Upper Bounds On The Coarsening Rates For Some Non-Conserving Equations by Nan Jiang A dissertation submitted to the Graduate Faculty of Auburn University in partial ful llment of the requirements for the Degree of Doctor of Philosophy Auburn, Alabama August 4, 2012 Keywords: coarsening, upper bounds, Allen-Cahn equation, Swift-Hohenberg equation, interpolation inequality, dissipation inequality Copyright 2012 by Nan Jiang Approved by Dmitry Glotov, Assistant Professor of Mathematics Wenxian Shen, Professor of Mathematics Yanzhao Cao, Professor of Mathematics Abstract In this thesis, we prove one-sided bounds on the coarsening rates for two models of non-conserved curvature driven dynamics by following a strategy developed by Kohn and Otto in [20]. In the rst part, we analyze the Allen-Cahn equation in one and two dimensions, with di erent choices of length scales. The analysis follows the framework of Kohn and Yan in [24]. In the one-dimensional domain, by choosing an H 1-type length scale, our analysis supports the assertion that the coarsening occurs at the rate t1=3. In the two-dimensional domain, we consider two types of length scales. First, we obtain the coarsening rate of t1=3 using an H 1-type length scale, and then, using another L2-type length scale yields that the energy decays no faster than the rate t 1=6. In all the cases, among the main ingredients, the interpolation inequality requires the most delicate analysis, and the dissipation inequalities are based on basic calculations using H older?s inequality. An ODE argument is adapted to combine these two components in each case. The well-posedness of the Allen-Cahn equation obtained using xed point method is presented in the appendix. For the Swift-Hohenberg equation, we again consider an L2-type length scale in a two- dimensional domain. The coarsening rate of t1=3 rate is established using an interpolation inequality which extends Kohn and Otto?s method. This rate is consistent with numerical results as an upper bound on coarsening rates. ii Acknowledgments My special thanks to all the people who helped me, encouraged me and made my time during my PhD so enjoyable at Auburn University. First of All, I would like to thank my advisor Professor Dmitry Glotov for all the support and guidance required to make my work possible and e cient. I have learned from Dr. Glotov not only how to do good mathematical research but also how to organize and write good mathematical papers. Dr. Glotov always gave me the freedom to make my own personal judgement on the path and direction of this thesis. I am also grateful for the help and direction of members of committee, Professor Wenxian Shen and Professor Yanzhao Cao. Besides the interesting and inspiring courses that I took from them, they also gave me some ideas of how to make my work more integrated. My special thanks go to Professor Konrad Patkowski for his careful and hard work, who read my dissertation many times, and provided a lot of valuable suggestions. I also want to thank the Department of Mathematics and Statistics for providing such a friendly environment to work in. Particularly thanks to my friends in our department. It has been helpful to study and have discussions with my friends. Finally, I appreciate the emotional support and encouragements from my parents and friends. iii Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Allen-Cahn Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Swift-Hohenberg Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Previous Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1 Cahn-Hillard Equations . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 Phase-Field Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.3 Epitaxial Growth Model . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.4 Discrete, Ill-posed Di usion Equations . . . . . . . . . . . . . . . . . 11 1.4 Outline of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Allen-Cahn Equation in One-Dimensional Space . . . . . . . . . . . . . . . . . 17 2.1 Introduction to the Main Result . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.1 Boundedness of Solutions of Allen-Cahn Equations . . . . . . . . . . 19 2.2.2 Boundedness of Solutions of Elliptic Allen-Cahn Equations . . . . . . 22 2.3 The Interpolation Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 The Dissipation Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.5 Upper Bound on The Coarsening Rate . . . . . . . . . . . . . . . . . . . . . 37 3 Allen-Cahn Equation in Two-Dimensional Space . . . . . . . . . . . . . . . . . . 44 3.1 Energy Decays No Faster than t 1=3 . . . . . . . . . . . . . . . . . . . . . . . 44 3.1.1 Introduction to the Main Result . . . . . . . . . . . . . . . . . . . . . 45 3.1.2 The Interpolation Inequality . . . . . . . . . . . . . . . . . . . . . . . 46 iv 3.1.3 The Dissipation Inequality . . . . . . . . . . . . . . . . . . . . . . . . 53 3.1.4 Upper Bound on the Coarsening Rate . . . . . . . . . . . . . . . . . . 55 3.2 Energy Decays No Faster than t 1=6 . . . . . . . . . . . . . . . . . . . . . . 56 3.2.1 Introduction to the Main Result . . . . . . . . . . . . . . . . . . . . . 56 3.2.2 Interpolation Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2.3 Dissipation Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2.4 Upper Bound on the Coarsening Rate . . . . . . . . . . . . . . . . . . 61 4 Swift-Hohenberg Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.1 Interpolation Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2 Dissipation Inequality and Upper Bounds on the Coarsening Rates . . . . . . 73 5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 A Well-posedness of Allen-Cahn Equation . . . . . . . . . . . . . . . . . . . . . . . 78 A.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 A.2 Existence and Uniqueness of Weak Solution . . . . . . . . . . . . . . . . . . 84 A.3 Existence and Uniqueness of Strong Solutions . . . . . . . . . . . . . . . . . 91 B Well-posedness of Swift-Hohenberg Equation . . . . . . . . . . . . . . . . . . . . 94 B.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 B.2 Existence of Strong Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 v Chapter 1 Introduction In various physical processes, domains that form in multi-stable systems slowly change in time, with the overall pattern becoming coarser. From the physical perspective, our particular point is to model the kinetic behavior for the systems whose spatial structure develops a pattern of domains or clusters that coarsen as time increases. The growth of single-crystal grains in polycrystalline materials, phase separation in alloys, and anti-phase boundary motion in antiferromagnetic materials are some important examples, [32]. For example, as in [5], it is a system in equilibrium, which is quenched from the symmetric (high temperature) phase into the symmetry breaking (low temperature) phase through some phase transition. Once the system sets into the ordered phase, it locally selects one, among all the possible, equilibrium con gurations. Di erent states are chosen at di erent locations and topological defects in the form of domain walls are created. In the course of time, the patches of ordered regions tend to grow while the density of topological defects diminishes. It is widely observed that for some coarsening processes described by di erent equations, some typical length scale that characterizes the distance between the topological defects increases and the length scale behaves as a temporal power law. And the questions will be whether we can nd the universal rates for the coarsening. It is di cult to expect all solutions coarsen at the same rate, because in the in nite-time limit the system should typically approach a stable equilibrium and stop coarsening. However, Kohn and Otto?s method in [20] provides a e ective way to nd an upper bound on the coarsening rates. Here in this dissertation, we mainly study the coarsening described by the Allen-Cahn equation 1 and the Swift-Hohenberg equation. So rst we introduce these two equations before we go into the details. 1.1 Allen-Cahn Equation We know that for the scalar ODE @tu = f(u); with f : R!R and f2C1, every solution t!u(t) is monotonic and every bounded solution converges as t!1. For the coarsening processes, one of the simplest mathematical models of this behavior arises as a modi cation of the model @tu(x;t) = f(u(x;t)); in [0;1] [0;1); which is a spatial variation of the above scalar ODE. For any bounded solution, u1(x) = limt!1u(x;t) exists for every x, with f(u1(x)) = 0 for all x. If f has multiple stable zeros, the limiting state u1 is typically non-constant, and the domains will form as time proceeds, corresponding to di erent limiting values of u1(x). The Allen-Cahn equation was originally studied by Allen and Cahn in [1]. Our focus is on the parabolic Allen-Cahn equation on domain [0;1) @u @t u 2u(1 u 2) = 0; in [0;1) u(x;t) = 0; on @ [0;1) u(x;0) = u0; in (1.1) where is an interval I in R or a square Q in R2. This PDE corresponds to the gradient ow of the energy E(u) = 12 Z Q jruj2 + (1 u2)2dx; 2 where R denotes the spatial average. We focus on the homogeneous Dirichlet boundary conditions. In the literature, this equation is also considered together with either periodic or homogeneous Neumann boundary conditions. Moreover, for unbounded domains, heteroclinic conditions at in nity are usually imposed. The latter condition ensures that there is at least one transition between phases and it guarantees that the energy has a lower bound. We note that, for the epitaxial growth model, the requirement of periodic boundary condition also ensures a lower bound on the energy. For our model, we could have chosen the Dirichlet boundary conditions u(0;t) = 1 and u(l;t) = 1 to mimic the heteroclinic condition at in nity, but we note the homogeneous Dirichlet boundary condition yields the same e ect of creating interfaces at the boundaries of the bounded domains and it extends naturally to dimension two or higher. We scale the system and prove a corresponding result in the unit interval I1 = [0;1]. With the length of I denoted by 1" = l, we de ne u"(x;t) = u x "; t "2 : Then u" solves the equation @tu" @2xu" 2"2u"(1 u2") = 0; in I1 [0;1) u"(0;t) = u"(1;t) = 0; t> 0 (1.2) Let W(u) = 12(1 u2)2, so that W0(u) = 2u(1 u2). We observe that W(u) is a double well energy density with equal minima at u = 1. As "!0 the solutions u" will converge almost everywhere to 1 or 1, [38], [37], [35]. For every t and same initial condition for each "> 0, the interval I1 will be partitioned as I1 = I11 [I 11 [Irest1 , where I 1 =fx2I1ju"(x;t)! as "!0g; 3 and Irest1 has measure 0. The interface between these two sets corresponds to the grain boundaries. The only stable states of this system are patternless constant solutions u = 1, [9]. The asymptotic behavior of solutions of (1.2) as t!1 has been well studied. As stated in [32], for any solution u(x;t), we expect that u1(x) = limt!1u(x;t) exists and satis es the equation of equilibrium: "2@2xu" W0(u") = 0: Hence, u1 is a stationary solution and for large t, typical solutions will be approximately piece-wise constant in space. In a variety of physical processes, domains that form in multi- stable systems change in time slowly. Similarly the solution to (1.2) changes extremely slowly after reaching a pattern of transition layers developed in a relatively short time. The solution will either grow up to 1 or bring down to 1, decreasing the part of the energy corresponding to the double-well potential. 1.2 Swift-Hohenberg Equation The equation considered in this section is proposed as a prototypical example of pattern forming systems [10]. It was rst derived by Swift and Hohenberg in [36] as a model for pattern-formation equation for a uid which is thermally convecting. These authors used weakly nonlinear analysis of the Boussinesq equations describing B enard convection with random thermal uctuations, as a simple model for the Rayleigh-B enard instability of roll waves. When spatially periodic patterns emerge in isotropic pattern-formation systems, random initial conditions will lead to patches of patterns with di erent orientation that are separated by sharp interfaces. The slow dynamics of those interfaces often govern the long-time behavior of systems far from equilibrium. 4 Here we consider the Swift-Hohenberg equation in the two-dimensional space ut = (1 +r2)2u+ u u3; in Q [0;T] u = @u@v = 0; on @Q [0;T] u(x;0) = u0; in Q; (1.3) where Q2R2. The Swift-Hohenberg equation describes the nonlinear interaction of plane waves. Most of the pattern forming systems described by the Swift-Hohenberg equation exhibit stationary stripe or roll patters, see for example [19]. We consider u as representing a grayscale image of the temperature at each point, that is, each coordinate has a temperature measurement u associated with that point. Hence, as in Figure 1.1, the image of u represents a set of convection rolls. Figure 1.1: Evolution of patterns in time, taken from [21]. We notice that the Swift-Hohenberg equation relates the temporal evolution of the pattern to the spatial structure of the pattern. plays the role of a temperature knob, measuring how far the temperature is above the minimum temperature di erence required for convection. Therefore, for < 0, the heating at the bottom of the uid is too small to cause convection, while for > 0, convection occurs. The term involving the gradient acts to smooth out sharp edges in the pattern. 5 In Figure 1.1, (a)-(d) are the images obtained from experiments involving the free surface of granular layers at t = 2;10;200;1000, where the bright parts correspond to the crests of the free surface and the dark parts corresponds to the troughs of the free surface. As time progressed, they locally align in parallel and create an increasingly ordered pattern and after a long time, a fully ordered striped pattern nally appears. Sub gures (e)-(h) are images obtained from the simulation of the two-dimensional Swift-Hohenberg equation in time for = 0:2. We can see that the coarsening dynamics of the striped pattern shows very similar spatiotemporal morphology in both the experiment and numerical calculation. During the formation of stripes [18], the width of the structure will decay in two stages. When t is small, the linear term in equation (1.3) dominates the system because of the small amplitude of the order parameter u. At this stage, the width decays rapidly. However, in the late-time region, nonlinear term e ects emerge and the width decays slowly. Moreover, when the correlation function of the local orientation order parameter is computed in the late-time region in real space, the characteristic length grows algebraically as L(t) tz, while the density E(t) of topological defects decays algebraically as E(t) t z, as in [21], [7], etc. 1.3 Previous Results The quantitative estimation of the coarsening rates was pioneered by Kohn and Otto in [20]. Their method, originally developed for the Cahn-Hilliard equations, involves the introduction of an auxiliary length scale and establishing and exploiting relations between this length scale and the energy. This method has subsequently been carried out for, among many models, an epitaxial growth model by Kohn and Yan in [24], for a discrete, ill-posed di usion equation by Esedo glu and Slep cev in [13], for a demixing model by Brenier, Otto and Seis in [4], for a phase eld model with arbitrarily complicated patterns of phases by Dai and Pego in [12]. We outline the method rst as it was implemented originally for the Cahn-Hilliard equation in [20] and then for the epitaxial growth models in [24, 13, 4, 12]. 6 Here we list some previous work with upper bounds on coarsening rates of di erent models and the outlines of their work. They all follow the method developed by Kohn and Otto in [20], but with speci c tools and techniques. All these models have conservation law structures, while the equations that we study have non-conserved curvature driven dynamics. 1.3.1 Cahn-Hillard Equations For Cahn-Hilliard equations, @m @t +r J = 0; with the associated energy E = Z 1 2 jrmj2 + (1 m2) dx; where, in the constant mobility case, = 2 and J := r@E@m and, in the degenerate mobility case, = 1 and J := (1 m2)r@E@m: Here, R denotes the spatial average, and m2( 1;1) with c = 12(1+m)2(0;1) standing for the relative concentration of the rst species. The focus in [20] is on the case of a \critical mixture", i.e, Z mdx = 0: 7 A physical scale L appropriate for this model is de ned as L := Z jr 1mjdx := sup Z m dxj is periodic with supjr j 1 : The mathematical interpretation of L is that it is the (W1;1) norm of m. The typical length scale is expected to behave as L(t) t1=3 in the constant-mobility Cahn-Hilliard equation and it should behave as L(t) t1=4 in the degenerate-mobility case The basic procedure is: 1. In the regime E 1, establish an interpolation inequality EL & 1. We use the notation & and throughout the paper as follows: & means C for some constant C > 0, and means is su ciently large. Thus, this assertion says there exists a constant C > 0 such that EL C in the regime where E 1C. This is the one-sided version of EL 1, since E is the interfacial area density which scales as \1/ length scale L". 2. Find a dissipation inequality between _E and _L for each of the constant and degenerate mobility cases. 3. Obtain an upper bound on the coarsening rate by an ODE argument based on the previous two results. The lower bound on energy corresponds to the upper bound on the length scale. To this end, we consider L as an absolutely continuous function of E and rewrite in the dissipation inequality _L = dLdE _E. Then an appropriate change of variables yields the desired lower bound on the rate of energy decay. 1.3.2 Phase-Field Model In 2004, Dai and Pego extended the method of Kohn and Otto and established an upper bound on the coarsening rate for a phase- eld model in [12]. The model is given by two equations in a non-dimensional form: "ut + l2 t = K ut; " t = " 1"g( ) + 2u; (1.4) 8 where g( ) = G0( ) = ( 2 1), and l;K; are non-dimensional parameters that represent latent heat, thermal di usivity and a relaxation time, respectively, and " measures the thickness of the transition layers between two phasesf +1gthe solid phase andf 1g the liquid phase, where " is small and "< K. This model describes the solid-liquid phase transition of a pure material in terms of the temperature u and an order parameter . The special domain is a large cubic cell Q = [0;a]n Rn with periodic boundary conditions. The associated energy is given by E(t) = Z Q " 2jr j 2 + 1 "G( ) + 2" l u 2 ; and the length scale is de ned as the H 1 norm of "u+ l2 , i.e., L(t) = Z Q jrvj2 1=2 ; where v is a periodic function that satis es v = "u+ l2 : Following the method of Kohn and Otto, there are three key steps. The rst is to nd a dissipation relation j_Lj2 KL4 ( _E), and this can be done by direct calculations. The second key step is to obtain the interpolation inequality. This is done by de ning periodic functions ! and that satisfy ! = u u; = ; where u = R u and = R , hence, rv = "r! + l2r . It is shown that, E(t)L(t) l2L1(t)E1(t) Ca r l" 2 E(t) 3=2; 9 where L1(t) = Z Q jr j2 1=2 and E1(t) = Z Q " 2jr j 2 + 1 "G( ) : Subsequently, Kohn and Otto?s technique can be applied to prove E1(t)L1(t) C. The ob- servation that _E(t) 0, together with the assumption that "0M and "0a2M3 are su ciently small yields E(t)L(t) & 1 whenever 0 <"<"0 and E(0) 0. The most technically delicate point in this paper is the pointwise interpolation inequality. For the continuous function v(x) = "u x " with period 1 in each independent variable, the assertion is, with Q1 denoting the unit square, Z Q1 "j vj2 +" 1(1 rvj2)2 Z Q1 v2 1 2 C; for some constant C. 1.3.4 Discrete, Ill-posed Di usion Equations For the coarsening phenomena in discrete, ill-posed di usion equations, upper bounds on the coarsening rate can also be found using a similar framework. Discrete, ill-posed di usion equations arise in the methods of granular ow, image processing, population dynamics, and many other applications. The speci c equation studied by Esedo glu and Slep cev in [13] is as follows vt = R(v) = R0(v) v +R00(v)jrvj2; where R( ) : R!R with R0( ) < 0 for all j j large enough. The unknown function v is de ned on the unit-space lattice L =f1;2;:::;Ngd, where d is the dimension. The l2 scalar product takes the form v w = X q2L vqwq: Denote by L = fv : L!Rg all real-valued lattice con gurations, and by P = fv : L! [0;1)g only the nonnegative con gurations. 11 On the set Z =fv2L: v = 0g, where v is the average value of v de ned as v = X q2L vq = 1jLj X q2L vq; the discrete H 1 norm is introduced in the following way: given s2Z there exists a unique, up to a constant, solution p of the discrete Poisson equation p = s: De ne the H 1 inner product by hs1;s2i= X q2L (r+p1)q (r+p2)q; where r+ is the forward nite di erence, r+v = (@+1 v;:::;@+d v) for (@+i v)q = vq+ei vq. Integration by parts giveshs1;s2i= s1 p2 = p1 s2. Then for s2Z, the H 1 norm is de ned by jjsjj= sup 6=const s q P q2Ljr+ qj2 : (1.5) We next outline the proof of the interpolation and dissipation inequalities following [13]. The energy is de ned as E(v) := X q2L f(vq); for v2P and f0 = R, and the associated length scale is de ned as L = 1pjLjjjv vjj; Then, by (1.5), L = sup 6=const P q2L(vq v) qq P q2Ljr+ qj2 : 12 The dissipation inequality ( _L)2 _ E can be easily obtained as follows: dL2 dt = 2L _L = 2 jLjh_v;v vi; so that _L = h_v;v vi jLj L = pjLjh_v;v vi jLjjjv vjj ; combining the Cauchy-Schwarz inequality h_v;v vi2 jj_vjj2jjv vjj2; with h_v; _vi= h R(v); _vi= R(v) _v = f0(v) _v = rE(v) _v; we have _L2 1 jLjh_v; _vi= 1 jLjrE(v) _v = _ E; where we used the chain rule in the last equality. To prove the interpolation inequality, for > 0 and 2[0;1), de ne F (z) = 8 >< >: 0 if 0 z z if z > The assumptions f F for some > 0 and v > are made to to avoid zero energy density. Suppose the associated energy E = E(v)jLj satis es E < ( v ) 2 72 v2(1 ) ; 13 Then by choosing an appropriate , it can be proved that, when d = 1, EL1 124 ( v )3(1 )2 +1 v 3(1 )2 ; and, when d 2, EL2(1 ) 124 d (1 )( v )3 2 v (1 ): Hence, in general, the interpolation inequality can be written as EL for some explicitly de ned > 0 and > 0. 1.4 Outline of This Thesis Our analysis follows the work of Kohn and Yan, and the work of Kohn and Otto. Speci cally, we prove that the energy averaged in time decays no faster than a power law. The energy that corresponds to the Allen-Cahn equation is E = Z 1 2(jruj 2 + (1 u2)2)dx: (1.6) This energy plays an important role in physics and has been well studied in [3]. In addition to the energy, we introduce an auxiliary length scale. Our main result is that the energy in the parabolic Allen-Cahn equation decays no faster than t 1=3 or t 1=6 with constant coe cients depending on the size of the domain in various ways. The energy that corresponds to Swift-Hohenberg equation is E(t) = Z Q 1 2j(1 +r 2)uj2 + 1 4u 2 1 2 u 2 + 1 4 2dx: Our main result is that the energy in the Swift-Hohenberg equation decays no faster than t 1=3. 14 We expect that the energy E(t) is concentrated mostly on the interfaces and has the same dimension as the H 1-type physical length scale L(t) so that E(t) 1=L(t). The interpolation inequality is a one-sided version of this relation and it takes the form of EL C for some positive constant C. And for the L2-type length scale, we establish a similar version of this relation E2(t) 1=L(t). The interfacial area decreases, that is, _E 0, because the motion is surface-energy-driven. However, we need a more accurate energy-dissipating structure of the dynamics. This re ned structure is obtained in the form of the dissipation inequality. Finally, the interpolation and the dissipation inequalities are employed in the proof of an ODE lemma from which an upper bound on the time-averaged coarsening rate follows. Here is the basic strategy : 1. We assemble the well-posedness of the Allen-Cahn and Swift-Hohenberg equations from various sources in the appendix. Speci cally, for the Allen-Cahn equation, preliminaries include establishing various bounds for solution u, which can be done by standard parabolic estimates using the maximum principle. 2. The most important part is to obtain the dissipation inequality as well as the inter- polation inequality. The dissipation inequality can be obtained by an elementary method. We use two di erent strategies to prove the interpolation inequality according to di erent length scales: (a) First, obtain a uniform lower bound of the energy and then re ne this lower bound to relate it to the length scale, as in section 2.3 and section 3.1.2; (b) An alternative way to relate E to L is to separate the L2 norm of u into two parts: one for E and the other for L, since we have 1 Z u2dx = Z (1 u2)dx ( Z (1 u2)2dx)1=2 .E1=2; 15 for the Allen-Cahn equation, and Z u2dx = Z ( u2)dx ( Z ( u2)2dx)1=2 .E1=2; for the Swift-Hohenberg equation. The right-hand-sides of both inequalities decay to zero as E 1. This technique is implemented in section 3.2.3 and section 4.1. 3. Various versions of the ODE lemma from Kohn and Otto?s paper are given. Two distinct versions appear in section 2.5 and section 3.2.4. 16 Chapter 2 Allen-Cahn Equation in One-Dimensional Space In this chapter, we look at the upper bound on the coarsening rates for the Allen-Cahn equation in the one-dimensional space. 2.1 Introduction to the Main Result We consider the solutions of the parabolic Allen-Cahn equation in the domain I [0;1) @tu @2xu 2u(1 u2) = 0; in I [0;1) u(0;t) = u(l;t) = 0; t> 0 u(x;0) = u0(x); in I (2.1) where I = [0;l] R and u0 is bounded. Because we focus on the upper bounds on the coarsening rates of Allen-Cahn equation, we will give the well-posedness results in Appendix A. First, let us look at the process of domain wall formation that occurs for this system and how coarsening by domain wall motion and annihilation can be described. We expect that starting with a bounded initial condition, u rapidly approaches 1 where u > 0, and 1 where u < 0 at the initial stage of coarsening. Domain walls or transition layers form between these domains at positions corresponding roughly to zeros in the initial data. Fusco and Hale [17] developed a rigorous geometric theory of domain wall dynamics. Their idea for a geometric description of these slow dynamics is to describe solutions containing N domain walls in terms of an N dimensional manifold of \metastable" states in X. Using a restricted gradient ow approach, given N domain walls initially located at given positions 17 h1 < h2 < < hN in (0;1), the positions will evolve in time according to equations well-approximated by exponentially small nearest neighbor interactions: @thj = 12 e hj+1 hj e hj hj 1 ; where h0 = h1 and hN+1 1 = 1 hN are obtained by re ection through boundaries. Equation (1.2) corresponds to the gradient ow of the energy E(u) = 12 Z I j@xuj2 + (1 u2)2dx; (2.2) where R denotes averaging over the interval. To construct the length scale, we let v(x) = Z x 0 u(z)dz; and de ne L = Z I v2(x)dx 1 2 = 1 jIj Z I u(z)dz 1 2 ; (2.3) where jIj stands for the length of interval I. In our proof we will employ that u"2H10 (I1) for each xed time, according to the well-posedness in Appendix A. We now present our main results. We state a special case as Theorem 2.1 and then the general case as Theorem 2.2. Theorem 2.1. Suppose the initial energy is E0 and the initial length scale is L0. Then we have Z T 0 E2dt& Z T 0 (t 13 )2dt for T L30 1 E0: Here, we use R to denote averaging over the time interval [0;T]. This theorem states that E & t 1=3 in a L2 time-averaged sense and it also holds for some time average of the other norms of E, as well as E replaced by E L (1 ). 18 Theorem 2.2. Suppose the initial energy is E0 and the initial length scale is L0. For any 0 1, suppose r satis es r< 3; r> 1 and (1 )r< 2. Then we have Z T 0 E rL (1 )rdt& Z T 0 (t 13 )rdt for T L30 1 E0: Both of the two inequalities above in Theorem 2.1 and Theorem 2.2, respectively, depend on the size of the domain I, and the speci c dependence will be given in the ODE Lemma 2.13 later. Notice that when = 1, it permits 1 < r < 3, and the minimum possible permitted is 13. The conclusion of the theorem is strongest when and r are smallest, i.e., for values close to the curve r = 1. Indeed, if the estimate holds for a given r0 < 3 then it holds for all r between r0 and 3 by an application of Jensen?s inequality, and if the estimate holds for a given 0 < 1, then it holds for all > 0 by an application of the interpolation inequality, [20]. Theorems 2.1 and 2.2 are valid in the two-dimensional domain with the energy and length scale de ned in (3.2) and (3.3). The two-dimensional analogs of these two results will be proved in chapter 3. 2.2 Preliminary Results In this section, we show the boundedness of solutions of the elliptic and parabolic Allen- Cahn equations, which will be used in the following proof for interpolation and dissipation inequalities. 2.2.1 Boundedness of Solutions of Allen-Cahn Equations In this section, rst we prove that the solution of the parabolic Allen-Cahn equation (A.1) is uniformly bounded in domain Q, where we suppose the domain is Q = [0;l]n and Q1 = [0;1]n with n = 1 or 2. We prove this by pointwise parabolic estimates. We will use 19 the boundedness to prove dissipation inequality for a L2 length scale for the two-dimensional space case. The technique for the proof of the following lemma appears in [3] in the context of the Ginzburg-Landau equation. Lemma 2.3. Let u be a solution of (A.1) with a bounded initial condition, then for t "2 and x2Q1 = [0;1], ju(x;t)j p2: Proof. By the dilation scaling as above, the space domain is Q = [0;1="]n. Set (x;t) := ju(x;t)j2 1 and multiply equation (A.1) by u, to get d dt + 2jruj 2 + 4 ( + 1) = 0: (2.4) Now consider the ODE y0(t) + 4y(t)(y(t) + 1) = 0 (2.5) which is the space independent version of (2.4). We verify directly that y0(t) = e 4t 1 e 4t is a solution for equation (2.5) for t> 0. Let ~ (x;t) = y0(t), then d~ dt ~ + 4~ (~ + 1) = 0: (2.6) Subtracting (2.4) from (2.6) gives us d dt(~ ) (~ ) + 4(~ )(~ + + 1) = 2jruj 2 0: (2.7) 20 Since 1 + + ~ =juj2 + ~ 0, then, by the maximum principle in [14] ~ (x;t) (x;t) 0 for all t 0 and x2Q: Therefore, (x;t) y0(t); for all t> 0 and x2Q; so that ju(x;t)j2 = (x;t) + 1 2; for all t 14 and x2Q: Remark 2.4. We can also notice that jru"(x;t)j C"; jdu"dt (x;t)j C"2: Indeed, ju(x;t)j p2 for t 14, we have ju(1 u2)j p2 for t 14: Let p> 2 be xed. It follows from the standard regularity theory for the linear heat equation that for each compact set F Q [14;1) we have jjdudtjjLp(F) C(F) and jj ujjLp(F) C(F) In particular, by Sobolev embedding and the L1 bound for u we have jjujjC0; (I [1 2;1)) C; 21 where = 12(1 1p). Moreover, jju(1 u2)jjC0; (I [1 2;1)) C: By the C0; regularity theory, we have jjujjC0; (I [1;1)) C: Hence, jru"(x;t)j C"; jdu"dt (x;t)j C"2 for t "2 and x2Q: 2.2.2 Boundedness of Solutions of Elliptic Allen-Cahn Equations In this section we will establish uniform bound on the solution to the elliptic Allen-Cahn equation and its derivative. This will be used to prove that, in the one-dimensional case, the energy is uniformly bounded from below. The boundedness of energy will be used to prove the interpolation inequality. We consider the solution of equation u" 2"2u"(1 u2") = 0; in Q u" = 0; on @Q; (2.8) and de ne E"(u") = Z Q "jru"j2 + 1"(1 u2")2: If u"2H10 (Q) is a minimizer of E"(u), then u" is a solution to (2.8). Indeed, i( ) = E"(u+ v) = Z Q "(ru+ rv)2 + 1"(1 (u+ v)2)2: 22 Then i0( ) = Z Q 2"(ru+ rv)rv 4"(1 (u+ v)2)(u+ v)v: Integration by parts and evaluation at = 0 yields i0(0) = 2" Z Q ( u+ 2"2u(1 u2))v = 2"( u+ 2"2u(1 u2);v)L2; so that the critical points of the energy satisfy equation (2.8). The next lemma shows us that the solution of the equation (2.8) and the gradient of the solution are bounded from above. Lemma 2.5. Let u" be a solution of (2.8). Then ju"j 1 and jru"j C" on Q. Proof. First, we observe that equation (2.8) has the weak form Z Q ru" rv 2"2 (1 ju"j2)u"vdx = 0; (2.9) for all v2H10 (Q). Note that the boundary conditions are incorporated into (2.9). Now, denote Q+ = fju"j> 1g and let v = (sgnu")(ju"j 1)+, where x+ = max(x;0): Then v2H10 (Q), i.e., v is a test function for (2.9). We rst compute rv = (sgnu")rju"j in Q+. Substituting v into (2.9), we have Z Q+ jru"j2 + 2"2 (ju"j2 1)ju"j(ju"j 1)dx = 0; that is, Z Q+ jru"j2 + 2"2 (ju"j 1)2ju"j(ju"j+ 1)dx = 0; (2.10) 23 since the integrand of (2.10) is non-negative, we must have measQ+ = 0, and therefore, ju"j 1 almost everywhere in Q. Second, let v" = u" w; where v" is the solution of the equation v" = 1"2u"(1 ju"j2) on Q v" = 0 on @Q; and w is the solution to the equation w = 0 on Q w = 0 on @Q: Then, it follows from the elliptic estimates as appearing in [2] and the fact thatju"j 1that jjrv"jjL1 C"jjv"jjL1 C" (jju"jjL1 +jjwjjL1) C": 2.3 The Interpolation Inequality In this section, we prove that EL& 1 when E 1; (2.11) where E and L are de ned in (2.2) and (2.3), and the constants implicit in (2.11) are independent of the size of I. 24 With the scaling as in the previous section u"(x;t) = u x "; t "2 ; u" solves the equation @tu" @2xu" 2"2u"(1 u2") = 0; in I1 [0;1) u"(0;t) = u"(1;t) = 0; t> 0: (2.12) Let v"(x) = Z x 0 u"(z)dz = "v x " : (2.13) Then E and L may be rewritten as, E = 12 Z I1 "2j@xu"j2 + (1 u2")2dx; and L = 1" Z I1 v2"dx 1 2 ; respectively. Correspondingly, and (2.11) becomes Z I1 "j@xu"j2 + 1"(1 u2")2dx Z I1 v2"dx 1 2 & 1 when Z I1 "2j@xu"j2 + (1 u2")2dx 1: The last statement is a consequence of the following theorem. 25 Theorem 2.6 (Interpolation). There is a constant c > 0 with the property that for any function u"2H10 (I1) and any "> 0, Z I1 "j@xu"j2 + 1"(1 u2")2 Z I1 v2" 1 2 + Z I1 "2j@xu"j2 + (1 u2")2 c : In the next few lemmas, we establish various lower bounds on the energy E" that lay the groundwork for proving Theorem 2.6. In the rst lemma, we show that E" is uniformly bounded below. Lemma 2.7. De ne E"(u") = Z I1 "j@xu"j2 + 1"(1 u2")2; where I1 is the unit interval in R. There exist constants a0 > 0 and "0 > 0 such that for any " "0 and any u"2H10 (I1), we have E"(u") a0: Proof. We rst prove this lemma for solutions. We claim speci cally that, when u0" is a solution to equation (2.8), there exist > 0 and "0 > 0 such that if 1 " Z I1 (1 ju0"j2)2 ; with "<"0; (2.14) then ju0"(x)j 12 8x2I1: This is a contradiction with the regularity and the homogeneous Dirichlet boundary condi- tions. Next, we prove this claim. By Lemma 2.5, we have j@xu0"j C"; 26 where C does not depend on ". Therefore, ju0"(x) u0"(y)j C"jx yj; 8x;y2I1: Assume, by contradiction, that ju0"(x0)j< 12 for some x0 2I1. Then, ju0"(x) u0"(x0)j C"jx x0j; and ju0"(x)j 12 + C" in I1\J (x0); where J is an interval of length 2 with 1. We choose in such a way that C" = 14, i.e., = "4C, so that 1 ju0"(x)j 14 in Q1\J (x0); and, consequently, (ju0"(x)j2 1)2 116 I1\J (x0): Also, meas(I1\Jh) h 8x2I1 and 8h 1: Hence, Z I1 (1 ju"j2)2 Z I1\J (x0) (1 ju0"j2)2 "64C: Therefore, 1 " Z I1 (1 ju0"j2)2 164C: Let < 164C and "0 = 4C so that we arrive at a contradiction with (2.14). Hence, there existsa0 > 0 such thatE"(u0") a0. In particular, this lower bound is valid for the minimizers of the energy E". 27 Finally, suppose u" is any function. Let u0" be a minimizer of E" and, therefore, also a solution of (2.8). Then E"(u") E"(u0") a0: The next lemma is based on a result from Modica [29]. It guarantees compactness in L2 of a sequence u"j with uniformly bounded energy by taking advantage of the polynomial structure of the nonlinear part of the energy W(u) = (1 u2)2. Lemma 2.8. Suppose f"jg is a sequence such that "j ! 0 and fu"jg is a sequence for which E"j(u"j) is uniformly bounded. Then there exists a subsequence of f"jg, without loss of generality also denoted by f"jg, such that fu"jg converges to a function u0 in L2(I1) as j!1. Proof. Recall that I1 = [0;1] and x "> 0. De ne (t) = Z t 0 j1 s2jds and w"(x) = (u"(x)): Since there exist t0 > 1;c1 > 0 and c2 > 0 such that c1t4 (1 t2)2 c2t4; for t t0; we see that (t) Z t0 0 j1 s2jds+ Z t t0 pc 2s2ds Z t0 0 j1 s2jds+ pc 2 3 t 3: Then, for some c3 > 0 and c4 > 0, we have (t) c3 +c4(1 t2)2; for t t0: 28 Therefore, Z I1 w"dx = Z I1 (u")dx c3 +c4 Z I1 (1 u2")2dx c3 +c4"E"(u"); and we conclude that fw"g is bounded in L1(I1). On the other hand, @xw"(x) = 0(u"(x))@xu"; and Z I1 j@xw"jdx = Z I1 j1 u2"jj@xu"jdx 12 Z I1 "j@xu"j2 + 1"(1 u2")2dx = 12E"(u") c5; for some c5 > 0, so the compactness yields that there is a sequencef"hgof positive numbers converging to 0 such that fw"g converges in L1(I1) to a function w0. Now, let be the inverse function of and de ne u0(x) = (w0(x)). We have 0(t) = j1 t2j pc1t20 for every t t0. Hence is Lipschitz continuous on [ (t0);1) and uniformly continuous on the entire real line. It follows that u"j = w"j converges in measure on I to u0 as j!1. Since Z I1 u4"jdx t40 + 1c 1 Z I1 j1 u2"j(x)j2dx t40 + "jc 1 E"j(u"j); that is, fu"g is bounded in L4(I1), hence, fu"g converges to u0 in L2(I1). In the next lemma, we re ne the result in Lemma 2.7 and claim that the bound may be taken to be any constant c with c 12 as long as the length scale is su ciently small. 29 Lemma 2.9. Let E"(u") be as in Lemma 2.7 and v" be de ned as in equation (2.13). For any c 12, there exists > 0 such that for all " 1, if RI1 v2" , then E"(u") c. Proof. We prove this lemma by contradiction. Suppose, for some c 12, there exist sequences fv"jg and f"jg such that Z I1 v2"j 1j but E"j(v"j) 0, we choose a subsequence, without loss of generality, also denoted as f"jg, such that inf "j > 0, so that fRI1j@xu"jj2g is bounded, hence, fu"jg is pre-compact in L2(I1). In both cases, for a further subsequence, u"j !u1 in L2(I1) for some u1 in L2(I1). On the other hand, (2.15) implies limj!1v"j = v1 = 0, therefore @xv1 = u1 = 0. But by compactness of fu"jg and Fatou?s lemma, 1 = Z I1 (1 u21)2 lim inf j Z I1 (1 u2"j)2 lim inf j c"j 12: This contradiction shows that the lemma is true. In the proof of the conclusion of Theorem 2.6, namely, (energy density ) ( length scale ) & 1; we note that the most di cult case is when "! 0. Before proving Theorem 2.6, we need to establish the following proposition rst. It states that when the length scale is bounded above by some constant depending on the length of the interval, the energy has a lower bound. This will be used to prove Theorem 2.6 by contradiction by employing the following technique. Mainly, we assume the length scale is small, and we divide the domain into a mesh of subintervals. On most of the subintervals, the length scale in relatively small. At 30 the same time, by Lemma 2.9, the energy is large on those subintervals where the length scale is small. The proposition below gives a rigorous basis for this argument. Proposition 2.10. For any c0 12, there exists a constant c1 > 0 with the following property. Consider any interval I R with length l and any u"2H10 (I) satisfying Z I v2"dx c1l3; where v" is de ned as in (2.13). Then we have Case A : Z I "j@xu"j2 + 1"(1 u2")2 c0 if l "; (2.16) Case B : Z I "j@xu"j2 + 1"(1 u2")2 c0l" if l "; (2.17) Proof. For Case A, we de ne ul(x) = u"(lx). Then, with vl(x) = Z x 0 ul(z)dz = Z x 0 u"(lz)dz = 1lv"(lx); (2.16) is equivalent to proving that for any c0 12, there exists c1 such that if " l 1; Z I1 v2l c1; then Z I1 " l j@xulj2 + l " (1 u2l )2 c0: Since "l 1, this is exactly the result of Lemma 2.9. 31 We prove case B by contradiction. Suppose for some c0 12, there exist sequences "k;Ik;lk;u"k satisfying 8 >>> >>> < >>> >>> : "k lk 1; 1 l3k R Ikv 2 "k !0; R Ik" 2 kj@xu"kj 2 + (1 u2 "k) 2 0, then in the second term of (2.19), we have Z I1 j@xu"kj2 !0 and Z I1 u2"k !1: (2.20) But, by the Poincar e inequality, Z I1 u2"k C Z I1 j@xu"kj2 !0: 32 This is a contradiction with (2.20). Case 2: Suppose lim infk"k = 0 but RI1 v2"k is bounded away from 0. Without loss of generality, suppose "k!0. The convergence of the rst term in (2.19) to 0 gives us Z I1 "kj@xu"kj2 + 1" k (1 u2"k)2 !0; is in a contradiction with Lemma 2.7. Case 3: Suppose lim"k = 0 and limRI1 v2"k = 0. We use Proposition 2.10 to obtain a contradiction. Fix c0 and drop the subscript k to simplify the notation, and we write u" = u"k;v" = v"k. De ne = Z I1 v2" 1 2 : For any integer N > 1, we partition the unit interval I1 into N subintervals of length ! = 1N. The value of N will be determined later. If Z I! v2" c1!3; by applying Proposition 2.10 to I!, we have Z I! "j@xu"j2 + 1"(1 u2")2 c0 if ! "; or Z I! "j@xu"j2 + 1"(1 u2")2 c0!" if ! ": The choice of N depends on the relation between " and . Alternative 1: Suppose " . Then we choose N such that ! p ", that is, " ! . For any N line segment I! of length !, we say I!2A if RI! v2" c1!3 and let jAj be the 33 number of line segments in A. Since jAjc1!3 X I!2A Z I! v2" Z I1 v2" = 2; we have jAj 2 c1!3 1 ! = N: Therefore, the relation Z I! v2" c1!3 holds on most intervals. Since " !, for I! =2A, i.e., when Z I! v2" c1!3; Proposition 2.10 gives Z I! "2j@xu"j2 + (1 u2")2 c0!: Summing over all of these line segments, we have Z I1 "2j@xu"j2 + (1 u2")2 X I!=2A Z I! "j@xu"j2 + (1 u2")2 c0 X I!=2A !& 1 This is a contradiction with (2.19). Alternative 2: Suppose &". In this case, we choose N and ! = 1N such that ! = M , where M is a constant to be chosen later. We notice that ! & &". Again, considering N line segments I!, and we say I!2A if RI! v2" c1!3. Since jAjc1!3 X I!2A Z I! v2" Z I1 v2" = 2; then we have jAj 2 c1!3 = 1 c1M2 1 ! = N c1M2; 34 Now, we choose M such that M2c1 > 2. This choice guarantees that jAj< 12 1! = N2 , so at least half of the line segments I! are not in A. Recalling that we have !&", Proposition 2.10 yields Z I! "j@xu"j2 + 1"(1 u2")2 c0; for each I! =2A. Summing over all of these line segments gives us Z I1 "j@xu"j2 + 1"(1 u2")2 X I!=2A Z I! "j@xu"j2 + 1"(1 u2")2 c02! & 1! Therefore, Z I1 "j@xu"j2 + 1"(1 u2")2 Z I1 v2" 1 2 & ! & 1: This contradiction with (2.19) completes the proof of the theorem. 2.4 The Dissipation Inequality The dissipation inequality proved in this section provides the second ingredient for the ODE lemma in Section 2.5. It relates the rates of change with respect to the time of the energy and of the length scale. There are two critical points in the dissipation inequality. First, we notice that _E 0, that is, the energy is decreasing because the motion of the interfaces or transition layers is surface energy driven. Second, a re nement of this inequality involves _L which is controlled by _E because coarsening requires motion which dissipates energy. Lemma 2.11 (Dissipation). Suppose u is a solution of (2.1) and again let E and L be de ned as in (2.2) and (2.3), respectively. Then ( _L)2 jIj2 _E: 35 Proof. We rst derive an expression for the time derivative of the energy: _E = 1 2 Z I 2@xu @t@xu+ 2(1 u2)( 2u @tu)dx = Z I @2xu @tu 2u(1 u2)@tudx = Z I (@2xu+ 2u(1 u2))@tudx = Z I (@tu)2dx: Next, we turn to the time derivative of the length scale. Considering its square L2 = Z I v2dx; we obtain 2L_L = dL 2 dt = @t Z I v2dx = 2 Z I v@tvdx 2 Z I v2dx 1 2 Z I (@tv)2dx 1 2 : (2.21) On the other hand, @tv(x) = Z x 0 @tu(z)dz; so that, Z I (@tv)2dx = 1jIj Z I Z x 0 @tu(z)dz 2 dx: (2.22) By H older?s inequality Z x 0 @tu(z)dz x12 Z x 0 (@tu(z))2dz 1 2 jIj12 Z x 0 (@tu(z))2dz 1 2 ; and (2.22) becomes Z I (@tv)2dx = Z I Z x 0 (@tu(z))2dzdx jIj2 Z I (@tu(x))2dx 36 Hence, (2.21) gives us 2L_L 2jIjL Z I (@tu)2dx 1 2 = 2jIjL( _E)12; i.e., ( _L)2 jIj2( _E): 2.5 Upper Bound on The Coarsening Rate The next lemma is an ODE argument of Kohn and Otto [20]. Our proof carefully traces the dependence of the coarsening rate on the size of the domain, and makes precise change of the variables required for the speci c dissipation inequality listed in the hypothesis of the lemma. Before proceeding with the ODE lemma, we rst prove that L is absolutely continuous when viewed as a function of E. This fact will be invoked in the proof of the ODE Lemma 2.13. Proposition 2.12. Suppose E and L are continuously di erentiable on [0;T] and ( _L)2 _E; on [0;T]: (2.23) Then L E 1 : E([0;T])!R is absolutely continuous, i.e., 37 for any "> 0;there exists > 0 such that, if 0 s1 0 such that for any t 2 [0;T] and any h 2 [0;h0] with t+h2[0;T], (L(t+h) L(t))2 4(E(t+h) E(t))h: Proof of claim: Assume that there exist ftkg [0;T] and hk!0 with tk +hk2[0;T] such that (L(tk +hk) L(tk))2 > 4(E(tk +hk) E(tk))hk; or, equivalently, L(t+h) L(t) hk 2 > 4(E(t+h) E(t))h k ; without loss of generality, tk !t2 [0;T]. Taking the limit and using the continuity of _E and _L, we get ( _L(t))2 > 2 _E(t); a contradiction with (2.23). To prove (2.24), let "> 0 and take = " 2 4T . Suppose 0 h0, let sk;j = sk+jh0, j = 0;1;:::;nk 1; where nk =dtk skh 0 e so that sk = sk;0 0 with the following property: For any u"2H10 (Q1) and any "> 0, Z Q1 "jru"j2 + 1"(1 u2")2dx jju"jjH 1(Q1) + Z Q1 "2jru"j2 + (1 u2")2dx c : As in Section 2.3, we start by establishing a uniform lower bound on energy E". The proof is the same as that of Lemma 2.7, therefore we state the following lemma without proof. Lemma 3.2. De ne E"(u") = Z Q1 "jru"j2 + 1"(1 u2")2 47 where Q1 is the unit square in R2. There exit constants a0 > 0 and "0 > 0 such that for any " "0 and any u"2H10 (Q1), we have E"(u") a0: The next lemma claiming compactness in L2(Q) of the set of admissible functions with uniformly bounded energy is the same as Lemma 2.8 of one-dimensional case as well, and we omit its proof. Lemma 3.3. Suppose E"(u") is uniformly bounded and f"jg is a sequence such that "j!0, then there exist a subsequencef"jgof positive numbers such thatfu"jgconverges to a function u0 in L2(Q) as j!1. The following lemma and proposition share the techniques of their proofs with Lemma 2.9 and Proposition 2.10, respectively, but the use of a di erent length scale warrants separate proofs. Lemma 3.4. Let E"(u") be as in Lemma 3.2. For any c 12, there exists > 0 such that for all " 1, if jju"jjH 1(Q1) , then E"(u") c. Proof. We prove this lemma by contradiction. Suppose, for some c 12, there exist sequences fu"jg and f"jg such that jju"jjjH 1(Q1) 1j but E"j(u"j) 0, we choose a subsequence, without loss of generality, also denoted as f"jg, such that inf "j > 0, so that fRQ1jru"jj2g is bounded, hence, fu"jg is pre-compact in L2. In both cases, for a further subsequence, u"j !u0 in L2 for some u0 in L2. 48 On the other hand, (3.6) implies limj!1u"j = u0 = 0, since jjrp"jjjL2 !0 as j!1, i.e., limj!1rp"j =rp0 = 0, and u0 = p0 = 0. But by compactness offu"jgand Fatou?s lemma, 1 = Z Q1 (1 u20)2 lim inf j Z Q1 (1 u2"j)2 c"j 12: This contradiction shows that the lemma is true. Proposition 3.5. For any c0 12, there exists a constant c1 > 0 with the following property. Consider any square Q R2 with side length l and any u"2H10 (Q) satisfying jju"jjH 1(Q) c1jQj: Then we have Case A : Z Q "jru"j2 + 1"(1 u2")2 c0jQj1=2 if jQj1=2 "; (3.7) Case B : Z Q "jru"j2 + 1"(1 u2")2 c0jQj" if jQj1=2 "; (3.8) Proof. For Case A, we de ne u(x) = u"(lx), or, equivalently, u"(x) = u x l , and suppose without loss of generality that Q is centered at the origin. Then (3.7) is equivalent to proving that for any c0 12, there exists c1 such that if " jQj1=2 1; jjujjH 1(Q1) c1; then, Z Q1 " jQj1=2 jruj2 + jQj1=2 " (1 u2)2 c0: Since "jQj1=2 1, this is exactly the result of Lemma 3.4. 49 We prove Case B by contradiction. Suppose for some c0 12, there exists sequences "k;Qk;lk;u"k satisfying 8 >>>> >>> < >>> >>> >: "k jQkj1=2 1; 1 jQkjjju"kjjH 1(Qk) !0; R Qk" 2 kjru"kj 2 + (1 u2 "k) 2 0, then, in the second term of (3.10), we have Z Q1 jru"kj2 !0 and Z Q1 u2"k !1: (3.11) By the Poincar e inequality Z Q1 u2"k C Z Q1 jru"kj2 !0: 50 This is a contradiction with (3.11). Case 2: Suppose lim infk"k = 0 butjju"kjjH 1(Q1) is bounded away from 0. Without loss of generality, suppose "k!0. The convergence of the rst term in (3.10) to 0 gives us Z Q1 "2kjru"kj2 + (1 u2"k)2 !0; in a contradiction with Lemma 3.2. Case 3: Suppose lim infk"k = 0 and lim infkjju"kjjH 1(Q1) = 0. We use Proposition 3.5 to obtain a contradiction. Fix c0 and, dropping the subscript k to simplify the notation, we write u"k = u". De ne =jju"jjH 1(Q1): For any integer N > 1, we partition the unit square Q1 into N2 squares of side length ! = 1N. The value of N will be determined later. If jju"jjH 1(Q!) c1!2; by applying Proposition 3.5 to Q!, we have Z Q! "jru"j2 + 1"(1 u2")2 c0jQ!j1=2 if jQ!j1=2 = ! " or Z Q! "jru"j2 + 1"(1 u2")2 c0jQ!j" if jQ!j1=2 = ! " The choice of N depends on the relation between " and . Alternative 1: Suppose " . Then we choose N such that ! p" and therefore " ! . For the N2 squares Q!, we say Q!2A if jju"jjH 1(Q!) c1jQ!j= c1!2 and let 51 jAj be the number of squares in A. Since ! = 1N < 1 and jAjc21!4 X Q!2A jju"jj2H 1(Q!) jju"jj2H 1(Q1) = 2; we have jAj 2 c21!4 1 !2 = N: Therefore, the inequality jju"jjH 1(Q!) c1!2 holds on most, in particular on at least half, of the squares. Since " !, for Q! =2A, i.e, when jju"jjH 1(Q!) c1!2; Proposition 3.5 gives Z Q! "2jru"j2 + (1 u2")2 c0jQ!j Summing over all these squares, we have Z Q1 "2jru"j2 + (1 u2")2 X Q!=2A Z Q! "jru"j2 + (1 u2")2 c0 X Q!=2A jQ!j& 1 This is a contradiction with (3.10). Alternative 2: Suppose &". Then we choose N and ! = 1N such that ! = M , where M is a constant to be chosen later. Again, consider the N2 squares Q!, we say Q! 2A if jju"jjH 1(Q!) c1!2. Since jAjc21!4 X Q!2A jju"jj2H 1(Q!) jju"jj2H 1(Q1) = 2; we have jAj 2 c21!4; 52 and = !M 1M. Then we estimate jAj< 1M2c2 1 1 !2: Now, we choose M such that M2c21 2. This guarantees that jAj< 12 1!2 = N2 , so at least half of the squares Q! are not in A. Considering that we have !& &", by proposition 3.5, for each Q! =2A, Z Q! "jru"j2 + 1"(1 u2")2 c0jQ!j1=2: Summing over all these squares gives us Z Q1 "jru"j2 + 1"(1 u2")2 X Q!=2A Z Q! "jru"j2 + 1"(1 u2")2 c0jQ!j 1=2 2!2 & 1 !: Therefore, Z Q1 "jru"j2 + 1"(1 u2")2 jju"jjH 1(Q1) & ! & 1: This contradiction with (3.10) completes the proof. 3.1.3 The Dissipation Inequality Lemma 3.6 (Dissipation). Suppose u is a solution of (3.1) and again let E and L be de ned as in (3.2) and (3.3), respectively. Then ( _L)2 . jQj _E 53 Proof. As in Lemma 2.11, by replacing ux by ru, we again have that integration by parts yields _E = 1 2 Z 2ru rut + 2(1 u2)( 2u ut)dx = Z u2tdx; (3.12) where the boundary integrals are all vanishing due to the boundary conditions. Next, we compute 2L_L = dL 2 dt = 2 jQjhu; _uiH 1 2 jQjjjujjH 1(Q)jj_ujjH 1(Q): Therefore, _L2 jj_ujj 2 H 1(Q) jQj : (3.13) Let q be the solution to equation q = _u; in Q q = 0; on @Q: Then jjrqjj2L2(Q) =jj_ujj2H 1(Q) = Z Q q_udx jjqjjL2(Q)jj_ujjL2(Q) "2jjqjj2L2(Q) + 2"jj_ujj2L2(Q): (3.14) By the Poincar e inequality, jjqjjL2(Q) .jQj1=2jjrqjjL2(Q): 54 Thus (3.14) becomes jjrqjj2L2(Q) . "2jQj jjrqjj2L2(Q) + 2"jj_ujj2L2(Q): Choosing " = 1jQj in the inequality above, we have jj_ujjH 1(Q) =jjrqjjL2(Q) .jQj1=2jj_ujjL2(Q): (3.15) Combining (3.12), (3.13) and (3.15), we get _L2 .jj_ujj2L2(Q) = jQj _E; so that ( _L)2 . jQj _E; where the dependence on the system size included in the right hand side. 3.1.4 Upper Bound on the Coarsening Rate Now we can start to prove the ODE lemma by interpolation inequality and dissipation inequality as before. Lemma 3.7 (ODE). For any 0 1 and jQj& 1, suppose r satis es : r < 3; r > 1 and (1 )r< 2. Then EL& 1 and ( _L)2 jQj( _E) imply Z T 0 E rL(1 )rdt& 1 jQj r 3 T r3 for T L30 1 E0: (3.16) This ODE lemma is similar to Lemma 2.13 and hence we just give the statement without a detailed proof. The only di erence is the power of the domain size. In this lemma, we have jQj and in Lemma 2.13 we have jIj2, so we only need to adjust the corresponding power in the statement of this lemma. 55 3.2 Energy Decays No Faster than t 1=6 In this section, we will use another length scale which will improve our result in the sense that it allows us to see that the energy is dissipated at a slower rate. By choosing the new length scale, our interpolation inequality transforms into E2L & 1, so that L E 2, and when the energy decays as t 1=6, the coarsening rate for length scale is t1=3. The one-sided version of this statement that we prove here is that the energy decays no faster than t 1=6. 3.2.1 Introduction to the Main Result The auxiliary length scale that is employed in our analysis takes the form L = Z Q jW(u)jdx; (3.17) where we de ne W(u) = Z u 0 j1 s2jds: Here, we use R to denote averaging over the spatial domain. Now we state the main result which follows immediately from Lemma 3.12. Theorem 3.8. For the initial energy E0 and initial length scale L0 satisfying T L40 1 E0, it holds that Z T 0 E3dt& Z T 0 (t 14 )2dt: As in the one-dimensional case, this theorem follows from a more general and stronger statement that appears in the next theorem. Theorem 3.9. For any 0 1, suppose r satis es r < 4; r > 43 and (1 )r < 83. Then we have Z T 0 E32 rL 34(1 )rdt& Z T 0 t r4dt for T L40 1 E0: 56 3.2.2 Interpolation Inequality For the interpolation inequality, here we only use the de nition of E and L in the interfacial regime E 1, without considering the Allen-Cahn dynamics. Lemma 3.10 (Interpolation). If E and L are de ned as in (3.2) and (3.17), respectively, then E2L&k for E 1; where the constant number in the inequality k = 1jQj. Proof. De ne W(u) := Z u 0 j1 t2jdt; so that @W @u =j1 u 2j: Then, by Cauchy?s inequality, Z jr(W(u))jdx = Z jruj@W@u dx 12 Z jruj2 + (@W@u )2dx = E: This inequality was introduced by Modica and Mortola in [28]. We next introduce a smooth molli er ? that is radially symmetric, non-negative, and supported in the unit ball with RR2 ? = 1. Let the subscript " denote the convolution with the kernel ?"(x) = 1"2? x " ; that is, u" = u ?": The L2 norm of u may be split up as follows Z u2dx. Z (u u")2dx+ Z u2"dx: (3.18) 57 This inequality holds for any " > 0 and the precise value of " will be selected later in the proof. The rst term in (3.18) is estimated in terms of the energy: Z (u u")2dx sup jhj " Z (u(x) u(x+h))2dx . sup jhj " Z jW(u(x)) W(u(x+h))jdx ." Z jr(W(u))jdx: (3.19) For the second term in (3.18), we note that for any "> 0, since ju"(x)j= Z ?"(x y)u(y)dy = 1"2 Z ? x y " u(y)dy Z ?1=2" (x y)?1=2" (x y)ju(y)jdy Z ?"(x y)dy 1=2 Z ?"(x y)ju(y)j2dy 1=2 = Z ?"(x y)ju(y)j2dy 1=2 : Hence, Z u2"(x)dx Z Z ?"(x y)ju(y)j2dydx = Z Z ?"(x y)ju(x)j2dxdy = 1"2 Z ju(x)j2 Z ? x y " dy 2 dx 1"2 Z ju(x)j2dx: 58 Thus, Z u2"dx jQj"2 Z juj2dx .jQj"2 Z W(u)dx: (3.20) The last inequality holds since juj2 . W(u) for any u without considering the Allen-Cahn dynamics. Combining (3.18), (3.19) and (3.20), we have Z u2dx." Z jr(W(u))jdx+ jQj"2 Z W(u)dx "E + jQj"2 L: We now select " = jQjL E 1=3 , with E 1, to obtain "E + jQj"2 L =jQj1=3L1=3E2=3 +jQj1=3L1=3E2=3 = 2jQj1=3L1=3E2=3: Hence Z u2 .jQj1=3E2=3L1=3: We also have 1 Z u2dx = Z (1 u2)dx Z (1 u2)2dx 1=2 E1=2: Therefore, 1 .jQj1=3E2=3L1=3 +E1=2 which gives us Lemma 3.10 for E 1. 59 3.2.3 Dissipation Inequality The next lemma concerns with the rate at which L can change in relation to E and the rate of change of E. Lemma 3.11 (Dissipation). Suppose u is a solution of (3.1) and again let E and L be de ned as in (3.2) and (3.17), respectively. Then ( _L)2 . _EE: Proof. As in Lemma 2.11, by replacing ux with ru, we again have _E = 1 2 Z 2ru rut + 2(1 u2)( 2u ut)dx = Z u2tdx: Since L = Z W(u)dx; d dtL = Z utj1 u2jdx; so that j_Lj Z j1 u2jjutjdx Z u2tdx 1=2 Z (1 u2)2dx 1=2 j _Ej1=2jEj1=2: Hence, we have Lemma 3.11. 60 3.2.4 Upper Bound on the Coarsening Rate A variant of the following lemma appears in [20]. The version that we prove here employs the interpolation inequality proved in Lemma 3.10. Lemma 3.12 (ODE). For any 0 1 and jQj& 1, suppose r satis es: r < 4; r > 43 and (1 )r< 83. Then E2L&k and ( _L)2 .E( _E) imply Z T 0 E32 rL 34(1 )rdt&kr(3 1)4 T r4 for T L40 1 E0; (3.21) where the constant number in the inequality k = 1jQj. Proof. The energy E is a monotone function of time due to the di erential inequality ( _L)2 . E( _E). Indeed, since _E . _L2 E < 0, E is decreasing. Moreover, using a technique similar to the one used in the proof of Proposition 2.12, we can see that L is an absolutely continuous function of E, and we can write the dissipation inequality as dL de 2 ( _E)2 .Ej _Ej: (3.22) Here, the lower case e is used for the energy as an independent variable to distinguish it from E = E(t). From (3.22), we have 1 E dL de 2 j _Ej. 1: (3.23) When _E = 0, (3.23) is still true trivially. Multiplying (3.23) by any function f(E(t)) and integrating in time on the interval [0;T] gives Z T 0 f(E(t))dt& Z E(0) E(T) f(e) e dL de 2 de: 61 Taking f = e32 rL 34(1 )r and writing E0 = E(0), ET = E(T), we then have Z T 0 E32 r(t)L 34(1 )r(t)dt& Z E0 ET e32 r 1L 34(1 )r dL de 2 de (3.24) for all T > 0. Now we estimate the right hand side of (3.24). Consider the change of variables ^e = 12 3 2 r e2 32 r and ^L = 11 3(1 )r 8 L1 3(1 )r8 : The hypotheses r> 43; (1 )r< 83 imply > 13 (3.25) which will be used later. Since dL de 2 de = d^L d^e !2 dL d^L 2 d^e de d^e and d^e de = e 1 32 r; d^L dL = L 3(1 )r8 ; we can write the right hand side of (3.24) as: Z E0 ET de d^e d^L dL !2 dL de 2 de = Z ^E0 ^ET d^L d^e !2 d^e which is bounded below by the minimum over all functions ^L(^e) with boundary conditions ^L( ^E0) = 1 1 3(1 )r8 (L(0)) 1 3(1 )r8 ; ^L( ^ET) = 1 1 3(1 )r8 (L(T)) 1 3(1 )r8 : 62 To simplify the notations we denote these boundary conditions by ^L0 and ^LT, respectively. The extremal ^L is a linear function of ^e, so we have: Z T 0 E32 rL 34(1 )rdt& ( ^LT ^L0)2 ^E0 ^ET : (3.26) When T is such that L(T) 2L(0); the right side of (3.26) can be controlled since ^LT ^L0 & ^LT and ^E0 ^ET ^ET; so Z T 0 E32 rL 34(1 )rdt& ^L2T ^ET = L 2 34(1 )r T E 3 2 r 2 T : Rewriting the right hand side as L2 34(1 )rT E32 r 2T = [E32 T L 34(1 )T ]r 4[E2TLT]3 1; we conclude, using E2L&k and (3.25), that Z T 0 E32 rL 34(1 )rdt& [E32 T L 34(1 )T ]r 4k3 1 (3.27) provided L(T) 2L(0). Introducing h(T) := Z T 0 E32 rL 34(1 )rdt; we can write (3.27) as h& (h0)r 4r k3 1; 63 so we have: h r4 r(T)h0(T) &kr(3 1)4 r (3.28) provided L(T) 2L(0). Here we used r< 4. The previous method does not work when L(T) < 2L(0), but, for such T, we have E2(T) &L(T) 1 &L 10 ; which implies E32 (T)L 34(1 )(T) = (E2(T)L(T))34 L 34 (T) &k34 L 340 : Thus h0(T) &k34 rL 34r0 if L(T) < 2L0: (3.29) Combining (3.28) and (3.29), using r< 4, we have: d dt(h+L 3(4 r) 40 ) 44 r (h(t) +L 3(4 r) 40 ) r4 rh0(t) &kr(4 2)4 r for all t> 0. Indeed, for L(T) 2L(0), using (3.28), we have d dt(h+L 3(4 r) 40 ) 44 r (h(t) +L 3(4 r) 40 ) r4 rh0(t) &h r4 r(T)h0(T) &kr(3 1)4 r ; and for L(T) < 2L(0), by (3.29), d dt(h+L 3(4 r) 40 ) 44 r (h(t) +L 3(4 r) 40 ) r4 rh0(t) &L 3r 40 h0(T) &k3 r4 : By assumptions, r> 43 and jQj& 1, we have kr(3 1)4 r 0. Restricting attention to T 4 r4 L4 r0 , this becomes Z T 0 E rL(1 )rdt = h(T) &kr(3 1)4 T 4 r4 for T L40 which is precisely (3.21) as we need. As stated above, we establish the upper bounds on coarsening rates. Therefore obtaining the slower rates of the decay of the energy corresponds to an improvement. This improvement needs to be considered in conjunction with the dependence on the size of the domain in the coe cient appearing with the power of t in each case. For the length scale L =jjujjH 1(Q), it is jQj r=3 and, for the length scale L = R W(u)dx, it is jQj r(3 1)=4. 65 Chapter 4 Swift-Hohenberg Equation We study the coarsening of two-dimensional oblique stripe patterns of the Swift-Hohenberg equation. We expect the models to exhibit isotropic coarsening with a single characteristic length scale with the growth in time governed by a power law. Coarsening in the Swift- Hohenberg equation has been studied numerically, in [7]. Several numerical methods have been used to describe the pattern dynamics of the Swift-Hohenberg equation. Cross and Newell [8] proposed that higher order gradient terms in the phase equation would control the dynamics and suggested a growth rate t1=3 in [8]. Then Cross and Hohenberg [6] sug- gested t1=4 as an alternative. Elder, Vi~nals and Grant obtained numerical results in [15] showing the length scale increasing with time as t1=4 when the equation has a noise term (corresponding to a nite temperature thermodynamic system), and they observed a slower growth rate consistent with t1=5 without the noise term. In this section, we establish a one-sided version of this result, that is, an upper bound on the coarsening rate of the Swift-Hohenberg equation, and prove the system will coarsen no faster than a power law. As in the previous sections, we apply Kohn and Otto?s method in [20] to a properly chosen length scale and energy. Again, to focus on the coarsening rates, we provide the well-posedness of Swift-Hohenberg equation in Appendix B. We consider the Swift-Hohenberg equation ut = (1 +r2)2u+ u u3; (4.1) 66 with a variational formulation in terms of the energy functional E(t) = Z Q 1 2j(1 +r 2)uj2 + 1 4u 4 1 2 u 2 + 1 4 2 dx: (4.2) Here the potential function F is given by F (u) = 1 2 u2 + 14u4 + 14 2; since by (4.2), E(t) = Z Q 1 2j(1 +r 2)uj2 + 1 4(u 2 )2 dx = Z Q 1 2jr 2uj2 jruj2 + 1 4(u 2 )2 + 1 2u 2 dx: (4.3) Note that F is normalized so that F (0) = 14 2. The constant is viewed as an eigenvalue parameter. and if > 0, the potential function is a double well potential with the minima located at u = p . We also notice that the Swift-Hohenberg equation de nes a gradient ow so that ut = @E@u: As long as the minima of (4.2) are isolated, u(x;t)!U(x) as t!1. Using this variational formulation, it would be possible to predict that stationary solutions are stable by showing that they are minima of (4.2). We de ne a correlation length scale by L(t) = Z Q u2dx 1=2 : (4.4) 67 We will use the idea of Kohn and Otto?s method in [20] to obtain interpolation inequality EL& 1 and dissipation inequality _L2 . _E, and then, by the ODE lemma, to nd an upper bound on the coarsening rate. Now we state the main result which follows immediately from Lemma 4.5. Theorem 4.1. For initial energy E0 and initial length scale L0 satisfying T L30 1 E0, it holds that Z T 0 E2dt& Z T 0 (t 13 )2dt: A more general and stronger statement appears in the next theorem. Theorem 4.2. For any 0 1, suppose r satis es r< 3; r> 1 and (1 )r< 2. Then we have Z T 0 E rL(1 )rdt& Z T 0 t r3dt for T L30 1 E0: 4.1 Interpolation Inequality Lemma 4.3 (Interpolation). For E and L de ned in (4.2) and (4.4) respectively, provided there exists c0 > 0 such that >c0, then EL& 1 when E 1; where the constant number implied in the inequality depends on the system size jQj. Proof. The proof is similar to that of Lemma 1 in [20], however, we need a somewhat di erent treatment since we have di erent energy and length scale. For completeness and also for the purpose of tracking the constants, we here reproduce the details. Writing = ( u2) + u2, we rst focus on the rst term in the right hand side and estimate by the rst equation in (4.3) Z ( u2)dx Z ( u2)2dx 1=2 2E1=2: (4.5) 68 Next, we need to estimate R u2dx in terms of L and E. We de ne W(u) = Z u 0 j t2jdt; so that @W @u =j u 2j: By Young?s inequality Z jr(W(u))jdx = Z jruj@W@u Z 2jruj 2 + 1 2 @W @u 2 dx; (4.6) for any > 0. By the Poincar e inequality and using the boundary condition, we have Z jruj2dx = Z u udx 12 Z j uj2dx+ 2 Z u2dx 12 Z j uj2dx+ CjQj2 Z ru2dx; choosing = 1CjQj, Z jruj2dx CjQj Z j uj2dx: (4.7) Furthermore, Z jruj2dx = 2CjQj1 2CjQj Z 1 2 1 CjQjjruj 2dx Z jruj2dx 2CjQj1 2CjQj Z 1 2j uj 2dx Z jruj2dx = C Z 1 2j uj 2 jruj2dx; where C = 2CjQj1 2CjQj is close to 1 when jQj is large. 69 Then (4.6) becomes Z jr(W(u))jdx Z C 2 1 2j uj 2 jruj2 + 12 (u2 )2dx; by choosing = 2pC , we have Z jr(W(u))jdx pC Z 1 2j uj 2 jruj2 + 1 4(u 2 )2dx pC Z 1 2j uj 2 jruj2 + 1 4(u 2 )2 + 1 2u 2dx =pC E: We will use a smooth molli er which is radically symmetric, non-negative and sup- ported in the unit ball with RR2 = 1. Let the subscript denote the convolution with the kernel ( ) = 1 2 : The parameter will be determined later. Now we split the L2 norm R u2dx into two parts: Z u2dx 2 Z (u u )2dx+ 2 Z u2 dx: (4.8) Note that ju1 u2j2 8jW(u1) W(u2)j for all u1 and u2, therefore we obtain the following estimate Z (u u )2dx sup jhj Z (u(x) u(x+h))2dx 8 sup jhj Z jW(u(x)) W(u(x+h))jdx 8 Z jr(W(u))jdx 8p2C E: 70 For the second term of (4.8), we separately deal with ju j being either small or large: Z u2 = Z u2 minfu2 ; 2gdx+ Z minfu2 ; 2gdx: (4.9) Since F(u) = u2 minfu2; 2g is a convex function in u, by a version of Jensen?s inequality and the fact that R (x)dx = 1, we have F(u (x)) = F Z (y)u(x y)dy Z (y)F(u(x y))dy: The rst term of (4.9) becomes Z u2 minfu2 ; 2g Z Z (y)F(u(x y))dydx = Z (y) Z [u2(x y) minfu2(x y); 2g]dxdy = Z u2(x) minfu2(x); 2gdx 14( u2)2dx E: For the second term of (4.9), we have Z minfu2 ; 2gdx Z ju jdx: By duality, Z ju jdx = supf Z u (x) (x)dx : is Q periodic and j (x)j 1 a.e.g: Consider that is Q periodic with j (x)j 1 a.e., and write (x) = Z 1 2 x y (y)dy: 71 Hence, r (x) = 1 Z 1 2r x y (y)dy = 1 Z r (y) (x y)dy; and thus supjr j 1 supj j ; where = Rjr jdx. Therefore, Z ru (x) (x)dx = Z ru(x) (x)dx = Z u(x)r (x)dx Z u2dx 1=2 Z jr j2 1=2 L: Therefore, taking the superemum over all , we get Z ju jdx CjQj1=2 Z jru jdx CjQj1=2 L: Combining all the above estimates, we have Z u2dx 16p2C E + 2E + 2 CjQj1=2 L: Taking = CjQj1=2 L 8p2C E 1=2 to minimize the right hand side over , we get Z u2dx ~C(EL)1=2 + 2E; where ~C = 8(2CjQj1=2 p2C )1=2. Combining with estimate (4.5), we obtain ~C(EL)1=2 + 2E + 2E1=2; 72 which yields Lemma 4.3 for E 1. 4.2 Dissipation Inequality and Upper Bounds on the Coarsening Rates Next lemma relates the rate at which L can change when energy is dissipated to the rate of change of the energy. We nd a dissipation relation that bounds the growth rate in terms of a suitable measure of length scale. Lemma 4.4 (Dissipation). Suppose u is a solution of (1.3) and again let E and L be de ned as in (4.2) and (4.4), respectively. Then ( _L)2 _E: Proof. From (4.3), we have _E = Z r2u r2ut 2ru rut + (u2 + 1)(u ut)dx = Z r4u ut + 2r2u ut +u(u2 + 1)utdx = Z r4u+ 2r2u+u(u2 + 1) utdx = Z u2tdx: Since L2 = Z juj2dx; then 2L_L = 2 Z uutdx 2 Z juj2dx 1=2 Z jutj2dx 1=2 = 2L( _E)1=2 73 so that j_Lj2 _E Hence, we have Lemma 4.5. Since the proof of this lemma will be very similar with the proof of Lemma 2.13, we omit the details. Lemma 4.5 (ODE). For any 0 1, suppose r satis es: r< 3; r> 1 and (1 )r< 2. Then EL& 1 and ( _L)2 ( _E) imply Z T 0 E rL(1 )rdt&T r3 for T L30 1 E0; (4.10) where the constant number implied in the above inequality depends on the system size jQj. This lemma has a proof similar to that of Lemma 2.13 but with di erent dependence on the size of the domain. We apply this lemma with the interpolation inequality depending on the size of the domain instead of the dissipation inequality. By calculations similar to Lemma 2.13, we have K = ~C2 r(3 )3 , where ~C = 8(2CjQj1=2 p2C )1=2 in Lemma 4.3. 74 Chapter 5 Discussion Our accomplishment is the time-averaged lower bounds for energy E which corresponds to the time-averaged upper bounds on the coarsening rates for the Allen-Cahn equation and the Swift-Hohenberg equations. The lower bounds on the coarsening rates would depend on the geometry of the domain and cannot be established using the technique that we employed. It will be nice to nd pointwise-in-time bounds for both the energy E and the length scale L, but it will require some new ideas. We can see that the equations to which Kohn and Otto?s method has been applied pre- viously, including the Cahn-Hilliard equations, epitaxial growth model, phase- eld model, discrete ill-posed di usion equations, and many other models we did not list in this disserta- tion, all have conservation law structures in the equations. On the other hand, both of the equations that we have studied, namely, the Allen-Cahn equation and the Swift-Hohenberg equation, are non-conserving. In general, the equations with a conservation law structure can be written as du dt = r J: This structure provides certain advantages. For example, when proving the dissipation inequality for the Cahn-Hilliard equation, this property can be used for integration by parts to have Z t2 t1 Z du dt dxdt = Z t2 t1 Z J r dxdt Z t2 t1 Z jJjdxdt; where is periodic and supjr j 1: 75 This seems to be an important step in the proof. The same technique is also used in the proof of the dissipation inequality for discrete, ill-posed di usion equations, hdudt;siH 1 = h R(u);siH 1 = R(u) s = rE(u) s: On the other hand, if the equation has a conservation law structure, then the mean value u = 0 can be naturally maintained. This property is used in the proofs of interpolation inequality for both epitaxial growth model and phase- eld model. However, since the equations that we are looking at have no natural conservation law structure, we have to nd some other methods to approach the results that we need. In the proof of the interpolation inequality of the Allen-Cahn equation, we followed the frameworks of Kohn and Yan?s method, but the details have been proved by di erent methods. For example, to obtain the lower bound on the energy density, we used the idea that the solution to an elliptic the Allen-Cahn equation will give the minimum value of energy density which has a lower bound by regularity. We also adjust the interpolation inequality to be E2L & 1 for the two-dimensional Allen-Cahn equation with a length scale L = Z W(u)dx; with W(u) = Z u 0 j1 s2jds: This di erent interpolation inequality, combined with the dissipation inequality, improves the coarsening rates and the upper bound for the energy decay. But we will be very interested in looking for some universal and geometry-independent upper bounds on coarsening rates for non-conserving equations. This may require some new techniques. In this thesis, only two equations which are non-conserving were selected to study the upper bounds on coarsening rates, but there are many energy-driven dynamic systems that describe coarsening models in material science. For example, a natural extension would be 76 to look at a more general equation modeling epitaxial thin lm growth in [22], ut + 2u r (f(ru)) = g; in (0;T) @u @n = @ u @n = 0; on @ (0;T) u(x;0) = u0; in Q where n denotes the outward pointing unit normal to the boundary @ of the domain . While the energy associated with this equation appears in the introduction, a di erent choice of length scale alternative to the L2-norm of u employed in [24] may have the potential to yield more precise upper bounds on the coarsening rate. For example, the Hessian may be a candidate for such a choice of length scale. While the general framework of Kohn and Otto [20] may be followed, the implementation would require di erent techniques from what we have used. 77 Appendix A Well-posedness of Allen-Cahn Equation Here we prove the existence and uniqueness of the weak and strong solutions of the Allen-Cahn equation in a bounded domain 2 Rn, where n = 1;2. The proofs appear in [34] and cover the case of a more general non-linear term. For the completeness of the presentation, we adapt the theorems from [34] to apply speci cally to our models. We consider the parabolic Allen-Cahn equation on domain Q [0;1) @u @t u 2u(1 u 2) = 0; in Q [0;1) u(x;t) = 0; on @Q [0;1) u(x;0) = u0(x); in Q (A.1) where Q is the square in R2 with side length l. De ne W(u) = 12(1 u2)2, and let w(u) = W0(u) = 2u(1 u2): Notice that w(s) is a C1 function which satis es the bounds 1 2jsj4 w(s)s 1 jsj4; (A.2) and w0(s) 2; (A.3) both for all s2R. Indeed, 1 2s4 w(s)s = 2s2 2s4 1 s4 and w0(s) = 2 6s2 2. 78 We proceed to de ne the sense in which equation (A.1) holds. We sayu2L2(0;T;H10 (Q)) with dudt 2L4=3(0;T;H 1(Q)) is a weak solution of (A.1) if for any v2L4(0;T;H10 (Q)), we have hdudt;vi+a(u;v) =hw(u);vi (A.4) for almost every t2[0;T], where the bilinear form of a( ; ) is de ned as a(u;v) =h u;vi= mX j=1 (Dju;Djv); where ( ; ) stands for the inner product in L2(Q) and we use hv ;ui to denote the pairing between an element v 2H 1(Q) and an element u2H10 (Q), that is, there is an element v2H10 (Q) such that hv ;ui= (v;u)H10(Q): A.1 Preliminary Results In what follows, we write L2 for L2(Q), H10 for H10 (Q), H 1 for H 1(Q), etc. To prove the existence of solutions, the technique is essentially to construct a sequence that is weakly convergent and show that the limit is a solution. We will need some strong convergence of un and a weak version of the dominated convergence theorem [34]. Before proving the compactness theorem, we will need the following lemma, which is a special case of Ehrling?s lemma. Lemma A.1. For each > 0 there exists a constant c such that jjujjL2 jjujjH10 +c jjujjH 1; for all u2H10: 79 Proof. We prove this by contradiction, so suppose there exists > 0 such that for each n2Z+ there is a un with jjunjjL2 jjunjjH10 +njjunjjH 1: Let vn = unjju njjH10 , then jjvnjjL2 +njjvnjjH 1: (A.5) Since H10 L2 and vn is bounded in H10 with norm 1, vn is also bounded in L2. By (A.5), it follows that jjvnjjH 1 !0 as n!1. However, H10 L2 and vn!v in L2 imply that v = 0 which contradicts (A.5). We can now prove the compactness theorem. Theorem A.2. Suppose un is a sequence that is uniformly bounded in L2(0;T;H10 ), and dun dt is uniformly bounded in L 4=3(0;T;H 1). Then there is a subsequence that converges strongly in L2(0;T;L2). Proof. Since H10 is re exive, so is L2(0;T;H10 ). Since un is bounded in L2(0;T;H10 ), using Alaoglu compactness theorem, there is a subsequence un such that un *u in L2(0;T;H10 ): Next, we show that vn = un u!0 in L2(0;T;L2): To this end, we rst establish that vn!0 in L2(0;T;H 1) which is su cient to guarantee that vn!0 in L2(0;T;L2). Indeed, Lemma A.1 shows that for each > 0 there exists a c such that jjvnjj2L2(0;T;L2) jjvnjj2L2(0;T;H1 0) +c jjvnjj2L2(0;T;H 1); 80 and since vn is bounded in L2(0;T;H10 ), jjvnjj2L2(0;T;L2) C +c jjvnjj2L2(0;T;H 1): If vn!0 in L2(0;T;H 1), then lim sup n!1 jjvnjj2L2(0;T;L2) C for any > 0. Hence limn!1jjvnjj2L2(0;T;L2) = 0: To prove that vn ! 0 in L2(0;T;H 1), we observe that for vn 2 H1(0;T;H 1), we have vn2C([0;T];H 1) by Theorem 5.9.2 in [14], and max 0 t T jjvnjjH 1 CjjvnjjH1(0;T;H 1) C(jjvnjjL2(0;T;H10) +jjdvndtjjL2(0;T;H 1)) M: (A.6) Denoting _v = ddtv, we integrate equation vn(t) = vn(w) Z w t _vn( )d ; with respect to w from t to t+s to get vn(t) = 1s Z t+s t vn(w)dw Z t+s t Z w t _vn( )d dw = An +Bn; where An = 1s Z t+s t vn(w)dw; Bn = Z t+s t Z w t _vn( )d dw: 81 Now take > 0 and estimate jjBnjjH 1 Z t+s s jjdvndtjjH 1dw s1=4 Z t+s t jjdvndtjj4=3H 1dw 3=4 s1=4jjdvndtjjL4=3(0;T;H 1): We choose s such that jjBnjjH 1 2: (A.7) For this value of s, notice that Z t+s t vn(w)dw* 0 in H10: (A.8) Indeed, if is the indicator function of [t;t + s] and 2 H 1, then is an element of L2(0;T;H 1) and Z T 0 hvn(t); idt = Z t+s t hvn(t); idt =h Z t+s t vn(t)dt; i: Since vn * 0 in L2(0;T;H10 ), then (A.8) follows. Hence An * 0 in H10 , then An!0 in H 1. Therefore, for n large enough we have jjAnjjH 1 2 which together with (A.7) gives jjvnjjH 1 : Since vn(t) ! 0 in H 1 and vn(t) is bounded in H 1 for almost every t2 [0;T] by (A.6). Lebesgue?s dominated convergence theorem gives vn!0 in L2(0;T;H 1) and thus completes the proof. 82 Next we will prove a weak version of the dominated convergence theorem stating that if a sequence fujg is bounded in Lp and converges pointwise, then uj *g in Lp. Although this lemma will be used to prove the existence of solutions to the Allen-Cahn equation with the case p = 43, we prove a more general version for any p> 1. Lemma A.3. Let be a bounded open set in Rm and let uj be a sequence of functions in Lp( ) with jjujjjLp( ) C for all j2Z+: If u2Lp( ) and uj!u almost everywhere then uj *u in Lp( ). Proof. Let En =fxjx2 ;juj(x) u(x)j 1 for all j ng: These sets En increase with n and the measure of En increases to the measure of as n!1 since uj!u almost everywhere. Let n be the set of functions in Lq( ), where q is the H older conjugate of p, with support in En. Let = S1n=1 n. We can see that is dense in En. For 2Lq( ), take n = [En] , where [E] is the characteristic function of E. Then, since n ! almost everywhere and j n(x)j j (x)j, Lebesgue?s dominated convergence theorem gives n! in Lq( ). If we take 2 , then 2 n0 for some n0 and, for j n0, we have j (x)(uj(x) u(x))j j (x)j: Lebesgue?s dominated convergence theorem yields Z (uj u)dx!0 as j!1: 83 By the density of in Lq( ), for v2Lq( ), given > 0, choose 2 such that jjv jjLq( ) < 4C and N such that Z (uj u)dx< 2 for all j N: Then Z (uj u)(v + )dx< 2C( 4C) + 2 = ; which shows that uj *u in Lp( ). A.2 Existence and Uniqueness of Weak Solution We will obtain a solution by using the basis fwjg of eigenfunctions of the Laplacian to approximate the equation by systems of ODEs. We prove the existence and uniqueness of the approximations of (A.1) using the corresponding results for ODEs. Then the Alaoglu compactness theorem will guarantee the existence of a weak limit. Finally, the limit is shown to satisfy (A.1). Theorem A.4. Equation (A.1) has a unique weak solution given by (A.4): for any T > 0 given u(0) = u0 2L2(Q) there exists a solution u with u2L2(0;T;H10 (Q))\L4(Q (0;T)); u2C0([0;T];L2(Q)); and u0 7! u(t) is continuous on L2(Q). Equation (A.1) holds as an equality in the space L4=3(0;T;H 1(Q)). Proof. Let the functions wk = wk(x), k = 1;2; be eigenfunctions of . It is known that fwkg1k=1 are smooth and form an orthogonal basis of H10 (Q). For a xed n> 0, de ne 84 Pn as the orthogonal projection in L2 onto the span of fwkgnk=1: Pnu = nX j=1 (u;wj)wj: We will look for the n dimensional approximation un(t) = nX j=1 unj(t)wj; of the solution u, with the coe cients unj(t) (0 t T;1 j n) that satisfy unj(0) = (un(0);wj) = (u0;wj) and (dundt ;wj) + ( un;wj) = (w(un);wj); j = 1;:::;n that is, dunj dt unj = (w(un);wj): We could also write this in a vector form as dun dt un = Pnw(un); un(0) = Pnu0: (A.9) Since the nonlinearity in (A.9) is locally Lipschitz, then the nite-dimensional system has a unique solution on some nite time interval. Multiplying (A.9) by un and integrating over Q, we get 1 2 d dtjjunjj 2 L2(Q) + ( un;un) = (Pnw(un);un): (A.10) Since ( un;un) =jjrunjj2L2(Q); 85 and jjunjj2H1 0(Q) is equivalent to jjrunjj2L2(Q), we have (Pnw(un);un) = (w(un);Pnun) = (w(un);un) Z Q 1 junj4dx; then (A.10) becomes 1 2 d dtjjunjj 2 L2(Q) +jjunjj 2 H10(Q) + Z Q junj4dx jQj: Integrating both sides in time from 0 to T gives 1 2jjun(T)jj 2 L2(Q) +jjunjj 2 L2(0;T;H10(Q)) +jjunjj 4 L4(Q (0;T)) 1 2jju0jj 2 L2(Q) +TjQj: With the time interval being nite, the domain being bounded, and u0 2L2(Q), we see that un is uniformly bounded in L1(0;T;L2(Q)); un is uniformly bounded in L2(0;T;H10 (Q)); un is uniformly bounded in L4(Q (0;T)): (A.11) Since jw(s)j (1 +jsj3), jjw(un)jj4=3L4=3(Q (0;T)) = Z T 0 Z Q jw(un)j4=3dx dt 4=3 Z T 0 Z Q (1 +junj3)4=3 dt Z T 0 C Z Q junj4 + 1dx dt: We have w(un) is uniformly bounded in L4=3(Q (0;T)). Next, note also that, by the Sobolev embedding theorem, H1(Q) L4(Q); 86 that is, v2H10 (Q) implies that v2L4(Q). Then, it follows that, if u2L4=3(Q), the dual of L4(Q), then u2H 1(Q), the dual of H10 (Q), so that L4=3(Q) H 1(Q): Therefore, L2(0;T;H 1(Q)) and L4=3(0;T;L4=3(Q)) are continuously embedded in the space L4=3(0;T;H 1(Q)). It follows by (A.9) that we have dun dt is uniformly bounded in L 4=3(0;T;H 1(Q)): (A.12) By Banach-Alaoglu weak- compactness theorem, we can extract a convergent subse- quence fung converging weakly in the following spaces un *u in L2(0;T;H10 (Q)); un *u in L4(Q (0;T)); w(un) * in L4=3(Q (0;T)); (A.13) for some 2L4=3(Q (0;T)). Furthermore, since un is uniformly bounded in L2(0;T;H10 (Q)) by (A.11), and dundt is uniformly bounded in L4=3(0;T;H 1(Q)) by (A.12), and the following chain of embeddings holds H10 (Q) L2(Q) H 1(Q) with H10 (Q) being re exive, by Theorem A.2, we can extract a further subsequence such that un!u in L2(0;T;L2(Q)): 87 Actually, we need Pnw(un) * in L4=3(Q (0;T)). Therefore we write Z Q (0;T) (Pnw(un) )?dxdt = Z Q (0;T) (w(un) )?dxdt Z Q (0;T) Qnw(un)?dxdt; for all ?2L4(Q (0;T)), where Qn = I Pn. The rst terms tends to zero by (A.13). For the second term, let ? = nX j=1 j(t)?j with j 2L4(0;T) and ?j 2C1c (Q). Such functions ? are dense in L4(Q (0;T)) and the following identity holds for them Z Q (0;T) Qnw(un) nX j=1 j(t)?j ! dxdt = Z Q (0;T) w(un) nX j=1 j(t)Qn?j ! dxdt: Since Pnu!u in L4(Q) for u2L4(Q), we have Qnu! 0 in L4(Q), that is, Qn?j ! 0 in L4(Q) for each j. Hence we have the convergence Pnw(un) * in L4=3(Q (0;T)) as required. It follows that every term in (A.9) converges in the weak- topology of the dual space of V = L2(0;T;H10 (Q))\L4(Q (0;T)) which is V = L2(0;T;H 1(Q)) +L4=3(Q (0;T)). Then du dt u = holds in L2(0;T;H 1(Q)) +L4=3(Q (0;T)) L4=3(0;T;H 1). It remains to show that = w(u). Since un!u in L2(Q (0;T)), there is a subsequence unj such that unj(x;t)!u(x;t) for a.e. (x;t)2Q (0;T). It follows, using the continuity of w, that w(unj(x;t))!w(u(x;t)) for a.e (x;t)2Q (0;T). With the uniform bound on w(unj) 2L4=3(Q (0;T)), we deduce that w(unj) * w(u) in L4=3(Q (0;T)) by Lemma A.3. By the uniqueness of the limit, w(u) = . 88 To prove the continuity of u(t) from [0;T] into L2(Q), notice that u2L2(0;T;H10 (Q))\ L4(Q (0;T)) and that du dt = u+w(u)2L 2(0;T;H 1(Q)) +L4=3(Q (0;T)): By extending u outside [0;T] by zero and setting um = (u) 1 m , a molli ed version of u with respect to variable t, we can approximate u by a sequence um 2 C1([0;T];H10 (Q)) which converges to u in the sense that um!u in L2(0;T;H1(Q)); dum dt ! du dt in L 2(0;T;H 1(Q)): Then for any t0, jjum(t)jj2L2(Q) =jjum(t0)jj2L2(Q) + 2 Z t t0 hddsum(s);um(s)ids: Choosing t0 such that jjum(t0)jj2L2(Q) = 1T Z T 0 jjum(t)jj2L2(Q)dt; we estimate jhddtum(s);um(s)ij jjdum(s)dt jjH 1(Q)jjum(s)jjH10(Q); to obtain, jjum(t)jj2L2(Q) 1T Z T 0 jjum(t)jj2L2(Q)dt+ 2 Z T 0 jjdumdt jjH 1(Q)jjumjjH10(Q)dt: Hence, sup t2[0;T] jjum(t)jj2L2(Q) C(jjumjjL2(0;T;H10(Q)) +jjdumdt jjL2(0;T;H 1(Q))); where the constant C depends on T. 89 Since um is a Cauchy sequence in L2(0;T;H10 (Q)) and dumdt is a Cauchy sequence in the space L2(0;T;H 1(Q)), it follows that um is a Cauchy sequence in C0([0;T];L2(Q)) and therefore u2C0([0;T];L2(Q)). Next we need to show that u(0) = u0. Choose some ?2C1([0;T];H10 (Q)\L4(Q)) with ?(T) = 0. We note that, in particular, ?2L2(0;T;H10 (Q))\L4(Q (0;T)). Integrating by parts the following equation in the variable t, we have Z T 0 hu;?0i+ ( u;?)ds = Z T 0 hw(u(s));?ids+ (u(0);?(0)): Performing the same step in the Galerkin approximation yields Z T 0 hun;?0i+ ( un;?)ds = Z T 0 hPnw(un(s));?ids+ (un(0);?(0)): Since un(0) = Pnu0 !u0, we take limits to have Z T 0 hu;?0i+ ( u;?)ds = Z T 0 hw(u(s));?ids+ (u0;?(0)): Hence, u(0) = u0. To prove uniqueness and continuous dependence, let u0 and v0 be in L2(Q) and consider h(t) = u(t) v(t). Then, dh dt h = w(u) w(v); h(0) = u0 v0; and multiplying by h and integrating over Q gives 1 2 d dtjjhjj 2 L2(Q) +jjhjj 2 H10(Q) = (w(u) w(v);h): 90 Note that we have w0(s) 2 , for all s2R, so (w(u) w(v);h) = Z Q (w(u(x)) w(v(x)))(u(x) v(x))dx = Z Q Z u(x) v(x) w0(s)ds ! (u(x) v(x))dx Z Q 2ju(x) v(x)j2dx =2jjhjj2L2(Q): We therefore obtain 1 2 d dtjjhjj 2 L2(Q) 2jjhjj 2 L2(Q); and by Gronwall?s inequality jju(t) v(t)jjL2(Q) e2tjju0 v0jjL2(Q): Hence we have uniqueness if u0 = v0 and continuous dependence on initial conditions other- wise. A.3 Existence and Uniqueness of Strong Solutions Now, we will show how increasing the regularity of u0 results in more regular solutions. In particular, if u0 2H10 \Lp, then u(t) is in this space for all t 0, and the solutions are continuous into H10 . We call such solutions strong solutions. The uniqueness follows from the previous theorem, since a strong solution is automatically a weak solution. Theorem A.5. If u0 2H10 (Q)\L4(Q), then there exists a unique strong solution u(t)2C0([0;T];H10 (Q))\L1(0;T;L4(Q))\L2(0;T;D(A)); where A = and D(A) is the domain of A. 91 Proof. We take the inner product of ordinary di erential system dun dt +Aun = Pnw(un); un(0) = Pnu0; (A.14) with Aun to obtain 1 2 d dtjjrunjj 2 L2(Q) +jjAunjj 2 L2(Q) = Z Q Pnw(un) undx = Z Q w(un) undx = Z Q w0(un)jrunj2dx; using the boundary condition un = 0 on @Q for integration by parts and the fact that w(0) = 0. Therefore, we have 1 2 d dtjjrunjj 2 L2(Q) +jjAunjj 2 L2(Q) 2jjrunjj 2 L2(Q): Integrating both sides from 0 to T gives 1 2jjrun(T)jj 2 L2(Q) + Z T 0 jjAun(s)jj2L2(Q)ds Z T 0 jjrunjj2L2(Q)dt+ 12jjrun(0)jj2L2(Q); and so un is uniformly bounded in L2(0;T;D(A)) and L1(0;T;H10 (Q)), and we already know from the previous proof that un2L2(0;T;H10 (Q)). We now multiply (A.14) by dundt and integrate over Q. Simplifying the last term in the resulting identity as follows (Pnw;dundt ) = (w;Pndundt ) = (w;dundt ); 92 yields dun dt 2 L2(Q) + 12 ddtjjrunjj2L2(Q) = ddt Z Q W(un)dx; where W(s) = Rs0 w(t)dt. Integrating the above equation from 0 to t gives us Z t 0 dun dt 2 L2(Q) ds+ 12jjrun(t)jj2L2(Q) Z Q W(un(t))dx 12jjru0jj2L2(Q) + Z Q W(un(0))dx; and using 1 34jsj4 W(s) 1 14jsj4; we have Z t 0 dun dt 2 L2(Q) ds+ 12jjrunjj2L2(Q) + 14 Z Q jun(t)j4dx 2jQj+ 12jjru0jj2L2(Q) + 34 Z Q ju0j4dx: Hence, dundt is uniformly bounded in L2(0;T;L2(Q)) and un is uniformly bounded in the space L1(0;T;L4(Q)). Therefore we can extract the appropriate subsequence such that u2L1(0;T;L4(Q));u2L2(0;T;D(A));dudt 2L2(0;T;L2(Q)); which implies that u2C0([0;T];H10 (Q)). 93 Appendix B Well-posedness of Swift-Hohenberg Equation In this chapter, we will show the existence of strong solutions to equation (1.3). And we also show that all the solutions are bounded for uniformly bounded initial conditions. Our results are revised from [26] and [27], therefore, we omit the detailed proofs for the theorems. B.1 Preliminary Results First, we establish some notations that will be used in the following results. 1. Weighted norms. De ne the norm jjujjp; = Z Q ju(x)jp (x)dx 1=p ; (B.1) where : R2 ! (0;1) is a suitable weight with jr (x)j 0 (x) for some 0 <1 and 1 = RR2 (x) <1. 2. Uniformly local spaces Lplu(Q). De ne the norm jjujjp;lu = supfjjujjp;Ty : y2R2g; where Ty = (x y) is the translated weight and the translations are continuous with respect to the norm above. By the de nition above, jjujjp;Ty = Z Q (x y)ju(x)jpdx 1=p : 94 3. Space ~Lplu(Q) and associated Sobolev spaces ~Ws;plu (Q). De ne ~Lplu(Q) =fu2Lploc(Q) :jjujjp;lu <1g ~Ws;plu (Q) =fu2 ~Lplu(Q) : Dqu2 ~Lplu(Q) 8q2Nd0 with q1 + +qd sg for integers s. 4. Uniformly local Sobolev spaces Ws;plu (Q) = closure of C1bdd(Q) in ~Ws;plu (Q); where C1bdd(Q) is the set of all C1 functions which have all derivatives bounded in Q. This construction ensures the density of Ws+1;plu (Q) in Ws;plu (Q) for bounded domain. 5. Weighted Hilbert space. De ne W2;2 (Q) =fu2L2 (Q) :ru;r2u2L2 (Q)g; where L2 (Q) =fu2L2loc(Q) :jjujj 1g: We start with an elementary result about the lower bound on the Hk norms. Lemma B.1. For any weight with jr (x)j 0 and > 0, we have jjrkujj22; 2 jjrk 1ujj22; ( 20 + )jjrk 2ujj22; ; for any k 2. 95 B.2 Existence of Strong Solutions Here we intend to use the theorem proved by Levermore and Oliver in [25] for the existence of solutions to the Swift-Hohenberg equation. Theorem B.2. Let X;Y and Z be Banach spaces with Y Z X and let T(t) = eAtjt 0 be a semigroup on X with jjT(t)ujjY ct jjujjX; for all u2X; jjT(t)ujjY ct jjujjZ; for all u2Z; (B.2) for t2(0;1]. Moreover, let N : Y !Z be locally Lipschitz with jjN(u1) N(u2)jjZ c(jju1jj2 Y +jju2jj2 Y )jju1 u2jjY; for all u1;u2 2Y; (B.3) for some > 0, and without loss of generality we assume N(0) = 0. Assume the exponents ; ; satisfy 0 < 1; 0 (2 + 1) < 1; + 2 < 1; (B.4) then for every M > 0 there exists a time T(M) > 0 such that for every initial condition u0 2X with jju0jjX M there exists a unique solution u2C([0;T];X)\C([0;T];Y) to u(t) = T(t)u0 + Z t 0 T(t )N(u( ))d : (B.5) In addition, the mapping from u0 to u is locally Lipschitz continuous from X to C([0;T];X). 96 Notice that (B.5) is a xed point equation in Y norm. First, by contraction map- ping theorem, we can prove that there exists a unique solution u2E([0;T]) to the xed point problem (B.5). Then by the continuous embedding Z X and the continuity of the semigroup T(t), we can show that u2C([0;T];X). In order to use this theorem to the Swift-Hohenberg equation, we need to construct an analytic semigroup T(t) = eAtjt 0 for the linear part of the equation. Here, we de ne Au = (1 +r2)2u = r4u 2r2u u: (B.6) Theorem B.3. De ne the operator A : D(A) Lplu(Q) !Lplu(Q) by (B.6) for p2 [2;1) with D(A) = W4;plu \f boundary conditions g. Then for all admissible domains Q the resolvent (A I) 1 : Lplu(Q) !D(A) exists and satis es the estimate jj(A I) 1ujjp;lu C Rjjujjp;lu; for all u2Lplu(Q); where R depends only on 0 as de ned in equation (B.1) and C depends only on p. To prove this theorem, consider the boundary value problem r4u+ 2r2u+ u = f; for some which will be determined later, with the associated weak form B[u;v] = Z Q fvdx; for all v2C10 (Q); where B[u;v] = Z Q r2ur2v + 2r2urvr +vr2ur2 + 2 vr2u+ uvdx: 97 For any f2L2lu(Q) we can use Lax-Milgram theorem in the weighted Hilbert space W2;2 (Q) and this yields a unique solution u2W2;2 (Q). Finally, Hille-Yosida theorem with operator A0u = r4u 2r2u, that is, f = ( I A0)u, will complete the proof of this theorem. Next we can apply Theorem B.2 and Theorem B.3 to prove the existence of the solutions to the Swift-Hohenberg equation. Theorem B.3 shows that A = r4 2r2 1 is the in nitesimal generator of an analytic semigroup T(t) = eAtjt 0. Theorem B.4. For each admissible domain Q R2 and boundary conditions, and all initial conditions u0 2Lplu(Q) there is a unique strong solution u(t) = T(t)u0. Furthermore, for xed weight (x) = e jxj, there is a constant C such that for all admissible domains Q R2 and all initial conditions u0 2Lplu(Q) we have lim sup t!1 jju(t)jj1;4;lu C: We can construct an absorbing set in W3;dlu (Q) by the above conclusion. Let B0 =fu2 W3;plu (Q) :jjujj3;d;lu 2Cg, then Babs(Q) =[t>0T(t)(B0) W3;dlu (Q) is a bounded, invariant set, since the union can also be taken over a nite time interval. The above estimates imply that every bounded set in Lplu(Q) with p > d is absorbed in nite time into Babs. 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