Slow motion for the 1D Swift–Hohenberg equation
The goal of this paper is to study the behavior of certain solutions to the Swift–Hohenberg equation on a one–dimensional torus . Combining results from –convergence and ODE theory, it is shown that solutions corresponding to initial data that is –close to a jump function , remain close to for large time. This can be achieved by regarding the equation as the –gradient flow of a second order energy functional, and obtaining asymptotic lower bounds on this energy in terms of the number of jumps of .
1 Introduction, Motivation and Main Results
The fourth order partial differential equation
is a generalization of the Swift–Hohenberg equation introduced in 1977 by Swift and Hohenberg  as a model for the study of pattern formation, in connection with the Rayleigh–Bérnard convection, e.g. see ,. Among many different applications, the most famous ones in the literature are those in connection to pattern formation in vibrated granular materials , buckling of long elastic structures , Taylor–Couette flow , , and in the study of lasers . Moreover, in recent years great attention has been paid to models of phase transitions in the study of pattern–formation in bilayer membranes, see e.g.  where the Swift–Hohenberg equation turns out to be the gradient flow of Ginzburg–Landau type energies, with respect to the right inner product structure.
where is a one–dimensional torus. We assume that is a double–well potential with phases supported at and , and we study the long–time behavior of solutions when is sufficiently small. In particular, due to the presence of the small parameter in (1.2) the solutions are expected to develop interfacial structure driven by the minima of the potential . Equation (1.2) may be viewed as a gradient flow associated to a second order energy functional, and our main result consists of an asymptotic lower bound on the corresponding energy functional and the consequent bounds on the speed of evolution of the developed interfaces. Below we outline interfacial dynamics results for the lower order Allen–Cahn equation and its generalizations that provide much of the motivation for our analysis.
1.1 Allen–Cahn Equation and Generalizations to Higher Order
Equations displaying interfacial dynamics have been studied extensively in the last two decades. The prototypical example is the Allen–Cahn equation
(as well as its higher dimensional analog) seen as the –gradient flow of the energy
where is an interval. The special gradient–flow structure of (1.3) has allowed its analysis by a wide variety of methods and techniques.
In particular, it has been shown for the Allen–Cahn equation (see  and the references therein) that if the evolution from a sufficiently regular initial data occurs in four main stages. In the first stage, the diffusion term can be ignored and the leading order dynamics are driven by the independent ordinary differential equation . This is the time-scale in which interfaces develop, i.e., regions in the space domain that separate almost constant solutions corresponding to the stable equilibria of the ordinary differential equation. This stage, referred to as the generation of interface, has been analyzed for the Allen–Cahn equation first in , and subsequently in , , , , and other papers.
As the regions separating unequal equilibria decrease in length, the spacial gradient necessarily increases, and after time the dynamics are driven by a balance between the two terms on the right–hand side of (1.3). In particular, as shown in , after time the solution is exponentially close to the standing–wave profile
parametrized by the positions , where satisfies
The zeros of can be viewed as specifying the location of the interfaces. In particular, the residual is exponentially small and the corresponding third stage of the evolution proceeds on an exponentially slow time scale until two zeros of the solution of (1.3) collide and disappear as part of the fourth stage of the evolution.
The third stage, usually referred to as Slow Motion has been studied extensively. The most precise interface evolution results for the Allen–Cahn equation are given in , , , . Specifically, the zeros of the solution are approximated by , which at leading order move according to the evolution law
where , is a constant depending only on . The proof of this reduction involves invariant manifold theory and geometric analysis.
In  Bronsard and Kohn adopted a variational viewpoint to study the Allen–Cahn equation. While their method does not recover the evolution equation above, it does provide relatively simple energy arguments to obtain a bound on the speed of this evolution. In particular, Bronsard and Kohn first prove that for any there exists a constant such that, if is sufficiently close in norm to a step function taking values and having exactly jumps, and its energy satisfies
where , then
Using this energy estimate they prove that the solution of (1.3) with Dirichlet or Neumann boundary data, under the same conditions on the initial data , satisfies
for any . The limit in (1.10) may be viewed as providing an upper bound on the speed of the evolution of the transition layers of . Improvements of (1.9) have been obtained in  and . In particular, it has recently been established in  that for a sequence converging to a step function taking values and having exactly jumps, the Allen–Cahn functional admits the following multiple order asymptotic expansion
where are constants dependent on the potential and is the distance between consecutive transition layers of .
The gradient flow associated with the second order term in the above energy expansion gives, up to a multiplicative constant, the evolution equation (1.7), providing a crucial link between the variational and geometric approaches. Further insight into this connection can be seen as part of a general framework of –convergence of gradient flows developed in .
In regards to extensions to higher–order functionals, the problem has been studied in  in connection with a family of higher order functionals of the form
where stands for the –th spatial derivative of . Due to difficulties associated with higher order nature of the functional, in particular, the lack of exact solutions of the corresponding Euler–Lagrange equation, sharp bounds analogous to (1.1) have not been established. An important condition on in  is
Hypothesis 1: There exists constants such that for every interval with length and all
Under this hypothesis the authors prove that for any sufficiently close to a step function taking values and having exactly jumps,
where is a constant satisfying , for all eigenvalues of the linearization of
The initial value problem (1.2) can be seen as the –gradient flow of the second order energy functional
We note that the functional (1.15) does not satisfy Hypothesis 1 due to the negative term in the energy. We use recently established interpolation inequality (see  and ) to overcome this difficulty if is sufficiently small. Moreover, in the proof of an energy estimate analogous to (1.13), see Theorem 1.1, we do not assume any closeness condition on the functions we work with, we instead make an assumptions on the zeros of such functions.
Furthermore, inspired by , our analysis relies on the use of a particular test function, and on the study of the solutions of the Euler–Lagrange equation associated to (1.15) via hyperbolic fixed point theory, in particular through the work of Sell . Thanks to this approach we are able to improve the exponent in (1.13) and, consequently, obtain sharper bound on the speed of evolution for solutions of (1.2).
We recall that the –convergence of the energy functional has been proved in  for the case , and in  and  when is small. The asymptotic behavior of plays a crucial role in our analysis: we will use results from –convergence, together with a careful analysis of the minimizers of the associated Euler–Lagrange equation, to study the speed of motion of solutions of (1.2).
To conclude, we remark that the situation in the higher dimensional setting is quite different: solutions of the higher dimensional version of (1.3) and other classical gradient flow–type equations have been studied by many different authors, see, e.g., , , , , , . Due to the lack of results like (1.9), all of them use significantly different approaches to the one introduced in . A more recent work, see , closes the gap by making use of a –convergence result proved in  and doesn’t assume any specific structure of the initial data.
1.2 Statement of Main Results
We remark that a similar estimate can be obtained when the domain is an interval , with (1.16) replaced with
We highlight the fact that we are not requiring the function of Theorem 1.1 to be –close to a jump function, in contrast with , , , . On the other hand, it is easy to show that if is –close to a jump function taking values , then there exists an with the property that the zeros of are at least apart, as in the statement of Theorem 1.1.
The energy estimate above is a crucial ingredient to prove slow motion of solutions of (1.2), when the initial data is close in the norm to a function, as in , , . In particular, we will consider regular solutions of (1.2), whose existence is proved in the Appendix, see Theorem 4.1. Our analysis yields the following result.
To the best of our knowledge, only recently some regularity results for the Swift–Hohenberg equation have been proved, see . In the statement of Theorem 1.3 we assume that the solutions are sufficiently regular. In the Appendix we prove existence of solutions (though with weaker regularity) using De Giorgi’s technique of Minimizing Movements (see Theorem 4.1).
1.3 Outline of the Proof
A key step in proving the energy inequality (1.16) is a bound from below by the energy of an appropriately chosen test function. Given satisfying the assumptions of Theorem 1.1, we follow  to construct this test function by gluing together minimizers of the energy on each subinterval , where the admissible class now consists of functions that equal zero at the endpoints of . Thus,
where also solves a fourth order Euler–Lagrange equation corresponding to the energy functional.
This initial energy inequality has several key advantages. First, it assumes no assumptions about closeness of to a step function taking values . The required estimates can be proved for . Secondly, the additional property that solves a fourth order ODE on the whole subinterval is key in obtaining a sharper lower bound than the one established in . Specifically, in the middle of each subinterval , we can show that the minimizer , where the exponent is related to the linearization of the Euler–Lagrange equation. In fact, obtaining this bound is the central contribution of this paper, starting from Corollary 2.3 and culminating in Proposition 2.7. The proofs of Lemmas 2.4 and 2.5, which give the initial crude estimates on the ‘closeness’ of to , follow the ideas of  supplemented by the use of the interpolation inequality given in Lemma 2.2 and the use of instead of the original function . A point of departure is Lemma 2.6, in which the use of a Hartman–Grobman type theorem (see Theorem 5.4, from ), combined with the extra information on and the analysis of the linearized problem, allow us to obtain sharper exponential decay estimate.
Once these bounds on are obtained, we show that its energy is larger than the energy of the ‘optimal profile’ connecting the zeros of with and having energy . This is accomplished in the proof of Theorem 1.1.
In the remainder of the paper, we use the energy lower bound to obtain slow motion results in Section 3. Finally, in the Appendix we present a proof of existence of solutions for equation (1.2) in the more general case of a bounded domain , along with partial regularity results for the solutions themselves.
2 Preliminaries and Assumptions
Throughout this paper we will work with a double–well potential satisfying
A prototype for is given by
2.1 –convergence and Interpolation Inequalities
In this section we recall some properties of the energy
in the more general setting where is a bounded open set of with boundary, is a small parameter, and is a double–well potential, as in (2.5). In  Chermisi, Dal Maso, Fonseca and Leoni proved that the sequence of functionals , defined by
–converges as to the functional ,
We define the one–dimensional rescaled energy
and we introduce the set of admissible functions
We note that it was proved in , Section 5.1, that
so that in dimension we have
where is the number of jumps of the function . We further define
and remark that in our case of symmetric potential , . One of the key tools to prove the –convergence result is the following nonlinear interpolation inequality, see e.g. Theorem 3.4 in .
In particular, in the one dimensional setting, we will often use the following nonrescaled version of the previous result, see Lemma 3.1 in .
Let and be as in Lemma 2.2. Then there exist such that for every open interval , every , and every ,
for all .
Let and . We change variables , subdivide the resulting rescaled domain into subintervals, , of length between and (since ) and use Lemma 2.2 to obtain
The following lemmas established for a generalization of the Modica–Mortola Functional in  will be useful to prove our main result. While our energy does not satisfy the assumptions of , their argument is easily extended to our case with the help of the interpolation inequality (2.12). In particular, Lemma 2.4, shows that an function with a uniformly bounded energy, necessarily takes values close to and has small derivatives, except on a set of measure and Lemma 2.5 gives a characterization of the global minimizers for the energy , defined in (2.7), subject to small boundary conditions.
Let be an open interval, and . Then there exists a constant such that for any and every with the following property holds: there is a measurable set with such that
hold for all , where denotes the usual distance between a point and a set.
Then there exist constants such that the following holds. If and then the functional defined in (2.7) has a global minimizer on . This minimizer solves the Euler–Lagrange equation, and satisfies the estimates
We prove the proposition when , the case being identical.
We divide the proof into several steps. Moreover, we simplify the notation used for the norms when the domain of integration will be clear from the context.
Step 1. Fix . We claim that there exists such that if , then
To show this we note that, if satisfy for all , with , and , then the function
belongs to .
Using as a test function, (2.21) follows from Taylor’s formula for and the facts that and .
Step 2. Fix . We will show that there exists such that for every , with on and we have
Otherwise, there are points satisfying
and this proves (2.23).
Step 3. We claim that there exists and such that if and , with , then
By taking sufficiently small, we may assume that on . Indeed, since , if for some , then necessarily there exists such that , and so by (2.4),
which contradicts Step 1 for sufficiently small. Hence, Steps 1 and 2 imply (2.25).
Step 4. Finally, (2.12) with and standard compactness and lower semicontinuity arguments imply the existence of minimizer of and since by previous step for and
Furthermore, since is , from (2.26) and the Mean Value Theorem we have
The Euler–Lagrange equation
the bound from Step 3 and (2.27) imply
for some .
2.2 The Euler–Lagrange Equation
In this section we further analyze the behavior of the minimizers of the energy with the aid of the corresponding Euler-Lagrange equation, and we prove our main result, Theorem 1.1.
Consider the ordinary differential equation
where is a mapping satisfying for some . Assume has four eigenvalues , where and . Then for