THE CAUCHY PROBLEM AND STEADY STATE SOLUTIONS FOR A NONLOCAL CAHN-HILLIARD EQUATION

Transcription

1 Electronic Journal of Differential Equations, Vol. 24(24), No. 113, pp ISSN: URL: or ftp ejde.math.txstate.edu (login: ftp) THE CAUCHY PROBLEM AND STEADY STATE SOLUTIONS FOR A NONLOCAL CAHN-HILLIARD EQUATION JIANLONG HAN Abstract. We study the existence, uniqueness, and continuous dependence on initial data of the solution to the Cauchy problem and steady state solutions of a nonlocal Cahn-Hilliard equation on a bounded domain. 1. Introduction We are concerned with two different problems, the first being the Cauchy problem for a nonlocal Cahn-Hilliard equation u t = (ϕ(u) J u) in Rn (, T ), (1.1) u(x, ) = u (x), where ϕ(u) = u + f(u), f is bistable (e.g. f(u) = au(u 2 1) for some a > ), is convolution, and J = 1. R n The second problem is for the steady state equation J(x y)dyu(x) J(x y)u(y)dy + f(u) = C in, (1.2) u(x)dx =, where is a bounded domain, C is a constant. The case when = R or R n has been treated in [3, 5, 1, 11] and references therein. To derive equations (1.1) and (1.2), we consider the free energy E(u) = C J(x y)(u(x) u(y)) 2 dxdy + F (u(x))dx, (1.3) where C is a constant, F is the primitive of f, and u represents the concentration of one of the species of a binary material. Following [16], we consider the gradient flow for (1.3) in H 1 (), where H 1 is the space of distributions in the dual space of H 1 and with mean value zero. We do this since the total energy, E, decreases along the trajectories, and the average of u should be conserved. We have u t = grad H 1E(u). (1.4) 2 Mathematics Subject Classification. 35A5, 35M99. Key words and phrases. Phase transition; long range interaction; steady state solution. c 24 Texas State University - San Marcos. Submitted July 18, 24. Published September 29, 24. 1

2 2 JIANLONG HAN EJDE-24/113 Since the representative of grad E(u) in H 1 is grad H 1E(u) = (J(x y)dyu(x) J(x y)u(y)dy + f(u)), (1.4) gives u t = ( J(x y)dyu(x) J(x y)u(y)dy + f(u)). (1.5) In (1.3), making the approximation u(x) u(y) u(x) (x y), and assuming J to be isotropic, equation (1.4) leads to u = (d u f(u)), (1.6) t which is the classical Cahn-Hilliard equation. Equations (1.5) and (1.6) are important in the study of materials science for modelling certain phenomena such as spinodal decomposition, Ostwald ripening, and grain boundary motion. There is a lot of work on equation (1.6) (see for example [1, 2, 4, 9, 12, 13, 15, 14, 2] and references therein). For equation (1.5), there are very few results. In [6] and [7], we discussed the Neumann and Dirichlet boundary problems for (1.5). Here, we consider the Cauchy problem, where = R n and J = 1. Note that the steady state solutions for (1.5) in a bounded domain with no flux boundary condition satisfy the equation in (1.2) without the constraint. In this paper, we prove the global existence and uniqueness of solutions for equation (1.1). Also we prove the existence of nonconstant solutions for equation (1.2). The techniques used in the proof of the latter result can also be applied to the nonlocal phase field system discussed in [8]. We organize this paper as follows. In section 2, we establish the existence, uniqueness and continuous dependence on initial values for classical solutions of equation (1.1). In section 3, we prove that under certain conditions, there exists a discontinuous steady state solution for equation (1.2). 2. The Cauchy problem for the nonlocal Cahn-Hilliard equation For T >, let Q T = R n (, T ). We make the following assumptions: (D1) f C 2+β (R) and ϕ (u) c for some positive constants c and β, (D2) J C 2+β (R n ), J L 1 (R n ) L (R n ), and R n J = 1. First, we prove the uniqueness and continuous dependence of solutions on initial data. We have Proposition 2.1. Let u i (i = 1, 2) be two solutions of (1.1) with initial data u i (i = 1, 2). If conditions (D1) (D2) are satisfied, if u i C([, T ], L 1 (R n )) L (Q T ), and if u i L 1 (R n ) L (R n ) (i = 1, 2), then sup t T for some constant C(T ). R n u 1 u 2 dx C(T ) R n u 1 u 2 dx (2.1)

3 EJDE-24/113 THE CAUCHY PROBLEM 3 Proof. For any τ (, T ), and ψ C 2,1 (Q τ ), with ψ = for x large enough, after multiplying (1.1) by ψ, integrating over [, τ] R n, we have u i (x, τ)ψ(x, τ)dx R n τ τ = u i (x, )ψ(x, )dx + (u i ψ t + ϕ(u i ) ψ)dx dt ψ J u i dx dt. R n R n R n Set z = u 1 u 2, z = u 1 u 2, then the above equality gives τ z(x, τ)ψ(x, τ)dx = z (x)ψ(x, )dx + z(x, t)(ψ t + b(x, t) ψ)dx dt R n R n R n τ ψ J z(x, t)dx dt, R n (2.2) where { ϕ(u1) ϕ(u 2) b(x, t) = u 1 u 2 for u 1 u 2, (2.3) ϕ (u 1 ) for u 1 = u 2. Let g(x) C (R n ) have compact support, g(x) 1, and take λ >. We will choose ψ, above, to satisfy certain conditions. First, consider the following final value problem on a large ball () ψ t = b(x, t) ψ + λψ ψ = for x < R, < t < τ on x = R, < t < τ ψ(x, τ) = g(x) x R. (2.4) There exists a unique solution ψ C 2,1 ( () (, τ)) of of (2.4) which satisfies the following properties: τ () sup t τ ψ e λ(t τ), (2.5) b(x, t) ψ 2 dx dt C, (2.6) () ψ 2 dx C, (2.7) where the constant C depends only on g. To extend ψ to be zero outside of (), we define ξ R C (R n ) such that ξ R 1, ξ R = 1 if x < R 1, ξ R = if x > R 1 2, ξ R (x), ξ R (x) C (2.8) for some constant C which does not depend on R.

4 4 JIANLONG HAN EJDE-24/113 Let γ = ξ R ψ, where ψ satisfies (2.4) in () and is zero outside. Using γ instead of ψ in (2.2), we have z(x, τ)gξ R dx R n ξ R (x)z (x)ψ(x, )dx + R n ( J z λz)ξ R ψdx dt Q τ = b(x, t)z(x, t)(2 ξ R ψ + ψ ξ R )dx dt G(z, R). Q τ Since u 1 and u 2 belong to L (Q T ), and since b is positive, from estimates (2.5)-(2.7) and (2.8), we have τ G(z, R) (b u 1 u 2 ((2 ξ R ψ + ψ ξ R )) \ 1 τ C b( u 1 + u 2 )( ψ + 1)dx dt (2.9) \ 1 τ C ( u 1 + u 2 )dx dt. \ 1 Since u 1 and u 2 belong to L 1 (Q T ), letting R we have G(z, R). This implies τ z(x, τ)g(x)dx z (x) e λτ dx + ( J z λz e λ(t τ) dx dt. R n R n R n Letting λ and g(x) sign z + (x, τ), we obtain (u 1 u 2 ) + dx u 1 u 2 dx + C R n R n Interchanging u 1 and u 2 yields u 1 u 2 dx u 1 u 2 dx + C R n R n τ τ R n u 1 u 2 dx dt. (2.1) R n u 1 u 2 dx dt. Inequality (2.1) follows from the above inequality and Gronwall s inequality. Next we prove the existence of a solution to (1.1). Theorem 2.2. For any T >, if u (x) C 2+β (R n ), and if ϕ and J satisfy assumptions (D 1 ) (D 2 ), then there exists a unique solution of (1.1) which belongs 2+β 2+β, to C 2 (Q T ) L 1 (Q T ) L (Q T ). Proof. Since u (x) = for x large enough, we consider u t = (ϕ(u) J u) in () (, T ), u(x, t) = on () (, T ), u(x, ) = u (x). (2.11) 2+β 2+β, From [7, Theorem 2.4], there exists a unique solution u(x, t) C 2 ( () (, T )) of (2.11). Let u(x, t) = ve t in (2.11), we have e t v t + ve t = ϕ (u)e t v + ϕ (u) v 2 e 2t e t J v. (2.12)

5 EJDE-24/113 THE CAUCHY PROBLEM 5 Multiplying (2.12) by v and using v v = 1 2 v2 v 2, we obtain 1 2 (v2 ) t + v 2 = 1 2 ϕ (u) v ϕ (u) v v 2 e t ϕ (u) v 2 v (J v). (2.13) If there exists (P, t ) () (, T ] such that v 2 (P, t ) = max v 2, then we have v 2 (P, t ), v 2 (P, t ) =, v(p, t ) =, vt 2 (P, t ), and (2.13) yields v 2 (P, t ) J(P y)v(y, t )dyv(p, t ). (2.14) This yields max v M v(y, t ) dy (2.15) for some constant M which does not depend on R. Since u = is also a solution of (2.11) with initial data u =, by [7, Theorem 2.5], we have u(x, t) dx C(T ) u dx (2.16) for some constant C(T ) which does not depend on R. Inequalities (2.15) and (2.16) imply max v C(T ) u dx. (2.17) Since u L 1 (R n ), we have max v B(T ) (2.18) for some constant B(T ) which does not depend on R. This yields max u B(T )e T (2.19) for some constant B(T ) which does not depend on R. We have proved the solution of (2.11) is uniformly bounded, i.e., max u(x, t) C [,T ] for any R >, where C does not depend on R. A similar argument to that in the proof in [7, Theorem 2.2] yields u R 2+β C(K, T ) (2.2) for any R > K constant, where u R is a solution of (2.11) in (, T ) and C(K, T ) is a constant which does not depend on R ( 2+β is a Hölder norm defined in [19]). By employing the usual diagonal process, we can choose a sequence {R i } such that u Ri, Du Ri, and D 2 u Ri converge to u, Du, and D 2 u pointwise, and u satisfies equation (1.1). From (2.16) and (2.19), we also have u L 1 (Q T ) L (Q T ). Uniqueness follows from Proposition 2.1.

6 6 JIANLONG HAN EJDE-24/ Steady state solutions for the nonlocal Cahn-Hilliard equation In this part, we study equation (1.2). Proposition 3.1. Suppose R n is a closed and bounded set, J(x) and is continuous on R n, suppj B δ () for some positive constant δ, and f is nondecreasing. Then the only continuous solution of equation (1.2) is zero. Proof. Without loss of generality, we assume that f() =. If f(), we may use f(u) f() instead of f(u) in (1.2). Case 1: C in equation (1.2). If the conclusion is not true, since udx =, and u is continuous on, there exists P such that u(p ) = max u(x) >. Let A = {y : u(y) = max u(x)}. We claim: There exist P A and r > such that K := ( \ A) B r (P ) has positive measure. If this is not true, we have meas( \ A) =. This and u(x) = max u on A imply u = A u >. This contradicts u =. Since supp J B δ () implies supp J(P ) B δ (P ), choosing r 1 = min{δ, r} gives Inequalities (3.1)-(3.3) imply J(P y)(u(p ) u(y))dy This and f(u(p )) imply J(P y)u(p )dy meas(k B r1 (P )) >, (3.1) J(P y) > on K B r1 (P ), (3.2) u(p ) u(y) > on K B r1 (P ). (3.3) K B r1 (P ) J(P y)(u(p ) u(y))dy >. J(P y)u(y)dy + f(u(p )) >, (3.4) contradicting (1.2). Case 2: C > in (1.2). In this case, taking P such that u(p ) = min u < leads to a contradiction in a similar way. If f (u) changes sign, we make the following assumptions: (E1) = ( 1, 1) if dim = 1, = ( 1, 1) if dim > 1. (E2) J(x) = J( x ), J(x), and M sup J(x y)dy inf J(x y)dy m > x x for positive constants M and m. (E3) f C 1 (R) is odd, f(1) =, there exist δ > and a (, 1) such that f (u) δ on [a, ), and f( a) (1 + a)m. (E4) C = in (1.2). Remark 3.2. Condition (E3) implies that f( 1) =, f (u) δ on (, a], and f(a) (1 + a)m. Let = J(x y)dy. From (E 2), we have m M. (3.5)

7 EJDE-24/113 THE CAUCHY PROBLEM 7 Dividing (1.2) by, we consider u(x) 1 J(x y)u(y)dy + f(u(x)) =, u(x)dx =. (3.6) Theorem 3.3. If assumptions (E 1 ) (E 4 ) are satisfied, then there exists a solution of equation (3.6) such that { a for x M 1 (, 1), u(x) a for x M 2 ( 1, ) (3.7). Moreover, we have Proof. Following [3], we let 1 u(x) 1. (3.8) B = {u L () : u( x 1, x ) = u(x 1, x ), u(x) [a, 1] for x M 1 }. The definition of B implies that u(x) [ 1, a] for x M 2. Define 1 T u(x) = u(x) + h[ J(x y)u(y)dy u(x) 1 f(u(x))]. We want to show T : B B is a contraction map if h is small enough. In fact, since = J(x y)dy, with assumption (E2), we have j( x 1, x ) = j(x 1, x ). And if u(x) B, we have T (u( x 1, x )) = u( x 1, x h 1 ) + j( x 1, x J( x 1 y 1, x y )u(y 1, y )dy 1 dy ) 1 hu( x 1, x h ) + j( x 1, x ) f(u( x 1, x )) 1 = u(x 1, x h ) j(x 1, x J( x 1 + z 1, x y )u(z 1, y )dz 1 dy ) 1 + hu(x 1, x h ) j(x 1, x ) f(u(x 1, x )) 1 = u(x 1, x h ) j(x 1, x J(x 1 z 1, x y )u(z 1, y )dz 1 dy ) 1 + hu(x 1, x h ) j(x 1, x ) f(u(x 1, x )) 1 = (u(x 1, x h ) + j(x 1, x J(x 1 z 1, x y )u(z 1, y )dz 1 dy ) 1 hu(x 1, x h ) + j(x 1, x ) f(u(x 1, x ))) = T (u(x 1, x )). Choose h small enough such that (3.9) h 1 f (u) < 1 h (3.1)

8 8 JIANLONG HAN EJDE-24/113 for u [ 1, a] [a, 1] and x. This implies that u h[u + 1 f(u)] is increasing in u on [a, 1]. Since u(y) a for y M 1, and u(y) 1 for y M 2, we have for x M 1 T u(x) = h 1 J(x y)u(y)dy + u h[u + 1 f(u)] h 1 J(x y)u(y)dy + a ha h 1 f(a) = h 1 J(x y)u(y)dy + h 1 J(x y)u(y)dy + a ha h 1 M 1 M 2 f(a) ha 1 J(x y)dy h 1 J(x y)dy + a ha h 1 M 1 M 2 f(a) = a ha 1 J(x y)dy h 1 J(x y)dy h M 2 M 2 f(a) a h [(1 + a) J(x y)dy + f(a)] a M 2 by (E3). Also, T u(x) = h 1 h 1 J(x y)u(y)dy + u h[u + 1 f(u)] J(x y)u(y)dy + 1 h h 1 f(1) 1 (3.11) for x M 1. Estimates (3.9)-(3.11) imply that T maps B to B. For u, v B, choosing h small enough so that < 1 h(1 + δ 1 M ) < 1, we have T u T v = (u v) + h J(x y)(u(y) v(y))dy h(u(x) v(x)) h (f(u) f(v)) = (1 h hf (θu + (1 θ)v) )(u v) + h (1 h(1 + δ 1 M )) u v + h (u v) (1 hδ 1 M ) u v, J(x y)(u(y) v(y))dy where θ(x) (, 1) for all x. Here we used (E3) and the fact that for any x either u(x), v(x) a or u(x), v(x) a. Therefore, T is a contraction map from B to B. There exists a unique fixed point u(x) such that T u = u. Estimates (3.8) follows from the definition of B. Remark 3.4. If we just consider the solution to J(x y)u(x)dy J(x y)u(y)dy + f(u) = in (3.12)

9 EJDE-24/113 THE CAUCHY PROBLEM 9 without the condition udx =, then the conditions that f is odd and J(x) = J( x ) are not necessary. In this case, we can use a similar method to that in [3] to prove the existence of a discontinuous solution under conditions (E2), (E3), and (E4), where (E3) f C 1 (R), f( 1) = f(c) = f(1) = for c ( 1, 1), there exist δ >, a (, 1), b ( 1, ) such that f (x) δ on [a, ) (, b), f(a) (1 + a)m, and f(b) (1 + b)m, where M is defined in (E2). Acknowledgement. The author would like to thank Professor Peter W. Bates for his helpful discussions. References [1] N. D. Alikakos, P. W. Bates, and X. Chen; Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rat. Mech. Anal. 128 (1994), [2] P. W. Bates and P. C. Fife; The dynamics of nucleation for the Cahn-Hilliard equation, SIAM J. Appl. Math. 53 (1993), [3] P. W. Bates and A. Chmaj; An integrodifferential model for phase transitions: Stationary solution in Higher space dimensions, J. Stat. Phys. 95 (1999), [4] P. W. Bates and G. Fusco; Equilibria with many nuclei for the Cahn-Hilliard equation, J. Diff. Eq. 16 (2), [5] P. W. Batas, P. C. Fife, X. Ren, X. Wang; Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal. 138 (1997), [6] P. W. Bates and J. Han; The Neumann boundary problem for the nonlocal Cahn-Hilliard equation, to appear in J. Diff. Eq. [7] P. W. Bates and J. Han; The Dirichlet boundary problem for the nonlocal Cahn-Hilliard equation, submitted. [8] P. W. Bates, J. Han, and G. Zhao; The nonlocal phase field system, submitted. [9] J. W. Cahn and J. E. Hilliard; Free energy of a nonuniform system I, Interfacial free energy, J. Chem. Phys. 28 (1958) [1] A. Chmaj, X. Ren; Homoclinic solutions of an integral equation: existence and stability. J. Diff. Eq. 155 (1999), [11] X. Chen; Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Diff. Eq. 2 (1997), [12] J. G. Carr, M. E. Gurtin, M. Slemrod; Structured phase transitions on a finite interval, Arch. Rat. Mech. Anal. 86 (1984) [13] A. Novick-Cohen and L. A. Segel; Nonlinear aspects of the Cahn-Hilliard equation, Phys. D 1 (1984), [14] Q. Du and R. A. Nicolaides; Numerical analysis of a contunuum model of phase transition, SIAM J. Numer. Anal. 28 (1991), [15] C. M. Elliott and S. Zheng; On the Cahn-Hilliard equation, Arch. Rat. Mech. Anal. 96 (1986), [16] P. C. Fife; Dynamical aspects of the Cahn-Hilliard equations, Barett Lectures, University of Tenessee, [17] H. Gajewski and K. Zacharias; On a nonlocal phase separation model, J. Math. Anal. Appl. 286 (23), [18] G. Giacomin and J. L. Lebowitz; Phase segregation dynamics in particle systems with long range interactions II: Interface motion, SIAM J. Appl. Math. 58 (1998), [19] O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Uralceva; Linear and quasilinear equations of parabolic type, Volume 23, Translations of Mathematical Monographs, AMS, Providence, [2] B. Nicolaenko, B. Scheurer and R. Temam; Some global dynamical properties of a class of pattern formation equations, Comm. P. D. E. 14 (1989), Jianlong Han Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA address: hanjianl@math.msu.edu