. More generally, the non-local coupling through the term x y (n + β can take the form of a singular kernel k ( x, y []. The first term in ( is often use to efine the fractional Sobolev (semi-norm u an coincies with the usual graient in ( when β =. H β This work was supporte by the MURI/ ARO on Fractional PDEs for Conservation Laws an Beyon: Theory, umerics an Applications ( W9F-5--56. Corresponing author. E-mail aresses: Mark_Ainsworth@brown.eu (M. Ainsworth, zhiping_ mao@brown.eu (Z. Mao. The classical Cahn Hilliar equation is obtaine from ( by consiering a graient flow in H [7]. More generally, one can consier a graient flow in H α leaing to a fractional Cahn Hilliar equation of the form u t (x, t + ( α ( ε u (x, t + f (u (x, t = (3 which was stuie in []. Alternatively, by consiering the H - graient flow associate with (, we obtain, in the two imensional case, a ifferent type of Fractional Cahn Hilliar Equation (FCHE for / < β : u t (x, t (ε ( β u (x, t + f (u (x, t =, (x, t (, T ], u (, t is π- perioic for all t (, T ], u (x, = u (x, x, (4 where = (, π, ε is a positive constant, x = (x, x, u L ( H β per (, an f (u = F (u = u 3 u. The choice β = correspons to the classical Cahn Hilliar equation. Here, we consier the special case of perioic solutions in orer to minimize technical ifficulties. http://x.oi.org/.6/j.chaos.7.5. 96-779/ 7 Elsevier Lt. All rights reserve.
M. Ainsworth, Z. Mao / Chaos, Solitons an Fractals (7 64 73 65 In orer to efine the fractional Laplacian ( β, we use the Fourier ecomposition. For any u L per (, u (x = u ˆ kl e ikx + ilx, (5 k,l Z where i =, an the Fourier coefficients are given by u ˆ kl = u, e ikx + ilx = ue ( π ikx + ilx x. The fractional Laplacian is then efine by ( β u = (k + l β u ˆ kl e ikx + ilx. (6 k,l Z In the previous work [], we consier the FCHE given by (3 in which the usual integer orer erivative appears in the free energy but a graient flow in H α, < α is consiere. We showe in that case that the equation preserves mass for all positive values of fractional orer α an is well-pose, the solution is uniformly boune for all t > an the nature of the solution with a general α > is qualitatively closer to the behavior of the classical Cahn Hilliar equation than to the Allen Cahn equation. Here, we turn to the case of a free energy functional in which a fractional erivative appears. The two cases are essentially ifferent in that for (3 the ynamics are epenent on the fractional orer α but the steay state is not. However, for (4 the steay state will be epenent on the fractional orer β where the ynamics will not. Of course, one coul combine these cases if so esire. In the present work, we stuy the well-poseness of the equation (4 using energy type methos. We begin by establishing the existence an uniqueness of a Fourier-Galerkin approximation an then erive a number of a priori estimates for the Fourier-Galerkin scheme. Compactness arguments are then use to euce the existence an uniqueness of the solution to (4. An estimate for the rate of convergence of the Fourier-Galerkin approximation is then obtaine. Finally, we present some numerical illustrations of typical solutions to (4 an how they vary with the fractional orer β an the parameter ε. In particular, we show how the with τ of the iffuse interface epens on ε an β, an erive the scaling law τ = O (ε /β which is verifie numerically.. Well-poseness an Fourier-Galerkin approximation Let C per ( be the set of all restrictions onto = (, π of all real-value, π-perioic, C -functions on R. For each s, let H per s ( be the closure of C per ( in the usual Sobolev norm s with u s = k,l Z u ˆ kl ( + k + l s. ote that H per ( = L per (. In particular, we use to enote the L -norm. We collect a number of elementary technical results that were prove in [] an which are neee in the present article: Lemma. Let μ, ν then for any u, v H μ+ ν per (, it hols (see [, Lemma 8] (( μ+ ν u, v = (( μ u, ( ν v. (7 an (see [, Lemma 9] ( μ+ ν u = ( μ ( ν u = ( ν ( μ u. (8 If < μ ν an v H ν per (, then (see [, Lemma ] ( μ v ( ν v. (9 Moreover, if ν >, it hols (see [, Lemma 4] v C( v + ( ( ν / v ( (+ ν / v. ( The following estimate will be use in establishing the uniqueness of FCHE an the error estimate of the Fourier-Galerkin approximation. Lemma. (See [6, Lemma 8] Let X, B, Y be Banach spaces an satisfying X B Y with compact imbeing X B. Then for an η >, there exists a constant C η, such that for all v X, v B η v X + C η v Y. ( By testing FCHE (4 with a function v H + β per ( an using the fractional integration by parts (7, we arrive at the following weak form of (4: fin u L (, T ; H + β per ( with t u L (+ β (, T ; H (, such that per ( t u ( t, v + (ε ( (+ β / u (t, ( (+ β / v + ( f (u (t, v =, v H + β per (. ( One of our main results is the following proclaiming the wellposeness of the weak problem ( : Theorem. For / < β, let u H β per (. Then the FCHE (4 amits a unique weak solution u L (, T ; H + β per ( with t u L (+ β (, T ; H ( for all T, which aitionally satisfies per t u (t + ε ( (+ β / u (τ τ u + C (ε, u t an (3 u (t β C(ε( + u β, t >, (4 where C ( ε is a positive constant epens on ε but not on t. The proof of above theorem will be carrie out in Section 3 by means of Fourier-Galerkin iscretization, a priori estimates, an compactness arguments []... Fourier-Galerkin iscretization We consier the Fourier-Galerkin iscretization of (4, establish its existence an uniqueness, an erive various properties of the iscrete approximation using the similar arguments to those use in []. Let X = span { e ikx + ilx, k, l }, then the Fourier spectral-galerkin approximation for ( consists of fining u X, such that, ( t u (t, v + (ε ( (+ β / u (t, ( (+ β / v + ( f (u (t, v =, v X, (5 with initial conition u ( = u. Here is the usual L - projection operator in X, efine by (u u, v =, v X. (6 The next theorem shows that the Fourier-Galerkin approximation (assuming for now it exists preserves mass, reuces the energy monotonically, an is stable in the sense that u ( t β is uniformly boune. Theorem. Let u H β per (, then the Fourier spectral-galerkin metho preserves mass for all t, i.e., u (t x = u ( x = u x = u x. Furthermore, the free energy functional efine by ( ε E(v = ( β/ v + F (v, satisfies: for s < t
66 M. Ainsworth, Z. Mao / Chaos, Solitons an Fractals (7 64 73 t E(u (t + (ε ( β u (τ + f (u (τ τ = E(u (s, s (7 an hence E ( u ( t is monotonically ecreasing. Finally, the Fourier spectral Galerkin metho (5 is stable an, for all β >, there hols u (t β C(ε( + u β, t >, (8 where C ( ε C / ε for C a positive constant inepenent of t an. Proof. The mass conservation is prove by taking v = in (5. Eq. (7 is prove by choosing v = f (u + ε ( β u in (5, arguing as in the proof of Lemma in [] an integrating over ( s, t. The proof of (8 is essentially ientical with the proof of Theorem in []. In particular, we have ( β/ u (t ( β/ u ( + C u ( 4 β /ε, u (t C ( β/ u (t + C 3 u, an we arrive at (8 with C ( ε C / ε. We next show the existence an uniqueness of the Fourier- Galerkin solution. Theorem 3. For / < β, let u H β per (, then (5 amits a unique solution u ( t satisfying the following energy inequalities: t u (t + ε ( (+ β / u (τ τ u + C (ε, u t, t (9 t u (τ (+ β τ C (ε, u t ( for all t R +, where C ( ε, u an C ( ε, u are constants epening on ε an u but not on an t. Proof. Let u (t = k,l= u kl (t e ikx + ily, using a similar argument to the one use in Section in [] for the existence an uniqueness of Fourier-Galerkin problem, we arrive at the following orinary ifferential system for the evolution of the Fourier coefficients: t u kl (t = g kl (U(t, u kl ( = (u, e ikx + ily, k, l =,, ( where g kl (U(t := g kl (u, (t,..., u, (t ; ;u, (t,..., u (t an all g kl : R M R, M = ( +, are smooth an locally Lipschitz continuous. By virtue of the existence an uniqueness theorem of initial-value problems of orinary ifferential equations (see Theorem 5. in [3], we conclue that the initial-value problem ( amits a unique solution U ( t for t >. Moreover, it follows from (8 that k,l= u kl (t = u (t u (t β C(ε( + u β, t >. ow let us give the proof of the two estimates (9 an (. For each t >, taking v = u in (5, we get t u (t + ε ( (+ β / u (t + ( u 3 (t, u (t = ( u (t, u (t. ( oting that ( u 3, u = 3 u u, (3 an using (8 we have ( u, u = (u, u = (( ( β / u, ( (+ β / u Using the Cauchy Schwarz an Young inequalities, the above is boune by ε ( ( β / u + ε ( (+ β / u. (4 By virtue of (, (3 an (4, we obtain t u (t + ε ( (+ β / u (t ε ( ( β / u (t. Since ( β / < β/, we have from (9 that (5 ( ( β / u (t ( β/ u (t. (6 Then we obtain from (5 an the above equation t u (t + ε ( (+ β / u (t ε ( β/ u (t. The estimate (9 follows from integrating the above equation an the estimate (8. It follows from (5 that t u (t (+ β ε ( (+ β / u (t + f (u (t ε ( (+ β / u (t + ( + u (t 4 u (t. (7 Then using the estimate (, the above is boune by ε ( (+ β / u (t + C ( + u (t u (t + C ( ( β / u (t ( (+ β / u (t u (t Using (6 again, we obtain the estimate ( by integrating (7 an using the estimate (9. The estimates (9 an ( will be use later in the proof of Theorem establishing the existence an uniqueness of the original FCHE (4. 3. Proof of existence an uniqueness ( Theorem In this section, we turn to the proof of the existence an uniqueness of the weak solution of (. Proof. We firstly show the existence. Using Theorem 3 an [6, Theorem of Appenix D.4], an arguing as in the proof of Theorem 4 in [], we euce that there exists a sequence of functions { u : } with u L (, T ; H + β per ( with t u L (+ β (, T ; H per ( an u L (, T ; H + β per ( with t u L (+ β (, T ; H ( such that per u u in L (, T ; H + β per (, (8 t u t u in L (+ β (, T ; H per (, (9 for all T > as. Moreover, u satisfies the estimates (9 (, an hence the estimate (3 of Theorem hols. Let v H + β per ( an η C ([, T ]. Choose the test function v in (5 for each, then by multiplying both sies of the resulting ientity by η an integrating over t [, T ] we obtain T ( t u (t, η(t v t T + + T ( ε ( (+ β / u (t, η(t( (+ β / v t ( f (u (t, η(t v t =. We shall show that taking in the above equation gives
M. Ainsworth, Z. Mao / Chaos, Solitons an Fractals (7 64 73 67 T + + ( t u (t, η(t v t T T (ε ( (+ β / u (t, η(t( (+ β / v t ( f (u (t, η(t v t =. (3 The first two terms an the linear term involve in f can be treate irectly using (8 (9. ow let us show that T ( u 3 (t, η(t T ( v t u 3 (t, η(t v t, as. (3 ote that v = ( v v + v. Then the above expression is boune by T ( u 3 (t, η(t ( v v t T ( + (u 3 (t u 3. (t, η(t v t (3 By (7 an (8, the first term is equal to T ( ( ( β / u 3 (t, η(t( (+ β /. ( v v t Let C be a generic constant. By using the following estimate (see equation (6 of [] ( ν v 3 C v ( ν v, v H ν per (, we boun the first term of (3 as follows: C T C ( ( β / u (t u (t L ( t η ( (+ β / ( v v T u (t L ( t η ( (+ β / ( v v C u L (,T ;H + β per ( η ( (+ β / ( v v, as. (33 The first inequality of the above equation follows from estimates (6 an (8. Similarly, the secon term is estimate by ( T / C u (t + u (t u (t + u (t L ( t η(t u u L (,T ;H β ( ( (+ β / v, using Sobolev inequality (, (6 an (8, we have T u (t + u (t u (t + u (t L ( t T 3 ( u (t 4 L ( + u (t L ( u (t L ( + u (t 4 L ( t C T ( u (t H + β per ( + u (t H + β per ( u (t H + β per ( + u (t H + β per ( t. Consequently, the secon term is boune by ( C u L (,T ;H + β per ( + u / L (,T ;H + β per ( u / L (,T;H + β per ( + u L η(t (,T;H + β per ( u u L (,T ;H β ( v + β as. (34 Thus, (3 hols true an (3 follows. Since η C [, T ] in (3 is arbitrary, we euce that u ( t satisfies ( for all T >. Applying the same argument use in the proof of Theorem 4 of [] (see also the proof of Theorem 3. of [], it can be shown that u ( = u. This completes the proof of the existence of the weak solution for FCHE. We now give the proof of uniqueness which is similar to the arguments in [3]. Denote w = ε ( β u + f (u. Let u an u be weak solutions of the FCHE for the same initial ata u. Let ū := u u an w := w w. Then ( ū, w satisfies the following mixe form: t ū (t w (t =, w (t = ( β ū (t + u 3 (t u 3 (t ū. (35 By the mass conservation, we have ū x = u u x = u u x =. Consier the following ual problem: v = ū, (36 where ū =. By the Lax Milgram Lemma, problem (36 amits a unique weak solution v H per, v =, an moreover, v = iff ū =. Substituting for ū in (35 using (36, we have t ( v (t w (t =, w (t = ( + β v (t + u 3 (t u 3 (t + v. (37 Multiplying the first equation of (37 by v (t we euce t v (t + ( w (t, v (t = using (7. On the other han, testing the secon equation of (37 against v an recalling (36, it follows that ( w (t, v (t = ( + β/ v (t + (u 3 (t u 3 (t ū (t x v (t ( + β/ v (t v (t. By the above two estimates an the estimate (, we euce that t v (t + ( + β/ v (t v (t ( + β/ v (t + C v (t. Thus we get t v (t + ( + β/ v (t C v (t. Using the Gronwall lemma, we get v (t e Ct v (. Then, using the fact that ū ( = implies v ( =, then we obtain v (t = for all t. Consequently, v (t =, i.e., ū (t = for all t. The FCHE (4 enjoys similar properties to those of the Fourier- Galerkin approximation. In particular, we have the following corollary to Theorem : Corollary. Let / < β an u H β per (. Then the solution of FCHE satisfies u (t is constant for all t >, i.e., u (t x = u ( x = u x. an the energy is non-increasing, i.e., there hols for t > s >, E(u (t E(u (s E(u.
68 M. Ainsworth, Z. Mao / Chaos, Solitons an Fractals (7 64 73 Proof. The mass conservation can be obtaine by using the same argument use for the Fourier-Galerkin scheme in Section.. By estimate (8 an the compactness argument [6, Theorem 3 of Appenix D.4], we euce that there exists a sequence of functions { u : } with u H β per ( an u H β per ( such that u u in H β per (. Hence, since the embeings H β per ( L 4 per ( L per ( are compact (Rellich Konrachov Theorem, [6, 5.7], we euce u ( t converges strongly to u ( t in L 4 per ( an L per (. It follows from the uniqueness of limits that E(u (t E(u (t as, an then the energy monotonicity is obtaine by taking limits in the inequality E ( u ( s E ( u ( t which follows from (7. Thanks to above results, the stability estimate (4 now can be obtaine by applying the same argument as use for the Fourier- Galerkin problem (5 iscusse in Section.. 4. Discretization analysis 4.. Error analysis for the Fourier-Galerkin semi-iscretization In this section, we obtain an estimate for the convergence of the Fourier-Galerkin approximation to the solution of FCHE (4. Theorem 4. Suppose that the solutions of problem ( an (5 are uniformly boune an that u (t H r (, r > + β, t [, T ], then the error estimate hols for / < β : ( t u (t u (t Ce T/ε / r { u (t r + / } u (τ r, t [, T ]. (38 Proof. In orer to analyze the convergence of the Fourier-Galerkin scheme, we follow the usual approach an consier the function e (t X given by e (t = u (t u (t. Thus, for v X, by using the efinition of an noting that commutes with the fractional Laplacian operator (see [, Lemma 6], i.e., ( s u (x = ( s u (x, we have ( e t (t, v ( + ε ( (+ β / e (t, ( (+ β / v ( u = t (t, v ( + ε ( (+ β / u (t, ( (+ β / v ( u t (t, v ( ε ( (+ β / u (t, ( (+ β / v ( u = t (t, v ( + ε ( (+ β / u (t, ( (+ β / v ( + f (u (t, v ( = ( f (u (t f (u(t, v ( = ( ( β / ( f (u (t f (u(t, ( (+ β / v. Taking v = e in above equation, an using the Cauchy-Schwarz an Young inequalities, we get t e (t + ε ( (+ β / e (t ε ( ( β / ( f (u(t f (u (t + ε 4 ( (+ β / e (t which implies that t e (t + 3 ε ( (+ β / e (t ε ( ( β / ( f (u(t f (u (t C u (t + u (t u (t + u (t ( ( β / (u(t u (t + C ( ( β / (u (t + u (t u (t + u (t (u (t u (t. (39 The secon inequality follows from the following estimate (see equation (6 of [] : ( ( β / (w v [ C v ( ( β / w + w ( ( β / v ], v, w H β per (. Assuming that u ( t an u ( t are boune, then by the above estimate, (6, (4 an (8, we have that both u (t + u (t u (t + u (t an ( ( β / (u(t + u (t u (t + u (t are boune. We can then boun the right sie of (39 by: C ( ( ( β / (u (t u (t + ( ( β / e (t + C ( u (t u (t + e (t. Using the triangle inequality an estimate ( an noting that H + β L L an H + β H β L, we have C ( ( β / e (t ε Hence, 4 ( (+ β / e (t + C ε e (t, C e (t ε 4 ( (+ β / e (t + C ε e (t. C ( ( ( β / (u(t u (t + ( ( β / e (t + C ( u (t u (t + e (t C( ( ( β / (u (t u (t + u (t u (t + ε ( (+ β / e (t + C ε e (t, where C ε = C ε + C ε. Inserting this estimate into the earlier estimate (39, we get t e (t + ε ( (+ β / e (t C( ( ( β / (u (t u (t + u (t u (t + C ε e (t. Since e ( = u u =., it follows that { e C ε t e (t } Ce C ε t ( ( ( β / (u(t u (t t + u (t u (t, which in turn gives e (t C t + u (τ u (τ τ, e C ε (t τ ( ( ( β / (u (τ u (τ since e ( =. Finally, with the help of the following estimates (see [, Lemma 7] u (t u (t β c β ν u (t ν, ν > β, (4 an (see [4, Theorem.] u (t u (t c / ν u (t ν, ν > /, (4
M. Ainsworth, Z. Mao / Chaos, Solitons an Fractals (7 64 73 69 we have e (t C r e C ε t t u (τ r τ, t. By the triangle inequality an (4, we obtain the error estimate (38 for the Fourier-Galerkin approximation. -5 L error =.6 =.8 =. 4.. Time iscretization an its stability analysis For the time integration, we use a stabilize first-orer semiimplicit scheme in time [5]. For a given partition = t < t < < t K = T, t n = nδt, n =,,..., K, where δt is the time step, an K = T /δt, we use the Fourier-Galerkin scheme (5 in space, an the stabilize first-orer time semi-implicit scheme given by δt n + (u, v + ( (ε ( β u n + + f (u n n + + S(u, v =, v X, (4 where u n enotes the numerical approximation to u ( t at time t n an S is a positive stabilization constant. We now show the stability of the time scheme (4. The scheme (4 can be alternatively formulate in the following mixe form: v, φ X, δt ε (( β/ u n + n + (u, v + ( w n +, v =,, ( β/ φ + ( f (u n n + + S(u, φ = (w n +, φ. (43 The next result is similar to the result prove for the stanar Cahn Hilliar equation in [5], an shows that the scheme (43 is stable provie the stabilize constant is sufficiently large: Lemma 3. If S L /, where L = max { f (s } is assume to be boune, then the stabilize scheme (43 is unconitionally stable. Moreover, the solution of (43 satisfies the following energy law: E(u n + E(u n, n. (44 Proof. Taking v = δtw n + in the first equation of (43 an using the ientity (a b, a = a b + a b, we obtain (u n +, w n + + δt w n + =, on the other han, taking φ = u n + (43, we obtain ε ( ( β/ u n + ( β/ u n + ( f (u n, u n + By Taylor expansion, we get F (u n + F (u n = f (u n n + + S u n + (u in the secon equation of + ( β/ (u n + = (w n +, u n + + f (ζ n. (u n +, where ζ n is between u n n + an u. Therefore, by combining the above three equations, we obtain ε ( ( β/ u n + ( β/ u n + (F (u n + = f (ζ n F (u n u n + L + ( β/ (u n + n +, + S u + δt w n + u n +. We then conclue that the esire result hols true by the above estimate. Error in logscale - -5 3 4 5 6 7 8 9 Polynomial egree Fig.. Spatial L errors at time T = for FCHE with ε = / using initial conition (45. 5. umerical examples We present several numerical examples to emonstrate the accuracy of the Fourier-Galerkin scheme an to illustrate the behavior of the solutions to problem (4. We use the stanar ealiasing technique escribe in [, Section 4.3.] to compute the non-linear term involving f in the scheme (4. Example (Verification of spectral convergence rate. Let us first verify the convergence with respect to. We choose ε = / with a smooth initial conition u (x, x = cos x cos x. (45 The L ( errors at time T = for ifferent values of fractional orer β are shown in Fig. on a semi-log scale. The true solution is unknown an we therefore use the Fourier-Galerkin approximation in the case = 8 as a reference solution. We chose the time step to be sufficiently small as to rener the error in the temporal approximation negligible. As we can see in Fig., the convergence has spectral ecay as preicte in Theorem 4. Example (Two kissing bubbles []. ext, we consier a stanar example for the Cahn Hilliar equation involving the coalescence of two kissing bubbles. The initial conition is u (x, y = {, (x + + y or (x + y,, otherwise (46 an we wet ε = /, = 56, δt =., S =. We assume that the numerical solution provies an accurate approximation of the true solution since Theorem 4 shows that the scheme is spectrally convergent. We present plots of the solution for ifferent values of fractional orer β at various times in Fig. to illustrate the effect of changing the value of fractional erivative β on the character of the solution of FCHE. The profiles of the steay solutions with respect to x ( x = for ifferent values of fractional orer β are shown in Fig. 3, an a zoome figure is given in Fig. 4. Observe that in all cases, as the solution evolves, the two bubbles coalesce into a single bubble, an that the volume is preserve. In this respect, the results are similar to these observe
7 M. Ainsworth, Z. Mao / Chaos, Solitons an Fractals (7 64 73 Fig.. Evolution of solutions of FCHE with initial conition (5 for ifferent values of fractional orer β. From top to bottom t =,,..5 T = =.6 =.8 =. u(x,x =.5 -.5 - -4-3 - - 3 4 x Fig. 3. The solution of steay state to FCHE at time T =. for the alternative FCHE iscusse in []. On the other han, the with of the interface becomes thinner with ecreasing values of β. We contrast this with what we observe [] where the with of the interface is the same for all values of fractional orer. We can explain these observation as follows. Let E(u = E (u + E (u, where E (u := ε (u(x u (y x y n + β, E ( u := F (u. The ouble well potential, F tens to favor solutions that have regions corresponing to pure phases in which u = + or u =. Assume that the with of interface layer between two pure phases
M. Ainsworth, Z. Mao / Chaos, Solitons an Fractals (7 64 73 7.5 T = =.6 =.8 =. u(x,x =.5 -.5 -.6.8..4.6.8..4 x Fig. 4. Zoom of Fig. 3 ( x [.6,.4]..8.6.4. u(x -. -.4 -.6 -.8 - -.5 - -.5 - -.5.5.5.5 x Fig. 5. Steay state of the phase for Example 3 with β =. 8, ε =.. is τ, from which we euce that E (u = O (τ. (47 We wish to show how τ is relate to ε an the value of fractional orer β. Without loss generality, we consier the one imensional case since we are only intereste in escribing the with of the layer. In orer to estimate E ( u, we nee to know how the energy of a function with a layer of with τ varies with the fractional orer β an τ. To this en, we consier a smooth function u τ : R R having a layer of with τ near to x = at which the value of the function changes O ( over a istance τ (see Fig. 5 obtaine by using Example 3 an ecaying at infinity. The Fourier transform is given by u ˆ τ (ξ = + = iξ u τ (x e iξ x x = τ τ iξ u τ (x e iξ x x [ iξ u τ (x e iξ x ] u τ (x e iξ x x + O ( = iξ = τξ u ( sin ξτ + O (, u (s e iξ sτ s + O ( where u (x = u τ (sτ, s ( /, /, is a smooth function inepenent of τ. Using the efinition of fractional erivative efine
7 M. Ainsworth, Z. Mao / Chaos, Solitons an Fractals (7 64 73.9 = 4, t =., S =.4.8.7 in logscale.6.5.4.3.8.85.9.95..5..5. Fig. 6. Layer with τ against ε for ifferent values of β, = 4, δt =.. on R, i.e., ( ν/ v (ξ = ξ ν v ˆ (ξ, v (x L (R, ν >, we have that the Fourier transform of u τ (β is ( β/ u τ (ξ = ξ β u ˆ τ (ξ = τ ξ β u ( sin ξτ + O ( ξ β. Then, by Parseval s equality, we have u (β τ u ( sin ξτ τξ β = u ( τ β ξ + O ( ξ β sin (t/ t β t + O ( ξ β = O (τ β, for / < β < 3 /. behaves like Consequently, we euce that for a function with a layer of with τ, E (u = ε O (τ β, / < β < 3 /. (48 Since the steay state minimizes the free energy E ( u which is the sum of E ( u an E ( u, we expect that the E ( u an E ( u shoul be of the same orer. Otherwise, if say, E ( u E ( u then the system woul not be at a minimum since the state coul evolve further to reuce E ( u without a corresponingly large increase in E ( u. In other wors, thanks to (47 an (48, we have ε τ β = O (, which gives τ = O (ε /β. (49 In particular, for small ε the above equation suggests that a smaller value of β woul give a narrower interface between the two phases, which agrees with the qualitative observation. In orer to check for quantitative agreement with estimate (49, we consier a simple D example in which the behavior of the with of the layer can be more easily iscerne an from which measurement can be mae. Example 3 Interface layer. Consier the D version of problem (4 with initial conition {, x >, u (x = (5, otherwise an we approximate using Fourier-Galerkin with = 4, δt =., S =. 4. The solution at time T = for β =. 8, ε =. is shown in Fig. 5. By proucing plots of solutions for various choices of ε an β, an measuring the with of the layer τ, we arrive at the graph shown in Fig. 6, showing the layer with τ against ε with ifferent values of β =. 6,. 8,.. Observe that the slopes obtaine for ifferent values of β agree with estimate (49. 6. Conclusions We have shown that the Cahn Hilliar equation with a fractional free energy is well-pose in the sense that it amits a unique solution which epens continuously on the ata. Moreover, the mass is conserve an energy reuces monotonically, an the thickness τ of the iffuse interface epens on ε an β accoring to τ = O (ε /β. While the analysis carrie out for the fractional graient, the extension to a more general kernel is possible. References [] Abels H, Bosia S, Grasselli M. Cahn Hilliar equation with nonlocal singular free energies. Ann Mat Pura Appl 5;94(4:7 6. oi:.7/ s3-4- 4-9. [] Ainsworth M, Mao Z. Analysis an approximation of a fractional Cahn Hilliar equation. SIAM J umer Anal 7. To appear [3] Akagi G, Schimperna G, Segatti A. Fractional Cahn Hilliar, Allen Cahn an porous meium equations. J Differ Equ 6;6(6:935 85. oi:.6/j.je. 6.5.6. [4] Cahn JW. Free energy of a nonuniform system. II. Thermoynamic basis. J Chem Phys 959;3(5: 4. [5] Cahn JW, Hilliar JE. Free energy of a nonuniform system. I. Interfacial free energy. J Chem Phys 958;8(:58 67. [6] Evans LC. Partial ifferential equations. Grauate Stuies in Mathematics, 9. American Mathematical Society, Provience, RI; 998. ISB -88-77-.
M. Ainsworth, Z. Mao / Chaos, Solitons an Fractals (7 64 73 73 [7] Fife PC. Moels for phase separation an their mathematics. Electron J Differ Equ. o. 48, 6 pp. (electronic. [8] Giacomin G, Lebowitz JL. Phase segregation ynamics in particle systems with long range interactions. I. Macroscopic limits. J Statist Phys 997;87( :37 6. oi:.7/bf8479. [9] Giacomin G, Lebowitz JL. Phase segregation ynamics in particle systems with long range interactions. II. interface motion. SIAM J Appl Math 998;58(6:77 9. oi:.37/s36399963346. [] He Y, Liu Y. Stability an convergence of the spectral Galerkin metho for the Cahn Hilliar equation. umer Methos Partial Differ Equ 8;4(6:485 5. oi:./num.38. [] Kopriva DA. Implementing spectral methos for partial ifferential equations: algorithms for scientists an engineers. Springer Science & Business Meia; 9. [] Liu C, Shen J. A phase fiel moel for the mixture of two incompressible fluis an its approximation by a fourier-spectral metho. Phys D 3;79(3 4: 8. oi:.6/s67-789(3 3-7. [3] Robinson JC. An introuction to orinary ifferential equations. Cambrige University Press, Cambrige; 4. oi:.7/cbo978584. ISB -5-865-; -5-5339-. [4] Shen J, Tang T, Wang L. Spectral methos: algorithms, analysis an applications. Springer;. [5] Shen J, Yang X. umerical approximations of Allen Cahn an Cahn Hilliar equations. Discrete Contin Dyn Syst ;8(4:669 9. oi:.3934/cs..8.669. [6] Simon J. Compact sets in the space L p (, T ; B. Ann Mat Pura Appl 987;46(4:65 96. oi:.7/bf7636.