On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow
We study the axisymmetric surface diffusion flow (ASD), a fourth-order geometric evolution law. In particular, we prove that ASD generates a real analytic semiflow in the space of -little-Hölder regular surfaces of revolution embedded in and satisfying periodic boundary conditions. Further, we investigate the geometric properties of solutions to ASD. Utilizing a connection to axisymmetric surfaces with constant mean curvature, we characterize the equilibria of ASD. Then, focusing on the family of cylinders, we establish results regarding stability, instability and bifurcation behavior, with the radius acting as a bifurcation parameter.
Key words and phrases:Surface diffusion, well posedness, periodic boundary conditions, maximal regularity, nonlinear stability, bifurcation, implicit function theorem
2000 Mathematics Subject Classification:Primary 35K93, 53C44 ; Secondary 35B35, 35B32, 46T20
The central focus of this article is the development of an analytic setting for the axisymmetric surface diffusion flow (ASD) with periodic boundary conditions. We establish well-posedness of ASD and investigate geometric properties of solutions, including characterizing equilibria and studying their stability, instability and bifurcation behavior. We establish and take full advantage of maximal regularity for ASD. Most notably, with maximal regularity we gain access to the implicit function theorem, a very powerful tool in nonlinear analysis and dynamical systems theory. We begin with a motivation and derivation of the general surface diffusion flow, of which ASD is a special case, and we introduce the general outline of the paper.
The mathematical equations modeling surface diffusion go back to a paper by Mullins  from the 1950s, who was in turn motivated by earlier work of Herring . Both of these authors investigate phenomena witnessed in sintering processes, a method by which objects are created by heating powdered material to a high temperature, while remaining below the melting point of the particular substance. When the applied temperature reaches a critical value, the atoms on the surfaces of individual particles will diffuse across to other particles, fusing the powder together into one solid object. In response to gradients of the chemical potential along the surface of this newly formed object, the surface atoms may undergo diffusive mass transport on the surface of the object, attempting to reduce the surface free energy. Given the right conditions – temperature, pressure, grain size, sample size, etc. – the mass flux due to this chemical potential will dominate the dynamics and it is the resulting morphological evolution of the surface which the surface diffusion flow aims to model. We also note that the surface diffusion flow has been used to model the motion of surfaces in other physical processes (e.g. growth of crystals and nano-structures). The article  contains the formulation of the model which we present below, which is set in a more general framework than the original model developed by Mullins.
1.1. The Surface Diffusion Flow
From a mathematical perspective, the governing equation for motion via surface diffusion can be expressed for hypersurfaces in arbitrary space dimensions. In particular, let be a closed, compact, immersed, oriented Riemmanian manifold with codimension 1. Then we denote by the (normalized) mean curvature on , which is simply the sum of the principle curvatures on the hypersurface, and denotes the Laplace–Beltrami operator, or surface Laplacian, on . The motion of the surface by surface diffusion is then governed by the equation
where denotes the normal velocity of the surface . If encloses a region we assume the unit normal field to be pointing outward. A solution to the surface diffusion problem on the interval , with , is a family of closed, compact, immersed hypersurfaces in which satisfy the equation
for a given initial hypersurface . It can be shown that solutions to (1.1) are volume–preserving, in the sense that the signed volume of the region is preserved along solutions. Additionally, (1.1) is surface–area–reducing. It is also interesting to note that the surface diffusion flow can be viewed as the -gradient flow of the area functional, a fact that was first observed in . This particular structure has been exploited in [52, 53] for devising numerical simulations.
For well-posedness of (1.1) we mention , where it is shown that (1.1) admits a unique local solution for any initial surface . Additionally, the authors of  show that any initial surface that is a small –perturbation of a sphere admits a global solution which converges to a sphere at an exponential rate. This result was improved in  to admit initial surfaces in the Besov space with and . For dimensions note that this allows for initial surfaces which are less regular than . An independent theory for existence of solutions to higher–order geometric evolution equations, which also applies to the surface diffusion flow, was developed in [37, 58]; see [54, 63] for a discussion of some limitations of these results.
More recent results for initial surfaces with low regularity are contained in [8, 42]. The author of  obtains existence and uniqueness of local solutions for various geometric evolution laws (including the surface diffusion flow), in the setting of entire graphs with initial regularity . Surface diffusion is also one of several evolution laws for which the authors of  establish solutions under very weak, and possibly optimal, regularity assumptions on initial data. In particular, their results guarantee existence and uniqueness of global analytic solutions, in the setting of entire graphs over , for Lipschitz initial data with small Lipschitz constant .
For interesting new developments regarding lower bounds on the existence time of solutions to the surface diffusion flow in and , we refer the reader to [54, 69, 70]. In particular, it is shown in  that the flow of a surface in which is initially close to a sphere in (that is, the -norm of the trace-free part of the second fundamental form is sufficiently small) is a family of embeddings that exists globally and converges at an exponential rate to a sphere. The results of [54, 69] regarding concentration of curvature along solutions may prove important in the analytic investigation of solutions approaching finite–time pinch–off.
In the context of geometric evolution equations, such as the mean curvature flow [35, 36], the surface diffusion flow, or the Willmore flow [43, 44], the underlying governing equations are often expressed by evolving a smooth family of immersions where is a fixed smooth oriented manifold and is the image of under . In this formulation, the surface diffusion flow is given by
where is the normal to the surface . This formulation is invariant under the group of sufficiently smooth diffeomorphisms of , and this implies that (1.2) is only weakly parabolic. A way to infer that (1.2) is not parabolic is to observe that if is a solution, then so is , for any diffeomorphism . Given a smooth solution one can therefore construct nonsmooth (i.e. non ) solutions by choosing to be nonsmooth. If (1.2) were parabolic, all solutions would have to be smooth, as was pointed out in  for the mean curvature flow.
Nevertheless, existence of unique smooth solutions for the mean curvature flow, for compact –initial surfaces , can be derived by making use of the Nash-Moser implicit function theorem, see for instance [32, 33].
The Nash-Moser implicit function theorem may also lead to a successful treatment of the surface diffusion flow (1.2). However, there is an alternative approach to dealing with the motion of surfaces by curvature (for example the mean and volume–preserving mean curvature flows, the surface diffusion flow, the Willmore flow) which removes the issue of randomness of a parameterization: if one fixes the parameterization as a graph in normal direction with respect to a reference manifold and then expresses the governing equations in terms of the graph function, the resulting equations are quasilinear and strictly parabolic. This approach has been employed in [26, 29, 30, 65], and also in . In the particular case of the surface diffusion flow, one obtains a fourth–order quasilinear parabolic evolution equation. One can then apply well–established results for quasilinear parabolic equations. The theory in [2, 12], for instance, works for any quasilinear parabolic evolution equation, no matter whether it is cast as a more traditional PDE in Euclidean space, or an evolution equation living on a manifold. This theory also renders access to well–known principles from dynamical systems.
The approach of parameterizing the unknown surface as a graph in normal direction has also been applied to a wide array of free boundary problems, including problems in phase transitions (where the graph parameterization and its extension into the bulk phases is often referred to as the Hanzawa transformation), see for example  and the references therein.
The literature on geometric evolution laws often considers the question of short-time existence standard and refers to the classical monographs [23, 24, 45]. However, when the setting is a manifold rather than Euclidean space, existence theory for parabolic (higher order) equations does not belong to the standard theory and requires a proof, a point that is also acknowledged in , see page 61.
1.2. Axisymmetric Surface Diffusion (ASD)
For the remainder of the paper, we focus our attention on the case of an embedded surface which is symmetric about an axis of rotation (which we take to be the –axis, without loss of generality) and satisfies prescribed periodic boundary conditions on some fixed interval of periodicity (we take and enforce periodicity, without loss of generality). In particular, the axisymmetric surface is characterized by the parametrization
where the function is the profile function for the surface . Conversely, a profile function generates an axisymmetric surface via the parametrization given above.
We thus recast the surface diffusion problem as an evolution equation for the profile functions . In particular, one can see that the surface inherits the Riemannian metric
from the embedding , with respect to the surface coordinates ; where the subscript denotes the derivative of with respect to the indicated variable . It follows that the (normalized) mean curvature of the surface is given by , where
are the azimuthal and axial principle curvatures, respectively, on . Meanwhile, the Laplace–Beltrami operator on and the normal velocity of are
Finally, substituting these terms into the equation (1.1) and simplifying, we arrive at the expression
for the periodic axisymmetric surface diffusion problem. To simplify notation in the sequel, we define the operator
which is formally equivalent to the right hand side of the first equation in (1.3).
The main results of this paper address
existence, uniqueness, and regularity of solutions for (1.3),
nonlinear stability and instability of equilibria for (1.3),
bifurcation of equilibria from the family of cylinders, with the radius serving as bifurcation parameter.
As mentioned in the introduction, we develop and take full advantage of maximal regularity for ASD. In this setting, the results in (a) follow in a straight forward way from . Existence results could also be based on the approach developed in [8, 42], but we prefer to work within the well-established framework of continuous maximal regularity. It provides a general and flexible setting for investigating further qualitative properties of solutions. The novelty of the results in (b) is analysis of the nonlinear structure of solutions. Corresponding results for linear stability and instability of equilibria are contained in  where a precise characterization of the eigenvalues of the linearized problem is given. Based on a formal center manifold analysis, the authors in  predict subcritical bifurcation of equilibria at the critical value of radius , but no analytical proof is provided. Thus, our result in (c) appears to be the first rigorous proof of bifurcation. In addition, we show that the bifurcating equilibria (which are shown to coincide with the Delaunay unduloids) are nonlinearly unstable. We note that previous results show linear instability of unduloids and we refer the reader to Remark 6.7 for a more detailed discussion.
The publication  has served as a source of inspiration for our investigations. It provides an excellent overview of the complex qualitative behavior of ASD, with results supported by analytic arguments and numerical computations.
The first investigations of evolution of an axisymmetric surface via surface diffusion can be traced back to the work of Mullins and Nichols [56, 57] in 1965, where one can already see some of the benefits of this special setting. Taking advantage of the symmetry of the problem, they developed an adequate scheme for numerical methods and they already predicted the finite time pinch–off of tube–like surfaces via surface diffusion, a feature similar to the mean curvature flow and a natural phenomenon to study in exactly this axisymmetric setting. Research continued to focus on pinch–off behavior using numerical methods, c.f. [10, 13, 14, 15, 19, 49, 50], wherein many schemes were developed to handle the continuation of solutions after the change of topology at the moment of pinch–off. Unlike the related behavior for the mean curvature flow, pinch–off for the surface diffusion flow remains a numerical observation that has yet to be verified analytically.
Much research has also focused on the numerical investigation of stability and instability of cylinders under perturbations of various types, see [10, 13, 15] for instance. In an important construction from , the authors observe destabilization of a particular perturbation of a cylinder (i.e. divergence from the cylinder) due to second–order effects of the flow, whereas the first–order (linear) theory predicts asymptotic stability. In fact, their formulation produces conditions under which a perturbation will destabilize due to –order effects, where –order analysis predicts stability. This result highlights the importance of studying the full nonlinear behavior of solutions to ASD.
We proceed with an outline of the article and description of our main results. In Section 2, we establish existence of solutions to (1.3) in the framework of continuous maximal regularity. In particular, we have existence and uniqueness of maximal solutions for initial surfaces which are –little–Hölder continuous. Solutions are also analytic in time and space, for positive time, with a prescribed singularity at time . Additionally, we state conditions for global existence of the semiflow induced by (1.3). We rely on the theory developed in  and the well–posedness results for quasilinear equations with maximal regularity provided in . We include comments on how we prove these well–posedness results in an appendix.
In Section 3, we characterize the equilibria of ASD using results of Delaunay  and Kenmotsu  regarding constant mean curvature surfaces in the axisymmetric setting. We conclude that all equilibria of (1.3) must fall into the family of undulary curves, which includes all constant functions (corresponding to the cylinder of radius ) and the two–parameter family of nontrivial undulary curves .
In Section 4, we prove that the family of cylinders with radius are asymptotically, exponentially stable under a large class of nonlinear perturbations, which maintain the same axis of symmetry and satisfy the prescribed periodic boundary conditions. In particular, given , we prove that any sufficiently small –little–Hölder regular perturbation produces a global solution which converges exponentially fast to the cylinder of radius . The value is determined by the volume enclosed by the perturbation, which may differ from the volume of the original cylinder. In proving this result, we note that the spectrum of the linearized equation at is contained in the left half of the complex plane, though it will always contain 0 as an eigenvalue. By reducing the equation, essentially to the setting of volume–preserving perturbations of a cylinder, we are able to eliminate the zero eigenvalue. We then prove nonlinear stability in the reduced setting, utilizing maximal regularity methods on exponentially weighted function spaces, and we transfer the result back to the (full) problem via a lifting operator.
In Section 5, we prove nonlinear instability of cylinders with radius . We take nonlinear instability to be the logical negation of stability, which one may interpret as the existence of at least one unstable perturbation, see Theorem 5.1 for a precise statement. This result makes use of a contradiction technique reminiscent of results from the theory of ordinary differential equations, c.f. . By isolating the linearization of the governing equation, one takes advantage of a spectral gap and associated spectral projections in order to derive necessary conditions for stable perturbations, which in turn lead to a contradiction.
We note that previous instability results for ASD have focused primarily on classifying stable and unstable eigenmodes of equilibria, which gives precise results on the behavior of solutions associated with unstable perturbations. However, these methods are limited to the behavior of solutions under the linearized flow.
Finally, in Section 6 we apply classic methods of Crandall and Rabinowitz  to verify the subcritical bifurcation structure of all points of intersection between the family of cylinders and the disjoint branches of unduloids. In particular, taking the inverse of the radius as a bifurcation parameter, we verify the existence of continuous families of nontrivial equilibria which branch off of the family of cylinders at radii , for all We conclude that each of these branches corresponds to the branch of –periodic undulary curves. Working in the reduced setting established in Section 4, it turns out that eigenvalues associated with the linearized problem are not simple, hence we cannot directly apply the results of . However, restricting attention to surfaces which are even (symmetric about the surface ), we eliminate redundant eigenvalues, similar to a method used by Escher and Matioc . In this even function setting, we have simple eigenvalues and derive bifurcation results, which we apply back to the full problem via a posteriori symmetries of equilibria.
Using eigenvalue perturbation methods, we are also able to conclude nonlinear instability of nontrivial unduloids, using the same techniques as in Section 5. We once more refer to Remark 6.7 for more information.
In future work we plan to investigate well-posedness of ASD under weaker regularity assumptions on the initial data. This will allow for a better understanding of global existence, and obstructions thereof. In particular, we conjecture that solutions developing singularities will have to go through a pinch-off.
We also plan to consider non–axisymmetric surfaces. In particular, we plan to investigate the stability of cylinders under non-axisymmetric perturbations.
Other interesting questions involve the existence and nature of unstable families of perturbations and reformulations of the problem to allow for different boundary conditions and immersed surfaces. In particular, reformulating ASD in terms of parametrically defined curves in would allow for consideration of immersed surfaces of revolution, a setting within which the branches of –periodic nodary curves would be added to the collection of equilibria. See Section 3 for a definition and graphs of nodary curves.
Throughout the paper we will use the following notation: If and are arbitrary Banach spaces, denotes the open ball in with center and radius and consists of all bounded linear operators from into . For an open set, we denote by the space of all real analytic mappings from into .
1.3. Maximal Regularity
We briefly introduce (continuous) maximal regularity, also called optimal regularity in the literature. Maximal regularity has received a lot of attention in connection with parabolic partial differential equations and evolution laws, c.f. [3, 4, 5, 12, 41, 48, 60, 64]. Although maximal regularity can be developed in a more general setting, we will focus on the setting of continuous maximal regularity and direct the interested reader to the references [3, 48] for a general development of the theory.
Let , for some , and let be a (real or complex) Banach space. Following the notation of , we define spaces of continuous functions on with prescribed singularity at 0. Namely, define
where denotes the space consisting of bounded, uniformly continuous functions. It is easy to verify that is a Banach space when equipped with the norm . Moreover, we define the subspace
and we set
Now, if and are a pair of Banach spaces such that is continuously embedded in , denoted , we set
where is a Banach space with the norm
It follows that the trace operator , defined by , is well-defined and we denote by the image of in , which is itself a Banach space when equipped with the norm
For a bounded linear operator which is closed as an operator on , we say is a pair of maximal regularity for and write , if
where denotes the space of bounded linear isomorphisms. In particular, is a pair of maximal regularity for if and only if for every , there exists a unique solution to the inhomogeneous Cauchy problem
2. Well-Posedness of (1.3)
When considering the surface diffusion problem, the underlying Banach spaces and in the formulation of maximal regularity will be spacial regularity classes which describe the properties of the profile functions . We proceed by defining these regularity classes. We define the one-dimensional torus , where the points and are identified, which is equipped with the topology generated by the metric
There is a natural equivalence between functions defined on and 2-periodic functions on which preserves properties of (Hölder) continuity and differentiability. In particular, we will be working with the so-called periodic little-Hölder spaces for . Definitions and basic properties of periodic little-Hölder spaces, as well as details on the connection between spaces of functions on and -periodic functions on can be found in  and the references therein. For the readers convenience, we provide a brief definition of below.
For , denote by the Banach space of -times continuously differentiable functions , equipped with the norm
Moreover, for and , we define the space to be those functions such that the -Hölder seminorm
is finite. It follows that is a Banach space when equipped with the norm
Finally, we define the periodic little-Hölder space
for and which is a Banach algebra with pointwise multiplication of functions and equipped with the norm inherited from . For equivalent definitions and more properties of the periodic little-Hölder spaces, see [46, Section 1].
In order to make explicit the quasilinear structure of (1.3), we reformulate the problem. By expanding the governing equation we arrive at the formally equivalent problem
where, for appropriately chosen functions ,
is a fourth-order differential operator with variable coefficients over and
is a -valued function over . Looking at these formal expressions, one can deduce several properties that the functions must satisfy in order to get good mapping properties for and . In particular, we want to choose such that for all , also we want that the spacial derivatives and make sense and the products , , , etc. have desired regularity properties. With these conditions in mind, we proceed with our well-posedness result.
2.1. Existence and Uniqueness of Solutions
We collect statements of well–posedness results and refer the reader to the appendix for comments on their proof. Fix and define the spaces of -valued little–Hölder continuous functions
where , for , denotes the continuous interpolation functor of Da Prato and Grisvard, c.f.  or . It is well-known that the little-Hölder spaces are stable under this interpolation method, in particular we know that
Before we can properly state a result on maximal solutions, we need to introduce one more space of functions from an interval to a Banach space , with prescribed singularity at zero. Namely, if for , i.e. is a right-open interval containing 0, then we set
which we equip with the natural Fréchet topologies induced by and , respectively.
Let . Then
where denotes the space of real analytic mappings between Banach spaces.
Proposition 2.2 (Existence and Uniqueness).
Fix and take so that . For each initial value , there exists a unique maximal solution
where denotes the maximal interval of existence for initial data . Further, it follows that
is open in and is an analytic semiflow on , i.e. using the notation , the mapping satisfies the conditions
The results in  also give the following conditions for global solutions.
Proposition 2.3 (Global Solutions).
Let for , such that , and suppose there exists so that, for all
then it must hold that , so that is a global solution. Conversely, if and , i.e. the solution breaks down in finite–time, then one, or both, of the conditions stated must fail to hold.
We can also state the following result regarding analyticity of the maximal solutions in both space and time.
Proposition 2.4 (Regularity of Solutions).
Under the same assumptions as in Proposition 2.2, it follows that
The preceding results can be slightly weakened to allow for arbitrary values of i.e. without eliminating the possibility that , by taking initial data from the continuous interpolation spaces , which coincide with the Zygmund spaces over .
3. Characterizing The Equilibria of ASD
We begin our analysis of the long-time behavior of solutions by characterizing and describing the equilibria of (1.3). For this characterization, we make use of a well-known, strict Lyapunov functional for the surface diffusion flow, namely the surface area functional, and a characterization of surfaces of revolution with prescribed mean curvature, as presented by Kenmotsu .
Recalling the operator , as expressed by (1.4) and taking it to be defined on , one will see that the set of equilibria of (1.3) coincides with the null set of . Although, from the well-posedness results of the previous section, we know that we can consider (1.3) with initial conditions in , upon which the operator is not defined, one immediately sees that all equilibria must be in (in fact, by Proposition 2.4, we can conclude that equilibria are in ). More specifically, if we define equilibria to be those elements , such that the maximal solution satisfies
then it follows immediately that and . Now, we proceed by characterizing the elements of the null set of .
Consider the functional
which corresponds to the surface area of . If is a solution to (1.3) on the interval , then (suppressing the variable of integration)
where we use integration by parts twice and eliminate boundary terms because of periodicity. Notice that the expression is non-positive for all times .
If is an equilibrium of (1.3) it follows that is identically zero on . Meanwhile, by definition of the operator , whenever . Hence, we conclude that is a strict Lyapunov functional for (1.3), as claimed, and we also see that the equilibria of (1.3) are exactly those functions for which the mean curvature function is constant on .
The axisymmetric surfaces with constant mean curvature have been
characterized explicitly by Kenmotsu in . All equilibria
of (1.3) are so-called undulary curves, and the unduloid
surfaces, which are generated by the undulary curves by revolution about the axis of symmetry,
are stationary solutions of the original surface diffusion problem (1.1).
Any complete surface of revolution with constant mean curvature is either a sphere, a catenoid, or a surface whose profile curve is given (up to translation along the axis of symmetry) by the parametric expression, parametrized by the arc-length parameter ,
Although the curves are well-defined for arbitrary values and , it is not difficult to see that, up to translations along the –axis, we may restrict our attention to values and , c.f. [39, Section 2]. However, in the sequel we will consider the unduloids in the setting of even functions on , for which we will benefit by allowing .
When , corresponds to a family of spheres controlled by the parameter . The spheres are a well-known family of stable equilibria for the surface diffusion flow, c.f. [26, 70], however their profile curves are outside of our current setting because they fail to be continuously differentiable functions on all of . Moreover, we note that the spheres represented by are in fact a connected family of spheres, or a chain of pearls (see Figure 1)111All figures were generated with GNU Octave, version 3.4.3, copyright 2011 John W. Eaton, and GNUPLOT, version 4.4 patchlevel 3, copyright 2010 Thomas Williams, Colin Kelley., for which even general techniques for (1.1) break down, as the manifold is singular at the points of intersection. These families of connected spheres may be interesting objects to investigate in a weaker formulation of ASD, but they fall outside of the current setting.
Catenoids, or more precisely the generating catenary curves (which are essentially just the hyperbolic cosine, up to scaling), fail to satisfy periodic boundary conditions, c.f. Figure 1.
In case , the curve is called a nodary (see Figure 2), which cannot be realized as the graph of a function over the -axis and hence falls outside the current setting. A reformulation of (1.3) to allow for immersed surfaces would permit nodary curves as equilibria. Such an extension of the current setting may prove beneficial to the investigation of pinch–off, as it may likely be easier to handle concerns regarding concentration of curvature for solutions near nodary curves, rather than embedded undularies.
For values , is the family of undulary curves, which generate the unduloid surfaces. The undulary curves are representable as graphs of functions over the -axis, which are strictly positive for in the given range (see Figure 3). In fact, the case corresponds to the cylinder of radius . Hence, by Theorem 3.1 above, we conclude that all equilibria of (1.3) fall into the family of undulary curves.
Notice that the curve is always periodic in both the parameter and the spacial variable . In order to ensure that the curve satisfies the -periodic boundary conditions enforced in (1.3) (which we emphasize is a condition regarding periodicity over the variable and not the arc-length parameter ), we must impose further conditions on the parameters and ; here we avoid because the curve trivially satisfies periodic boundary conditions. In particular, for , if and satisfy the relationship
then the curve is periodic in the variable, for . In the sequel, we will use the notation to denote the periodic undulary curve with free parameter and parameter fixed according to (3.2).
The role of the parameters and is clearly seen in the context of Delaunay’s construction. By rolling an ellipse with eccentricity along the –axis, the path traced out by one focus is an undulary curve. Here corresponds to a reassignment of major and minor axes in the associated ellipse. Further, it is clear that the ellipses are restricted to those with circumference , to match periodic boundary conditions.
4. Stability Of Cylinders With Large Radius
As seen above, the constant function , for , is an equilibrium of (2.1). Moreover, the constant function is associated to the cylinder with radius , which is a stationary solution of the original surface diffusion problem (1.1). In this section, we establish tools for and carry out the investigation of nonlinear stability for these equilibria.
4.1. Preliminary Analysis and Definitions
Throughout this analysis, we consider an arbitrary and , unless otherwise stated. Focusing on the properties of solutions near , we shift our equations, including the shifted operator
which maps to , where we consider , and is in the regularity class by Lemma 2.1; here we take . Now we consider the surface diffusion problem shifted by ,
where . We say that
is a solution to (4.1), with initial data , on the interval if satisfies (4.1) pointwise, for , and . We investigate the properties of around 0 in order to gain information about the stability of in (1.3).
Define the functional
which corresponds to the volume enclosed by the surface . It follows from the analyticity of multiplication and integration on little-Hölder spaces that is of class from to . The Fréchet derivative of is
for where the last equality holds by periodicity. Thus, conservation of along the solution follows by continuity of and convergence of to the initial data in . From these properties, it follows that
is a family of invariant level sets for (4.1). The following techniques are motivated by results of Prokert  and Vondenhoff , whereby one can take advantage of invariant manifolds in order to derive stability results.
First, we introduce the mapping
which defines a projection on . We denote by the image which exactly coincides with the zero-mean functions on in the regularity class , and we have the topological decomposition
In what follows, we equate the constant function with the value and we denote each simply as .
Consider the operator
which maps to and is of class , by regularity of the mappings and . Notice that and, using (4.2),
i.e. the Fréchet derivative of with respect to the first variable, at the origin, is a linear isomorphism. Hence, it follows from the implicit function theorem that there exist neighborhoods and and a function such that, for all ,
We can immediately state the following properties of , which follow directly from its definition and elucidate the relationship between and .
for all .
Given , it follows that
, for . This and the preceding remark follow from the fact that is injective when restricted to .
It follows from the identity and differentiating with respect to that . From this observation, and the fact that , it follows that
for . Hence, can be taken as a (local) parametrization of Moreover, from the preceding remark and the bijectivity of from to , we can see that is a Banach manifold over anchored at the point
For , we have the representation
and so we can see that can be realized (locally) as the graph of a -valued analytic function over the zero-mean functions .
Although depends upon the parameter , a priori, it follows easily from the preceding representation that
so that preserves the spacial regularity of functions regardless of the regularity parameter with which was constructed. However, notice that the neighborhood will remain intrinsically linked with the parameter which was used to construct .
With the established invariance and local structure of the sets , it follows that the dynamics governing solutions to (1.3) reside in the tangent space to the manifold . Hence, if we reduce (1.3) to a local system on , then we will have captured all of the dynamics of the problem. Remarks 4.1(d) is the first observation toward this reduced formulation. In fact, one can make use of the properties established in Remarks 4.1 to prove the following, even more general, result regarding the properties of the the tangent vectors to