SD Flow of Perturbed Cylinders

# On the Flow of Non–Axisymmetric Perturbations of Cylinders via Surface Diffusion

## Abstract.

We study the surface diffusion flow acting on a class of general (non–axisymmetric) perturbations of cylinders in . Using tools from parabolic theory on uniformly regular manifolds, and maximal regularity, we establish existence and uniqueness of solutions to surface diffusion flow starting from (spatially–unbounded) surfaces defined over via scalar height functions which are uniformly bounded away from the central cylindrical axis. Additionally, we show that is normally stable with respect to –axially–periodic perturbations if the radius ,and unstable if . Stability is also shown to hold in settings with axial Neumann boundary conditions.

###### Key words and phrases:
Surface diffusion, well posedness, unbounded surfaces, maximal regularity, nonlinear stability, implicit function theorem
###### 2000 Mathematics Subject Classification:
Primary 35K93, 53C44 ; Secondary 35B35, 35B32, 46T20

## 1. Introduction

The surface diffusion flow is a geometric evolution law in which the normal velocity of a moving surface equals the Laplace–Beltrami operator of the mean curvature. More precisely, we assume in the following that is a closed embedded surface in . Then the surface diffusion flow is governed by the law

 (1.1) V(t)=ΔΓ(t)HΓ(t),Γ(0)=Γ0.

Here is a family of hypersurfaces, denotes the velocity in the normal direction of at time , while and stand for the Laplace–Beltrami operator and the mean curvature of (the sum of the principal curvatures in our case), respectively. Both the normal velocity and the mean curvature depend on the local choice of the orientation. Here we consider the case where is embedded and encloses a region , and we then choose the outward orientation, so that is positive if grows and is positive if is convex with respect to .

The unknown quantity in (1.1) is the position and the geometry of the surface , which evolves in time. Hidden in the formulation of the evolution law (1.1) is a nonlinear partial differential equation of fourth order. This will become more apparent below, where the equation is stated more explicitly.

It is an interesting and significant fact that the surface diffusion flow evolves surfaces in such a way that the volume enclosed by is preserved, while the surface area decreases (provided, of course, these quantities are finite). This follows from the well-known relationships

 (1.2) ddtVol(t)=∫Γ(t)V(t)dσ=∫Γ(t)ΔΓ(t)HΓ(t)dσ=0

and

 (1.3) ddtA(t)=∫Γ(t)V(t)HΓ(t)dσ =∫Γ(t)(ΔΓ(t)HΓ(t))HΓ(t)dσ Missing or unrecognized delimiter for \right

where denotes the volume of and the surface area of , respectively. Hence, the preferred ultimate states for the surface diffusion flow in the absence of geometric constraints are spheres. 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 [20]. This particular structure has been exploited in [29, 30] for devising numerical simulations.

The mathematical equations modeling surface diffusion go back to a paper by Mullins [32] from the 1950s, who was in turn motivated by earlier work of Herring [24]. Since then, the surface diffusion flow (1.1) has received wide attention in the mathematical community (and also by scientists in other fields), see for instance Asai [6], Baras, Duchon, and Robert [8], Bernoff, Bertozzi, and Witelski [9], Cahn and Taylor [11], Cahn, Elliott, and Novick–Cohen [10], Daví and Gurtin [14], Elliott and Garcke [17], Escher, Simonett, and Mayer [18], Escher and Mucha [19], Koch and Lamm [25], LeCrone and Simonett [27], McCoy, Wheeler, and Williams [31], and Wheeler [38, 39].

 Cr:={(x,y,z)∈IR3:y2+z2=r2,x∈IR}

The main results of this paper address

• existence, uniqueness, and regularity of solutions for (1.1) for initial surfaces that are (sufficiently smooth) perturbations of ;

• existence, uniqueness, and regularity of solutions for (1.1) for initial surfaces that are (sufficiently smooth) perturbations of subject to periodic or Neumann–type boundary conditions in axial direction;

• nonlinear stability and instability results for with respect to perturbations subject to periodic or Neumann–type boundary conditions in axial direction.

In section 2 we derive an explicit formulation of (1.1) for surfaces which are parameterized over in normal direction by height functions , see (2.5). Equation (2.5) is a fourth–order, quasilinear, parabolic partial differential equation for which well–posedness in the unbounded setting can be established by virtue of recent results for maximal regularity on uniformly regular Riemannian manifolds (c.f. [36, 2]). A proof of well–posedness is carried out in section 3 of the current article. For other recent results regarding geometric evolution equations in unbounded settings, refer to Giga, Seki, Umeda [21, 22], wherein the mean curvature flow is shown to close open ends of non–compact surfaces of revolution given appropriate decay rates at the ends of the initial surface. The current setting differs from that of Giga et. al. in many ways; most notably, we consider surfaces which lack symmetry about the cylindrical axis; we control behavior at space infinity via regularity bounds (so surfaces do not become radially unbounded); and we consider the dynamics of surfaces which are uniformly bounded away from the cylindrical axis.

In [27], we studied the surface diffusion flow for the special case of axisymmetric and –periodic perturbations of . In this case it was shown that equilibria of (1.1) consist exactly of cylinders, –periodic unduloids and nodoids. In addition, we established nonlinear stability and instability results for cylinders as well as bifurcation results, where serves as a bifurcation parameter. More specifically, it was shown that cylinders are stable for , unstable for , and that a subcritical bifurcation of equilibria occurs at , with the bifurcating branch consisting exactly of the family of –periodic unduloids.

In this paper we show that the stability and instability results of [27] remain true for non–axisymmetric perturbations subject to periodic or Neumann–type boundary conditions in axial direction. In particular, we show that small –periodic perturbations of a cylinder with exist globally and converge exponentially fast to another nearby cylinder

 C(¯y,¯z,¯r):={(x,y,z)∈IR3:(y−¯y)2+(z−¯z)2=¯r2,x∈IR}.

It will be shown that the radius is uniquely determined by the volume of the solid enclosed by and bounded by the planes and . In order to prove this result, we shall show that all equilibria of (1.1) which are –periodic in axial direction and sufficiently close to (for ) are cylinders , with close to . This result, which is interesting by itself, is obtained by a center–manifold argument. In order to prove the stability result, we demonstrate that every cylinder with radius is normally stable with respect to –periodic, –regular perturbations. As discussed in section 5, these stability results also encompass perturbations satisfying Neumann–type boundary conditions in -direction.

## 2. Surface Diffusion Flow near Cylinders

Throughout this section, take fixed, and let be the (unbounded) cylinder of radius which is symmetric about the x–axis. Specifically,

 Cr:={(x,rcos(θ),rsin(θ)):x∈IR,θ∈T},

where denotes the one–dimensional torus with and identified and is equipped with the periodic topology generated by the metric

 dT(x,y):=min{|x−y|,2π−|x−y|},x,y∈T.

Further, we equip with the Riemannian metric , inherited from via embedding.

###### Remarks 2.1.

Two properties of are easy to see immediately:
(a) is a uniformly regular Riemannian manifold, as defined by Amann [4].
Indeed, this follows from [4, formula (3.3), Corollary 4.3, and Theorem 3.1], noting that is realized as the Cartesian product of the circle and . On a side line we note that the class of uniformly regular (closed) Riemannian manifolds coincides with the class of closed, complete manifolds of bounded geometry, see [15].
(b) is an equilibrium of (1.1) for every radius .
Indeed, noting that the mean curvature is constant throughout the manifold, it follows that

 ΔCrHCr≡0,

so that the normal velocity and the surface remains fixed in space.

Now, consider a scalar-valued function such that

 ρ(p)>−r,for allp∈Cr,

and define a new surface over the reference manifold as

 Γ:={p+ρ(p)ν(p):p∈Cr},

where denotes the unit outer normal field over . We also note that the points are uniquely determined via the surface coordinates , and so we also view the height function as a function of the variables and . Thus, abusing notation, we will interchangeably refer to and , where, in fact,

 ρ(x,θ):=ρ((x,rcos(θ),rsin(θ))),(x,θ)∈IR×T.

Extending this slight abuse of notation, one easily identifies an explicit parametrization for via

 (2.1) φ(x,θ):=(x,rcos(θ),rsin(θ))+ρ(x,θ)(0,cos(θ),sin(θ)),(x,θ)∈IR×T.

With the above construction, we have a correspondence between height functions and a particular class of embedded manifolds which we refer to as axially–definable — namely, those manifolds which admit an axial parametrization of the form (2.1). Within this class of manifolds, the geometry, time–dependent evolution, and regularity are determined by the height function itself. We turn to the task of explicitly expressing these relationships, and deriving the governing equation for surface diffusion, in the following subsection. For the purpose of expressing the necessary geometric quantities, we assume for the moment that is smooth enough for all following calculations to work. The precise desired regularity of will be addressed in detail when we discuss existence and uniqueness of solutions.

### 2.1. Geometry of Γ(ρ)

Utilizing explicit parametrizations of the form (2.1), we find the coefficients of the first fundamental form of as

 g11=(1+ρ2x),g12=g21=ρxρθ,g22=((r+ρ)2+ρ2θ),

and the metric is thus given by . Then,

 g11=(r+ρ)2+ρ2θG,g12=g21=−ρxρθG,g22=1+ρ2xG,

where the matrix is the inverse of and

 (2.2) G=G(ρ):=det(gij)=(r+ρ)2(1+ρ2x)+ρ2θ.

Likewise, the second fundamental form is expressed via the coefficients

 I11 =(r+ρ)ρxx√G, I12 =I21=(r+ρ)ρxθ−ρxρθ√G, I22 =(r+ρ)(ρθθ−(r+ρ))−2ρ2θ√G.

The mean curvature and Laplace–Beltrami operator are expressed using well–known formulas, c.f. [16, 34]. From [34, Equation (5)],

 (2.3) H(ρ)=ρ2θ−(r+ρ)[((r+ρ)2+ρ2θ)ρxx+(1+ρ2x)ρθθ−2ρxρθρxθ]G3/2 + 1G1/2 ,

and, from [34, Section 2.7],

 ΔΓ(ρ)=1√G{∂x[(r+ρ)2+ρ2θ√G ∂x−ρxρθ√G ∂θ]+∂θ[1+ρ2x√G ∂θ−ρxρθ√G ∂x]}.

The normal velocity of the surface is likewise

 VΓ(ρ)=(r+ρ)ρt√G(ρ).

Throughout, we often streamline notation by omitting explicit reference to dependence upon the spatial or temporal variables; e.g. and , etc.

### 2.2. The Surface Diffusion Flow

With the formulas from the last subsection, we can now express the governing equation for surface diffusion of as an evolution equation for the height function alone. Thus, defining the (formal) operator

 (2.4) Extra open brace or missing close brace

we arrive at the expression

 (2.5) {ρt(t,p)=[G(ρ(t))](p),for t>0,p∈Cr,ρ(0)=ρ0,on Cr,

for the surface diffusion flow for axially–definable surfaces .

###### Remark 2.2.

The notation reflects the functional–analytic framework we use to address equation (2.5). For each , we consider as an element of an appropriate Banach space of regular functions defined on . Then, maps to another function defined over which we then evaluate pointwise using the notation .

## 3. Existence and Uniqueness of Solutions

In this section we establish existence and uniqueness of solutions for (2.5). One essential tool that we use throughout is the property of maximal regularity, also called optimal regularity. Maximal regularity has received considerable attention in connection with nonlinear parabolic partial differential equations, c.f. [1, 5, 12, 28, 33].

### 3.1. Maximal Regularity

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 [1, 28] for a general development of the theory.

Let , for some , and let be a (real or complex) Banach space. Following the notation of [12], we define spaces of continuous functions on with prescribed singularity at 0 as

 (3.1) BUC1−μ(J,E):={u∈C(˙J,E):[t↦t1−μu(t)]∈BUC(˙J,E)and limt→0+t1−μ∥u(t)∥E=0},μ∈(0,1) ∥u∥B1−μ:=supt∈Jt1−μ∥u(t)∥E,

where denotes the space consisting of bounded, uniformly continuous functions. We also define the subspace

 BUC11−μ(J,E):={u∈C1(˙J,E):u,˙u∈BUC1−μ(J,E)},μ∈(0,1)

and we set

 BUC0(J,E):=BUC(J,E),BUC10(J,E):=BUC1(J,E).

If for , then we set

 C1−μ(J,E) :={u∈C(˙J,E):u∈BUC1−μ([0,T],E),T

which we equip with the natural Fréchet topologies induced by and , respectively.

If and are a pair of Banach spaces such that is continuously embedded in , denoted , we set

 (3.2) E0(J):=BUC1−μ(J,E0),μ∈(0,1],E1(J):=BUC11−μ(J,E0)∩BUC1−μ(J,E1),

where is a Banach space with the norm

 ∥u∥E1(J):=supt∈˙Jt1−μ(∥˙u(t)∥E0+∥u(t)∥E1).

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

 ∥x∥γE1:=inf{∥v∥E1(J):v∈E1(J)andγv=x}.

Given , closed as an operator on , we say is a pair of maximal regularity for and write , if

 (ddt+B,γ)∈Lisom(E1(J),E0(J)×γE1),μ∈(0,1),

where denotes the set of bounded linear isomorphisms. In particular, if and only if for every , there exists a unique solution to the inhomogeneous Cauchy problem

 {˙u(t)+Bu(t)=f(t),t∈˙J,u(0)=u0.

Moreover, in the current setting, it follows that , i.e. the trace space is topologically equivalent to the noted continuous interpolation spaces of Da Prato and Grisvard, c.f. [1, 12, 13, 28].

We shall establish well–posedness of (2.5) in the setting of little–Hölder regular height functions . These results seem to constitute the first existence and uniqueness results for the surface diffusion flow acting on unbounded (closed) surfaces. Note that we enforce minimal conditions on the ends of the initial surface, namely we look at surfaces which are uniformly bounded away from the axis of definition (i.e. the x–axis in our current setting) and satisfy minimal regularity assumptions.

For the convenience of the reader, we include a brief definition of the little–Hölder spaces on . For and open, we define to be the closure of the bounded smooth functions in the topology of the bounded Hölder functions . Further, for , we define to be the space of –times continuously differentiable functions such that the –order derivatives are in . We then define the space of little–Hölder regular functions on , , via an atlas of local charts and a subordinate localization system. For more details regarding function spaces on uniformly regular (and singular) Riemannian manifolds see [2, 36].

### 3.2. Quasilinear Structure and Maximal Regularity

Expanding terms of (2.4), it is straight–forward to see that is a fourth–order quasilinear operator of the form

 G(ρ)=−A(ρ)ρ+F(ρ):=−⎛⎝∑|β|=3,4bβ(ρ,∂1ρ,∂2ρ) ∂βρ⎞⎠+F(ρ,∂1ρ,∂2ρ),

where is a multi–index, its length, and , for , denotes the collection of all derivatives for . This quasilinear structure plays an important role in our well–posedness results below, for which we must look more closely at the fine properties of the variable coefficient linear operator .

Expanding terms, we have the following highest–order variable coefficients for the linear operator :

 b(4,0)(ρ) = ((r+ρ)2+ρ2θ)2G2, b(3,1)(ρ) =− 4ρxρθ((r+ρ)2+ρ2θ)G2, b(2,2)(ρ) = 2((r+ρ)2+ρ2θ)(1+ρ2x)+4ρ2xρ2θG2, b(1,3)(ρ) =− 4ρxρθ(1+ρ2x)G2, b(0,4)(ρ) = (1+ρ2x)2G2.

Therefore, the principal symbol of the linear operator satisfies

 σ[A(ρ)](p,ξ) :=∑j+k=4[b(j,k)(ρ)](p)(iξ1)j(iξ2)k =∑j+k=4[b(j,k)(ρ)](p)ξj1ξk2 (3.3) =1G2(((r+ρ)2+ρ2θ)ξ21+(1+ρ2x)ξ22−2ρxρθξ1ξ2)2 ≥1G2((r+ρ)2ξ21+ξ22)2.

With these quantities explicitly expressed, we will show that satisfies the property of continuous maximal regularity on . Well–posedness of (2.5) then follows by exploiting the quasilinear structure of the parabolic equation.

###### Proposition 3.1 (maximal regularity).

Fix , and take so that . Further, let be the set of admissible height functions. Then it follows that

 (A,F)∈Cω(Vμ,Mν(bc4+α(Cr),bcα(Cr))×bcα(Cr)),

where denotes the space of real analytic mappings between (open subsets of) Banach spaces.

###### Proof.

The regularity of and is a consequence of the analyticity of the following mappings: (which is easy to show using standard methods in nonlinear analysis and theory of function spaces)

 [ρ↦1/ρ] :V0→bcα(Cr), :bcσ+|β|(Cr)→bcσ(Cr), :bcα(Cr)×bcα(Cr)→bcα(Cr),

where . Additionally, one notes that the regularity of is determined by the regularity of the coefficients . Then, one is left only to show that is in the denoted maximal regularity class for . Note that is an open subset of (by [12, Lemma 2.5(a)], for instance); thus the regularity of mapping into is determined by the regularity of mapping into .

To show , by [36, Theorem 3.6], it suffices to show that is uniformly strongly elliptic on ; i.e. we need to show that there exist constants such that

 c1≤Re(σ[A(ρ)](p,ξ))≤c2,for (p,ξ)∈Cr×IR2,with |ξ|=1.

However, these bounds are obvious from the expression (3.2) and the assumption that admissible functions are uniformly bounded from below and above on . ∎

### 3.3. Well–Posedness of (2.5)

We now prove existence and uniqueness of solutions to (2.5). Due to the non–compact setting, we must take care of the behavior of the height function as the axial variable approaches . For this purpose, we concentrate on surfaces which remain uniformly bounded away from the x–axis (i.e. height function for and ). Also note that functions in the spaces are bounded from above by definition.

###### Proposition 3.2 (existence and uniqueness).

Fix and take so that . For each initial value

 ρ0∈Vμ:=bc4μ+α(Cr)∩[ρ>ε−r],

there exists a unique maximal solution to (2.5)

 ρ(⋅,ρ0)∈C11−μ(J(ρ0),bcα(Cr))∩C1−μ(J(ρ0),bc4+α(Cr)),

where denotes the maximal interval of existence for initial data . Further, it follows that

 D:=⋃ρ0∈VμJ(ρ0)×{ρ0}

is open in and the map defines an analytic semiflow on . Moreover, if the solution satisfies:

• , and

• there exists so that for all .

Then it must hold that and so is a global solution of .

###### Remark 3.3.

Condition (ii) of the proposition can also be interpreted by explicitly identifying the boundary of the admissible set . In particular, remains bounded away from if and only if there exists some such that, for all

• for all , and

• .

###### Proof.

Existence of maximal solutions is proved in the same way as Proposition 2.2 of [27], whereas the claim for global solutions differs slightly from Proposition 2.3 of [27] due to the fact that we do not have compact embedding of little–Hölder spaces over the non–compact manifold . In particular, well–posedness follows from [12, Theorems 3.1 and 4.1(c)], [36, Proposition 2.2(c)], and Proposition 3.1, while the semiflow properties follow from [12] when and from [5] in case . ∎

### 3.4. Well–Posedness: Axially Periodic Surfaces

In order to address the stability of cylinders under the flow of (2.5) — for which we will use explicit information regarding the spectrum of the linearization of  — we restrict our setting to one within which spectral calculations are tractable.

For fixed, define the axial shift operator

 Ta:=[x↦x+a]:IR→IR,

which naturally acts on functions as

 [Taρ](x,θ)=ρ(x+a,θ).

We define

 bcσa,per(Cr):={ρ∈bcσ(Cr):Taρ=ρ},

which we refer to as axially –periodic little–Hölder functions on . It is a straight–forward exercise to see that forms a closed subspace of , and is hence a Banach space. We thus consider the properties of the surface diffusion evolution operator as it acts on the axially periodic spaces.

If , then .

###### Proof.

It suffices to show that commutes with the nonlinear operator , since –periodic would then imply , as desired. The fact that indeed commutes with follows directly from the commutativity of with the operations:

 [ρ↦1/ρ],[ρ↦∂βρ],and[(ρ,h)↦ρh],

where, as before, . ∎

###### Remark 3.5.

The fact that preserves periodicity can also be seen directly from the geometric setting. In particular, recall that was constructed by modeling the evolution of the surface via (1.1), which depends only upon the geometry of the surface itself. Thus, if is periodic, then , and all relevant geometric quantities on must also be periodic, and hence the evolution cannot break this periodicity.

The well–posedness results for (2.5) simplify slightly when restricted to periodic surfaces, owing to the fact that the evolution is determined by the restriction to one interval of periodicity. In essence, periodic surfaces are expressed entirely in a compact setting.

###### Proposition 3.6 (periodic well–posedness).

Fix and take so that . For each initial value

 ρ0∈Vμ,per:=bc4μ+αa,per(Cr)∩[ρ>−r],

there exists a unique maximal –periodic solution to (2.5)

 ρ(⋅,ρ0)∈C11−μ(J(ρ0),bcαa,per(Cr))∩C1−μ(J(ρ0),bc4+αa,per(Cr)).

It follows that

 D:=⋃ρ0∈Vμ,perJ(ρ0)×{ρ0}

is open in and the map defines an analytic semiflow on . Moreover, if there exists so that, for all ,

1. ,

then it must hold that and is a global solution.

###### Proof.

By the assumption and periodicity, it is clear that is uniformly bounded away from the -axis. According to Theorem 3.2, the surface diffusion flow (2.5) admits a unique solution on a maximal interval of existence . It, thus, only remains to show that is -periodic for each . Let for . Then

 ∂tρa(t)=Ta∂tρ(t)=TaG(ρ(t))=G(Taρ(t)),t∈J(ρ0), ρa(0)=Taρ(0)=Taρ0=ρ0,

showing that and both are solutions of (2.5) with the same initial value. By uniqueness, , implying that is -periodic for all .

The global well–posedness result follows from compactness of the embedding and [12, Theorem 4.1(d)]. ∎

For we define the area of and the volume of the solid enclosed by , respectively, in a natural way by just considering the part of on an interval of periodicity in the -direction. More precisely, we set

 (3.4) Cr,a={(x,y,z)∈IR3:y2+z2=r2,x∈[0,a]},Γ(ρ,a)={p+ρ(p)ν(p):p∈Cr,a}.

Then we have the following result.

###### Proposition 3.7.

Fix and take so that . For each initial value

 ρ0∈Vμ,per:=bc4μ+αa,per(Cr)∩[ρ>−r],

let be the solution of (2.5). Then the surface diffusion flow preserves the volume of the region enclosed by and bounded by the planes and . Moreover, the flow reduces the surface area of .

###### Proof.

Although the assertions follow from (1.2)-(1.3) it will be instructive to give an independent proof that only relies on basic computations. A short moment of reflection shows that volume is given by

 (3.5) Vol(ρ(t))=12∫[0,a]×[0,2π](r+ρ(t))2dθdx.

Using the short form , and keeping in mind that is periodic in , one immediately obtains from (2.4)

 ddtVol(ρ(t))=∫(r+ρ)ρtdθdx=0.

Next we note that the surface area of is given by

 A(ρ(t))=∫[0,a]×[0,2π]√G(ρ(t))dθdx.

It follows from integration by parts (using periodicity) and formulas (2.2)–(2.4)

 ddtA(ρ(t)) =∫[(r+ρ)(1+ρ2x)√G−∂x((r+ρ)2ρx√G)−∂θ(ρθ√G)]ρtdθdx =∫(r+ρ)Hρtdθdx =