On Two Nonlinear Biharmonic Evolution Equations: Existence, Uniqueness and Stability

Transcription

1 On Two Nonlinear Biharmonic Evolution Equations: Existence, Uniqueness and Stability Ming-Jun Lai, Chun Liu, and Paul Wenston Abstract We study the following two nonlinear evolution equations with a fourth order (biharmonic) leading term: 2 u ɛ 2 ( u 2 )u = u t in R 2 or R 3 and 2 u + ɛ 2 (( u 2 ) u) = u t in R 2 or R 3 with an initial value and a Dirichlet boundary conditions. We show the existence and uniqueness of the weak solutions of these two equations. For any t [, + ), we prove that both solutions are in L (, T, L 2 ()) L 2 (, T, H 2 ()). We also discuss the asymptotic behavior of the solutions as time goes to infinity. This work lays the ground for our numerical simulations for the above systems in [Lai, Liu and Wenston 3]. AMS 2 Mathematics Subject Classifications: 35J35, 35K55, 35J6 Keywords and Phrases: nonlinear Biharmonic evolution equations, Ginzburg-Landau type equations, liquid crystals, existence and uniqueness, asymptotic behavior. Introduction In this paper, we are studying the following two biharmonic equations 2 u ɛ 2 ( u 2 )u = u t, in R 2 or R 3 (.) u(x, ) = u (x), x u(x, t) = u (x) x u n (x, t) = u 2(x) x Department of Mathematics, University of Georgia, Athens, GA 362. This author is supported by the National Science Foundation under Grant #DMS His address is mjlai@math.uga.edu. Department of Mathematics, Pennsylvania State University, University Park, PA 682. His address is liu@math.psu.edu. Department of Mathematics, University of Georgia, Athens, GA 362. His address is paul@math.uga.edu.

2 2 Biharmonic Evolution Equations and (.2) 2 u + ɛ 2 (( u 2 ) u) = u t, in R 2 or R 3 u(x, ) = u (x), x u(x, t) = u (x) x u n (x, t) = u 2(x) x, where is a bounded domain in R 2 or R 3 and n is the outward normal of the boundary. Equation (.) may be regarded as the gradient flow of the following energy functional: (.3) E(u) := 2 in the class ( u 2 + 2ɛ 2 ( u 2 ) 2 (.4) C = {u H 2 (), u = u, ) dx n u = u 2 }. Similarly, equation (.2) is the gradient flow of the energy functional (.5) E(u) := ( u 2 + ) 2 2ɛ 2 ( u 2 ) 2 dx in the class (.4). Much effort has been devoted to the study of the gradient flow of lower order energy, in particular, the following Ginszburg-Landau type energy, with u in both scalar and vector cases: ( (.6) min u 2 + ) u H () 2ɛ 2 ( u 2 ) 2 dx. u =u The Euler Lagrange equation is as follows: { (.7) u + ɛ 2 ( u 2 )u = u = u. in Such problem has been studied for many years (cf., e.g., [Allen and Cahn 79] and [Bronsard and Kohn 9]). The study of the above energy (.6) was partially motivated by [Allen and Cahn 79] to understand the motion of free interfaces between different phases. The vector Ginzburg-Landau equation (cf. [Du, Gunzburger and Peterson 92]) is very important in the understanding the vortices in the superconductivities. Our study is also motivated by the following problem in liquid crystals. Recall that the usual nematic molecule configuration is determined by minimizing the following Oseen Frank energy: E(n) = k (divn) 2 + k 2 (n curl(n)) 2 + k 3 (n curl(n)) 2 + (k 4 + k 2 )(trace( n) 2 (divn) 2 ) 2

3 M. J. Lai, C. Liu, and P. Wenston 3 which is equal to n 2 in the case of equal constants. Here, the unit vector n stands for molecule orientation. A special type of liquid crystals is called smectic. The molecule posses certain positional order (layer structure). In this case, n = φ where the level sets of φ will represent the layer structure in the sample. In order to study this new energy φ 2 dx under constraint φ =, it is natural to introduce the above penalized energy (.5). We hope that as ɛ, the minimizer of the above energy, or the solution of the Euler Lagrange equation (.2) will be convergent to the corresponding one with the nonlinear constraint φ = (cf. [Kinderlehrer and Liu 96] and [E 97]). Such a singularly perturbed variational problem also arises in the study of thin film blisters (cf. [Ortiz and Gioia 94]), where the scalar function φ will stand for the height of the blistered film and φ 2 represents the bending energy. One well-known conjecture is to show that the limit solution is φ(x) = dist(x, ) (cf. [Jin 97] and [Kohn 96]). Thus, for both equations, it is important to study the asymptotic behavior as ɛ. However, little is known about the variational problem involving the second order derivative term as those in (.3) and (.5). One of the main difficulties is the lack of the maximum principle. Our study of the prototype problems (.) and (.2) is in the direction of trying to understand the effectiveness of the second order singular perturbation. Most of our estimates depend on the size of ɛ. In 2, we first obtain the existence and uniqueness results of the weak solution of (.) and then establish the regularity of such weak solutions. After that we let t + and get the existence of the solution of the steady state case. The uniqueness of the weak solution of the steady state case can be achieved when ɛ is not very small. We also obtain the existence results of the weak solution of (.2) in 3. If the weak solution is the exact solution, as t +, we show that the time dependent solution converges to the solutions of the steady state case. We also perform numerical simulations to the equations (.) and (.2) together with their steady state equations. The detail of our numerical experiments and related analysis will be presented in [Lai, Liu and Wenston 2]. There we have applied a special bivariate spline space to numerically solve these equations. We notice many interesting phenomena, for instance, indications of possible nonuniqueness of the stationary solutions and selfsimilar structures. We include some examples in The Study of Equation (.) In this section, we study the existence, uniqueness and regularity of the solutions of (.). We also discuss the long time behavior and hence the stability property of the solutions. We start with the energy law of the system (.). It comes from the natural of being the gradient flow of the system. Lemma 2.. Suppose that u(x, t) is a smooth solution of (.). Then the following 3

4 4 Biharmonic Evolution Equations equality holds: (2.) ( d u 2 + ) 2 dt 2ɛ 2 ( u 2 ) 2 dx = u t 2 dx. Proof. Multiplying equation (.) by u t and integrating over the domain, we get u 2 t dx = ( 2 u + ɛ 2 ( u 2 )u)u t dx = d ( u dt 2ɛ 2 ( u 2 ) 2 )dx. The second equality employed integration by parts and the fact that the left hand side of (.) is the variation of (2.2) E(u) = 2 ( u 2 + 2ɛ 2 ( u 2 ) 2 )dx with respect to u. We also point out that since the boundary values of u does not change with respect to time t, we have u t =. We have thus established the identity. Lemma 2.2. Let u(x, t) be a smooth solution of (.) with u = u 2 =. Then (2.3) d u 2 dx + ( u dt ɛ 2 ( u 2 ) 2 + ɛ 2 u 2 )dx = ɛ 2. Moreover, the L (, T, L 2 ()) and L 2 (, T, H 2 ()) norms of u are bounded. Proof. Equality comes from by multiplying (.) by u and integrating by parts. Then we have d u 2 dx = u 2 dx 2 dt ɛ 2 ( u 2 ) u 2 dx = ( u 2 + ɛ 2 ( u 2 ) 2 )dx ɛ 2 ( u 2 )dx which yields (2.3). Moreover, integrating (2.3) with respect to t [, T ], we get (2.4) 2 u 2 dx + = T ɛ T u 2 dx. ( u 2 + ɛ 2 ( u 2 ) 2 + ɛ 2 u 2 )dxdt It follows that the L (, T, L 2 () and L 2 (, T, H 2 () norms of u are bounded. completes the proof. This 4

5 M. J. Lai, C. Liu, and P. Wenston 5 Remark. In case of non-homogeneous boundary conditions (.8), we let φ(x) H 2 () solve the following biharmonic problem with Dirichlet boundary conditions 2 φ = in (2.5) φ(x) = u (x) n φ(x) = u 2(x) on on The existence of φ is obvious (cf. [Grisvard 85]). Subtracting φ from u, we see the new solution called u again satisfies the following (.) u t + 2 u + ɛ 2 ( u 2 )u = f(u), u(x, ) = u (x), u = n u =. where f(u) comes from the nonlinear term in (.) and in fact in x f(u) = 3 ɛ 2 φu2 3 ɛ 2 φ2 u ɛ 2 φ3 + ɛ 2 φ. Then Lemma 2.2 is also true for the u which satisfies the nonhomogeneous boundary condition. In fact we have Lemma 2.3. Let u(x, t) be a smooth solution of (.). C(ɛ, T ) independent of u such that (2.6) u 2 dx + T ( u 2 + 2ɛ 2 u 4 )dxdt C(ɛ, T ). Proof. Multiplying u both sides of (.), and integrating over, we have Then there exists a constant d u 2 dx + ( u dt ɛ 2 ( u 2 )u 2 ) = f(u)udx. That is, letting m = max φ(x) < by Sobolev Embedding Theorem, x d u 2 dx + ( u dt ɛ 2 u 4 )dx ( + 3m2 ) ɛ 2 u 2 dx + 3 ɛ 2 m u 3 dx + (m3 + m) ɛ 2 udx C (ɛ, m) u 2 dx + 3m ɛ 2 ( u 4 dx) /2 ( u 2 dx) /2 +. 5

6 6 Biharmonic Evolution Equations It follows that d u 2 dx + 2 dt u 2 dx + ( ɛ 2 δ) u 4 dx C 2 (ɛ, m) u 2 dx + for an arbitrary small δ and for a constant C 2 dependent only on ɛ and m. Then a standard argument implies ( ) d e C 2(ɛ,m)t u 2 dx + dt ( 2e C 2(ɛ,m)t u 2 dx + ( ) ɛ 2 δ) u 4 dx e C2(ɛ,m)t and hence, e C 2(ɛ,m)t t u 2 dx + 2 e C 2(ɛ,m)(t s) u 2 dx + t ( u 2 + ( ɛ 2 δ) u 4 ) dx e C 2(ɛ,m)(t s) ds. This completes the proof. We now introduce the weak formulation for (.): find u L 2 (, T, H 2 ()) such that u t vdx+ u vdx + ɛ 2 (u 2 )uvdx = f(u)vdx for all v H 2 (). Let V N H 2 () be a finite dimensional space for each N Z + and for all u H 2 () there exists a sequence u N V N such that u n u in H 2 () norm. We remark here that there are many choice of these spaces V N s. For instance, since Lv = 2 v is self-adjoint with zero boundary conditions, we can choose V N to be the space spanned by the first N eigen functions of L which has an orthonormal basis. For our practice, we shall use the bivariate spline space v N = S3r( + r N ) where ( + N ) is the N th uniform refinement of triangulated quadrangulation +. This will transform (.) into a system of ordinary differential equations in the coefficients of u V N with respect to time variable t. This is the exact spline space we implemented as finite elements in [Lai, Liu and Wenston 2]. The spline space mentioned above is defined as S r 3r( +) = {S C r () : s t P 3r, t +}, where P 3r is the space of all polynomials of degree 3r and + is a triangulation of the domain R 2 which is obtained from a non-degenerate quadrangulation of by adding two diagonals of each quadrilateral in (cf. [Lai and Schumaker 99]). It is known that for any finite dimensional ODE system always, a local solution always exists. To prove the global existence of the finite dimensional ODE system, we only need to prove the following a priori estimate 6

7 M. J. Lai, C. Liu, and P. Wenston 7 Lemma 2.4. Let u N V N be the weak solution satisfying (2.7) = [ t (u N ) vdx + u N v + ] ɛ 2 ((u N) 2 )u N v dx f(u N )vdx, for t (, T ) and for all v V N with u N = n V N = and u N (x, ) = u,n (x) which is the projection of u (x) in V N. Then for arbitrary t (, ), (2.8) t u N 2 dx + 2 e C 2(ɛ,m)(t s) u 2 dxds e C 2(ɛ,m)t u,n 2 dx + C(t, ɛ, ) for a positive constant C(t, ɛ, ). Proof: We replace u with u N in the proof of Lemma 2.3. This completes the proof. It follows from (2.8) that u N L (, T, L 2 ()) and u N L 2 (, T, L 2 ()). These in fact imply that u N L 2 (, T, H 2 ()) by the Poincaré inequality. (2.8) also implies the global existence of the solution of (2.7). That is, for any fixed T >, there exists a smooth solution u N V N such that (2.7) holds for v V N and all t (, T ). Next, by Rellich s Compactness Theorem (cf.[adam 75]), there exists a subsequence in {u N } N Z+, say {u N } itself, convergent weakly in L 2 (, T, H 2 ()) as well as weakly in L 2 (, T, H ()) to u. Since the nonlinearity in each term of (2.7) is below the critical power of the Sobolev Embedding Theorem, by passing the limit in the integration of (2.7) with respect to t over [, T ], we know that the limit u is the weak solution of (2.6). Therefore, we have obtained the following Theorem 2.(Existence). for any fixed T >, there exists a weak solution u of (.) such that (2.9) u L (, T, L 2 ()) L 2 (, T, H 2 ()). Remark 3. Theorem 2. implies that there exists a weak solution u for the original system (.). Next we study the uniqueness of the weak solution for each ɛ >. Theorem 2.2(Uniqueness). The weak solution in Theorem 2. is in fact unique. Proof: Suppose that u and u 2 are two different solutions of (.) and that both u and u 2 satisfy (2.9). Then letting w = u u 2, we have d dt (w)vdx + w vdx + G(u, u 2 )wvdx = 7

8 8 Biharmonic Evolution Equations for all v H 2 (), where G is a polynomial in u and u 2 of degree at most 2. Taking v = w, we have d (2.) w 2 dx + 2 w 2 dx C(ɛ) w 2 dx dt for a constant C(ɛ) dependent on the L () norm of u and u 2. Since w t= =, (2.) implies w 2 dx for t >. That is, u u 2. This completes the proof. The above result is not necessarily true for the steady state case. See Theorem 2.4 for a partial answer. For a fixed ɛ >, and when the boundary of the domain is smooth, the weak solution u we get in Theorem 2. is in fact a classical C smooth solution. This is the consequence of (2.9) and the regularity theory of the biharmonic heat equation (cf. [Grisvard 85]). Furthermore, let us discuss the asymptotic behavior of (.) as t +. We have Theorem 2.3. Suppose that u H 2 (). The solution of (.) subsequently converges to the solution of the steady state problem as t +. That is, for almost all sequence {t j } with t j +, there exists a subsequence convergent to a weak solution of the steady state problem. Proof: By Lemma 2., we have T u t 2 dxdt = u(x, T ) ɛ 2 ( u(x, T ) 2 ) 2 )dx + ( u(x, ) ɛ 2 ( u 2 ) 2 )dx ( u(x, ) ɛ 2 ( u 2 ) 2 )dx Since the solution u is smooth, we have u t L 2 (, +, L 2 ()). This implies u t (x, t j ) 2 dx for almost all sequence {t j } j Z+ (, + ) with t j +. Moreover, by Fatou s Lemma, u t (x, t j ) converges weakly to zero in L 2 (). Furthermore, Lemma 2. also implies ( u 2 + ɛ 2 ( u 2 ) 2 )dx ( u 2 + ɛ 2 ( u 2 ) 2 )dx for any t (, + ). That is, u L (, +, L 2 ()). Since u 4 dx = ( u 2 + ) 2 dx 2 ( u 2 ) 2 dx + 2, we have u 4 dx C(ɛ) < + for any t (, + ). Thus, u L (, +, L 4 ()). 8

9 M. J. Lai, C. Liu, and P. Wenston 9 It now follows that u L (, +, H 2 ()). There exists a subsequence {u(x, t j ), j Z} convergent weakly in H 2 (). For simplicity, let us say u(x, t j ) u (x) weakly in H 2 (). Moreover, u(x, t j ) u (x) in L p (), p > 2. By passing t j + in the following equation, t u(x, t j)v(x)dx + = f(u(x, t j ))v(x)dx, we have obtained [ u(x, t j ) v(x) + ] ɛ 2 ( u(x, t j) 2 )u(x, t j )v(x) dx ( u (x) v + ) ɛ 2 ( u 2 )u v dx = f(u )v(x)dx or u is a solution of the steady state problem. This completes the proof. We now study the properties of the weak solution u of the steady state problems satisfying (2.) for v H 2 (). We have [ u vdx + ] ɛ 2 ( u 2 )uv dx = f(u)vdx Lemma 2.5. Let u be a weak solution satisfying (2.). Then, for a positive constant C(ɛ), (2.2) u 2 dx C(ɛ) Proof: Let v = u in (2.) and recall m = max φ <, we have x u 2 dx+ ɛ 2 u 4 dx + 3m2 ɛ 2 u 2 dx + 3m ɛ 2 u 3 dx + m( + m2 ) ɛ 2 + 3m2 ɛ 2 u 2 dx + 3m ( ) ( ɛ 2 u 4 2 dx u 2 dx + m( + ( ) m2 ) ɛ 2 u 2 2 dx C ɛ 2 u 2 dx + 2ɛ 2 u 4 dx + C 2 ɛ 2 9 u dx ) 2

10 Biharmonic Evolution Equations for some constants C and C 2. Thus u 2 dx + 2ɛ 2 u 4 dx C ɛ 2 C ɛ 2 2ɛ 2 ( u 2 dx + C 2 ɛ 2 u 4 dx) /2 + C2 u 4 dx + C2 2ɛ 2 + C 2 ɛ 2. The inequality (2.2) thus follows. Regarding the uniqueness of the weak solution u of the steady state problem, we have Theorem 2.4. If ɛ is not very small, i.e., ɛ > K 2, then the weak solution u of (2.) is unique, where K is the Poincaré constant dependent on. Proof: Suppose that there exists two solutions u and u 2 in H 2 (). Let a 2 (φ, ψ) = φ ψdx and b(θ, φ, ψ) = ɛ 2 ( θ 2 )φψdx. Then we have a 2 (u u 2, ψ) + b(u, u, ψ) b(u 2, u 2, ψ) = 3 ɛ 2 φ(u 2 u 2 2)ψdx + 3 ɛ 2 φ 2 (u u 2 )ψdx. Let ψ = u u 2 H 2 (). Then ɛ 2 (a 2 (u u 2, ψ) + b(u, u ψ) b(u 2, u 2, ψ)) =ɛ 2 (u u 2 ) 2 dx + u u 2 4 dx u 2 u dx and 3 φ(u 2 u 2 2)ψdx 3 φ 2 (u u 2 )ψdx 3 u 2 u Combining the above equality and inequality, we have ɛ 2 (u u 2 ) 2 dx + u u 2 4 dx 4 u u 2 2 dx K 4 (u u 2 ) 2 dx u u 2 2 dx where we have used the Poincaré inequality twice. Thus, if ɛ > K 2, then u u 2. This completes the proof. We will end the section by looking at the self-dual solutions of the equation (.) in the D stationary case. For simplicity, we set ɛ =. That is, since u = u 2, we have 2 (u ) 2 = 3 u3 u + C.

11 M. J. Lai, C. Liu, and P. Wenston Clearly, the above solution can not be an odd function. If the boundary conditions are and, this hints that the sharp interface is unlikely to appear as ɛ. This is different from the solution of the self-dual of the energy of (.6) in the -D case. Our numerical simulation of the stationary (.) in the 2-D boundary value problem also is consistent with the above observation. 3. The Study of Equation (.2) In this section, we study the existence, uniqueness, and regularity of the solutions of nonlinear biharmonic equation (.2). The higher order nonlinearity of (.2) than (.) will play a crucial role in the analysis as well as in our numerical simulation (cf. [Lai, Liu and Wenston 2]). Again let ϕ H 2 () be the weak solution of the following biharmonic equation 2 ϕ = x Then û = u ϕ satisfies the following ϕ(x) = u (x) n ϕ(x) = u 2(x) 2 û + ε 2 (( û 2 ) û) x x. + ε 2 (( û 2 ) ϕ) + 2 (( û ϕ) û) ε + 2 ε 2 (( û ϕ) ϕ) + ε 2 ( ϕ 2 û) + ε 2 ( ϕ 2 ϕ) = û t For convenience, write u = û and let f(u) = ε 2 ( u 2 ϕ) + 2 ( u ϕ u) ε2 + 2 ε 2 ( u ϕ ϕ) + ε 2 ( ϕ 2 u) ε 2 ϕ + ε 2 ( ϕ 2 ϕ). Then we may rewrite the above nonlinear biharmonic equation in the following simplified form u t + 2 u (3.) ε 2 (( u 2 ) u) = f(u) u(x, ) = u (x) ϕ(x) x u(x, t) =, nu(x, t) = on. We now introduce the weak formulation: Find u L 2 (, T, H 2 ()) s.t. u(x, ) = u (x) ϕ(x) and [ (3.2) u t vdx = u v + ] ε 2 ( u 2 ) u v dx + f(u)vdx

12 2 Biharmonic Evolution Equations for all v H 2 (). Suppose that has smooth boundary and u H 3/2+σ ( ), u 2 H /2+σ ( ) with σ >. Then m = max ϕ(x) < + x (cf. [Grisvard 85]). We are now ready to prove the following L 2 estimate. Notice the differences between Lemma 2.2 and the following Lemma 3. due to the nonlinearity of (.2). Lemma 3.. Suppose that u is a weak solution of (3.2). Then there exists a constant C(ε, m) > such that t u 2 dx + e C(ε,m)(t s) u 2 dxds e C(ε,m)t + u 2 dx. Proof: Let v = u in (3.2). We have d u 2 dx = 2 dt + That is, Note that We have 3m ε 2 d u 2 dx + 2 dt = ( u 2 + ε 2 ( u 2 ) u 2 ) dx f(u)udx. u 2 dx + ε 2 f(u)udx + ε 2 u 4 dx u 2 dx. f(u)udx = 3 ε 2 u 2 ϕ udx 2 ε 2 u ϕ 2 dx ε 2 ϕ 2 u 2 dx + ε 2 ϕ udx ε 2 ϕ 2 ϕ udx 3m ε 2 u 2 u dx 2 ε 2 u ϕ 2 dx ε 2 ϕ 2 u 2 dx + m ε 2 u dx + m3 ε 2 u dx. u 2 u dx 3m ( ε 2 ε 2 ) /2 ( u 4 dx u 4 dx + 9m2 4ɛ 2 2 ) /2 u 2 dx u 2 dx

13 and m + m 3 ε 2 M. J. Lai, C. Liu, and P. Wenston 3 u dx m + m3 ( ) /2 ε 2 u 2 dx m + m3 2ε 2 + m + m3 2ε 2 u 2 dx. Thus, we put the above inequalities together to have d u 2 dx + u 2 dx + 2 dt ε 2 ϕ 2 u 2 dx + 2 ε 2 u ϕ 2 dx m + ( m3 9m 2 2ε 2 + 4ɛ 2 + m + ) m3 2ε 2 u 2 dx. Note that Thus, we have u 2 dx C u 2 dx + C 2 u 2 dx. d u 2 dx + u 2 dx C(ε, m) + C(ε, m) u 2 dx. dt A standard argument now completes the proof of Lemma 3.. Let V N = S3r( +) r N ) H 2 () be the finite dimensional subspace of H 2 () introduced before. Let u N V N be a weak solution satisfying (3.3) t (u N)vdx = ( u N v + ε 2 ( u N 2 ) u N v)dx + f(u N )vdx for all v V N with initial value u N (x, ) = u,n (x) ϕ N (x) which is a spline approximation of u ϕ, e.g., u N (x) ϕ N (x) = Q N (u ϕ) as in [Lai and Schumaker 98]. The equation (3.3) is a finite dimensional ODE system. We know that u N exists for t [, T N ) with T N T. We need to show T N T. Otherwise, T N < T implies that u N will blow up when t T N. However, the proof of Lemma 3. shows that Lemma 3.2. Let u N V N be a solution of (3.3) for t [, T N ). Then (3.4) t u N 2 dx + e C(ε,m)(t s) u N 2 dxds e C(ε,m)t + u,n ϕ N 2 dx. Since u,n ϕ u ϕ in L 2 (), we have u N L (, T, L 2 ()) and u N L 2 (, T, L 2 ()). 3

14 4 Biharmonic Evolution Equations These facts imply u N L 2 (, T, H 2 ()). We know, by Rellich s compactness theorem, there exists a subsequence in {u N } N Z+, say {u N } convergent weakly in H 2 () and strongly in W,p () to u for p < +. Note that u N 2 u N vdx u 2 u vdx u N 2 ( u N u) vdx + ( u N 2 u 2 ) u vdx ( ( + ) /2 ( u N 4 dx ) /2 ( u N u 2 dx ) /4 ( u N u 4 dx ) /4 v 4 dx u N + u 2 u v 2 dx as N + for each t [, T ). We next claim that f(u N )vdx f(u)vdx. First of all, we have u N 2 ϕ vdx u 2 ϕ udx ( ) /2 ( ) /2 u N u 2 dx u N + u 2 ϕ u 2 dx ( ( ( ) /2 ( u N u 2 dx ) ( /2 ( u N u 2 dx ) /8 ( ϕ 8 dx v 8 dx, as N. u N + u 4 ) /4 ( /4 ( u N dx) 4 + ) /8 ) /2 ) /4 ϕ v 4 dx ) ) /4 u 4 dx Secondly, we have u N ϕ u N vdx u ϕ u vdx u N ϕ( u N u) vdx + ( u N u) ϕ u vdx ( ) /2 ( ) /4 ( /8 2 u N u 2 dx u N 4 dx ϕ dx) 8 ( ) /8 v 8 dx, as N +. 4

15 M. J. Lai, C. Liu, and P. Wenston 5 Thus, we have the claim. This completes the proof of the following Theorem 3. (Existence). For any fixed T >, there exists a weak solution u satisfying (3.2). By Lemma 3., u L (, T, L 2 () L 2 (, T, H 2 ()). Hence, there exists a weak solution u for the original system (.2). Next we study the uniqueness of the weak solution for each ε >. We have Theorem 3.2(Uniqueness). The weak solution in Theorem 3. is in fact unique. Proof: Suppose that u and u 2 are two different solutions of (.2). Letting w = u u 2, we have t (w)vdx + w vdx + ε (3.5) 2 u 2 u vdx ε 2 u 2 2 u 2 vdx ε 2 w vdx = for all v H 2 (). That is, we have t (w)vdx + w vdx + ε 2 u 2 w vdx + ε 2 = ε 2 w vdx ε 2 u 2 w u 2 vdx w u u 2 vdx. Let v = w. We have ( ) d w 2 dx + w 2 dx + 2 dt ε 2 u 2 w 2 dx + ε 2 = ε 2 w 2 dx ε 2 ( w u w u 2 )dx. Similarly, letting w = u 2 u, we have ( ) d w 2 dx + w 2 dx + 2 dt ε 2 u 2 2 w 2 dx + ε 2 = ε 2 w 2 dx ε 2 ( w u w u 2 )dx. u 2 w 2 dx u w 2 dx Since w = w, we add the above two equations together to get ( ) d w 2 dx + 2 w 2 dx + dt ε 2 u 2 w 2 dx+ ε 2 u 2 2 w 2 dx + ε 2 u w 2 dx + ε 2 u 2 w 2 dx = 2 ε 2 w 2 dx 2 ε 2 ( w u w u 2 )dx 2 ε 2 w 2 dx + ( ) ε 2 ( w u ) 2 dx + ( w u 2 ) 2 dx. 5

16 6 Biharmonic Evolution Equations That is, we have ( d dt ε 2 2 ε 2 ) w 2 dx + 2 w 2 dx+ u 2 w 2 dx + ε 2 u 2 2 w 2 dx w 2 dx C(ɛ) w 2 dx + w 2 dx for a constant C(ɛ) dependent only on ɛ. Thus, we multiply e C(ɛ)t both sides of the inequality above to get ) d (e C(ɛ)t w 2 dx. dt Since w 2 dx = when t = we have e C(ɛ)t w 2 dx for all t [, T ]. Hence w 2 dx = for all t [, T ]. That is, u u 2. We have thus complete the proof. Furthermore, let us discuss the asymptotic behavior of (.2) as t +. We have Lemma 3.3. Let u(x, t) be a smooth solution of (.2). Then the following equality holds (3.6) d ( u dt 2ε 2 ( u 2 ) 2 )dx = u t 2 dx The proof is the same as that of Lemma 2.. We may omit the details. With the above lemma, we can prove the following Theorem 3.3. Suppose that u H 2 (). Then the smooth solution of (.2) subsequently converges to the solution of the steady state problem on t +. Proof: By Lemma 3.3, we have T u t 2 dxdt ( u 2 + 2ε 2 ( u 2 ) 2 ) dx which is true for any t [, + ). This implies u t L 2 (, +, L 2 ()). That is, u t (x, t j ) 2 dx as t j +. This implies u t (x, t j ) converges to zero weakly in L 2 (). 6

17 Another application of Lemma 3.3 yields M. J. Lai, C. Liu, and P. Wenston 7 ( u(x, t) 2 + ε 2 ( u(x, t) 2 ) 2 )dx ( u 2 + ε 2 ( u 2 ) 2 )dx for any t [, + ). That is, u L (, +, L 2 ()). It also follows that u L (, +, L 4 ()). By Poincaré s inequality we have u 2 dx K u 2 dx K /2 ( ) /2 u 4 dx which implies u L (, +, L 2 ()). Together with the facts u L (, +, L 2 ()) and u L (, +, L 2 ()) we know that u L (, +, H 2 ()). The boundedness of {u(x, t j )} j Z+ in H 2 () implies that there exists a subsequence in {u(x, t j )} j Z+, say {u(x, t j )} j Z+ converges to u H 2 () weakly. By Rellich s compactness theorem, { u(x, t j )} j Z+ converges to u in L p (), p 2. Recall the fact that u j (x, t j ) converges to zero weakly in L 2 (). By passing t j + in the following we obtain u t (x, t j )vdx + = [ u(x, t j ) v + ) ε 2 ( u(x, t j) 2 ) u(x, t j ) v dx f(u(x, t j ))vdx, [ u v + ( u ε 2 2 ) ] u v dx = f(u )vdx. That is u is a solution of the steady state problem. This completes the proof. We notice that in general the weak solution of (.2) may not be smooth enough to make a sense of (3.6) in Lemma 3.3. We next study the properties of the weak solution u of the steady state problems satisfying (3.7) for v H 2 (). We have the following [ u vdx + ] ɛ 2 ( u 2 ) u v dx = f(u)vdx Lemma 3.4. Let u be a weak solution satisfying (3.7). Then, for a positive constant C(ɛ), (3.8) u 2 dx C(ɛ). The proof is similar to that of Lemma 2.5. We omit the detail. Regarding the uniqueness of the weak solution u of the steady state problem, we have 7

18 8 Biharmonic Evolution Equations Theorem 3.4. If ɛ is not very small, i.e., ɛ > K 2 then the weak solution u of (3.7) is unique, where K is the Poincaré constant which is dependent on. The proof is again similar to that of Theorem 2.4. We leave the detail to the interest reader. 4. Some Numerical Examples We want to end the paper by presenting some numerical examples in this section. The examples here are the stationary solutions of equations (.) and (.2). The solutions indicate many interesting nonlinear effects. Example. We simulate the solution of 2 u + ɛ ( u 2 )u = 2 u 2 = on the upper half domain u 2 = on the lower half domain u n = (4.) with = [, /2] [, ]. The following graphs are numerical solution for ɛ =. and ɛ = Fig. 4.. Numerical solution of equation (4.) and the middle cross section with ɛ =. We observed that the surfaces pop up in the middle of which are not like the solution of (.7) which is subject to the maximum principle. We also observe that the portion of the surfaces transiting from to does not become narrow rapidly as ɛ. This is another indication of the possible absence of the sharp interface, even with the help of the prescribed boundary conditions in (4.). Example 2. We simulate the solution of 2 u + ɛ ( u 2 ) u =, 2 u = n u = 8 (4.2)

19 M. J. Lai, C. Liu, and P. Wenston Fig Numerical solution of equation (4.) and the middle cross section with ɛ = Fig The contour plot of the numerical solutions of equation (4.2) with ɛ =. with as shown in the figures below. The following graphs are numerical solution for ɛ =. and ɛ =.. In the case of the L-shape domain (Fig 4.3 and Fig 4.4), the numerical solution indicates that as ɛ approaches zero, the solution approaches to the distance function to the domain boundary. We want to point out this result does not contradict to the results in [Carme Calderer, Liu and Voss 97] since all the boundaries of the domain are straight lines. In particular, the level set plot illustrates the defect lines coming out of the 5 corners and bisect the angles. Moreover, the center defect line consists of 2 straight line with 2 parabola segments. This is consistent with the special solution constructed in [Carme Calderer, Liu and Voss 98]. References [] S. Allen and J. Cahn, A microscopic theory for the antiphase boundary motion and 9

20 2 Biharmonic Evolution Equations Fig The contour plot of the numerical solutions of equation (4.2) with ɛ = Fig The contour plot of the numerical solutions of equation (4.2) with ɛ =. its application to antiphase domain coarsening, Acta Metallurgica, 27 (979), pp [2] S. Angenet and M. Gurtin, Multiphase thermo-mechanics with interfacial structure. Evolution of an isothermal interface, Acta Rational Mech. Math. Anal. 8(989), pp [3] L. Bronsard and R. Kohn. Motion by mean curvature as the singular limit of Ginzburg- Landau dynamics, J.D.H. Eqs. 2(99), pp [4] M. Carme Calderer, C. Liu and K. Voss, Radial configurations of smectic A materials and focal conics, Physica D, 24(998), pp

21 M. J. Lai, C. Liu, and P. Wenston Fig The contour plot of the numerical solutions of equation (4.2) with ɛ =. [5] M. Carme Calderer, C. Liu and Karl Voss, Smectic A Liquid Crystal Configurations with Interface Defects Submitted, 998. [6] Du, Q., M. D. Gunzburger, J. S. Peterson, Analysis and approximation of the Ginzburg Landau model of superconductivity, SIAM Review, 34(992) pp [7] W. E, Nonlinear Continuum Theory of Smectic-A Liquid Crystals, Preprint, 997. [8] P. Grisvard, Elliptic Problems on Nonsmooth Domains, Pitman, Boston, 985. [9] W.M. Jin, Singular Perturbations, Fold Energies and Micromagnetics, Ph. D. thesis, Courant Institute, 997. [] D. Kinderlehrer and C. Liu, Revisiting the Focal Conic Structures in Smectic A. Proc. Symposium on Elasticity to honor Professor J. L. Ericksen 996, ASME Mechanics and Materials Conference. [] R. Kohn, A singular perturbed variational problem describing thin film blisters, IMA workshop, Feb. 5 9, 996. [2] M. J. Lai and L. L. Schumaker, On the approximation power of splines on triangulated quadrangulations, SIAM J. Num. Anal., 36(999), pp [3] M. J. Lai, C. Liu and P. Wenston, Numerical simulations on two nonlinear biharmonic evolution equations, to appear in Applicable Analysis, 23. [4] M. J. Lai and P. Wenston, Bivariate spline method for the steady state Navier-Stokes equations, Numerical Methods for PDE, 6(2), pp [5] Ortiz, M and G. Gioia, 994, The morphology and folding patterns of buckling-driven thin-film blisters, J. Mech. Phys. Solids, 42(994), pp [6] J. Rubinstein, P. Sternberg and J. Keller. Fast reaction, slow diffusion and curve shortening. SIAM J. Appl. Math., 49(989), pp [7] R. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A (989), pp [8] P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational. Mech. Anal, (988), pp