UR Scholarship Repository UR Scholarship Repository On quasilinear parabolic equations and continuous maximal On quasilinear parabolic equations and continuous maximal regularity regularity

. We consider a class of abstract quasilinear parabolic problems with lower–order terms exhibiting a prescribed singular structure. We prove well–posedness and Lipschitz continuity of associated semiﬂows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diﬀusion ﬂow in various settings.


Introduction
In this paper, we consider abstract quasilinear parabolic evolution equations given by u + A(u)u = F 1 (u) + F 2 (u), for t > 0, for which we extend previous well-posedness and global existence results in the setting of continuous maximal regularity. As a particular feature, we admit nonlinearities F 2 with a prescribed singular structure.
In the last decades, there has been an increasing interest in finding critical spaces for nonlinear parabolic partial differential equations. As a matter of fact, there is no generally accepted definition in the mathematical literature concerning the notion of critical spaces. One possible definition may be based on the idea of a 'largest space of initial data such that the given PDE is well-posed.' Critical spaces are often introduced as 'scaling invariant spaces,' provided the underlying PDE enjoys a scaling invariance. It has been shown in [17] that the concept of critical weight and critical space introduced there (and also used in this paper) captures and unifies the idea of largest space and scaling invariant space. In more detail, it has been shown in [17] that E µcrit is, in a generic sense, the largest space of initial data for which the given equation is well-posed, and that E µcrit is scaling invariant, provided the given equation admits a scaling.
Our approach for establishing well-posedness of (1.1) relies on the concept of continuous maximal regularity in time-weighted spaces and extends previous results by Angenent [2], Clement and Simonett [5], Lunardi [13], and Asai [3]. The results parallel those in [9,17,20], where well-posedness of (1.1) is studied by means of maximal L p -regularity in time-weighted function spaces.
Leveraging the singular structure of F 2 , along with inequalities from interpolation theory and continuous maximal regularity, we prove local well-posedness of (1.1) via a fixed point argument. Allowing for rough initial values in E µ with µ ≥ µ crit , we prove Lipschitz continuity of the associated semiflow on V µ and derive conditions for global well-posedness and asymptotic behavior of solutions near normally stable equilibria. A key feature of our results is that dynamic properties of solutions are controlled in the topology of E µ , rather than requiring further control in stronger topologies of E β or E 1 as solutions regularize.
In particular, we prove that a priori bounds in the topology of E µ yield global existence. Moreover, we extend the generalized principle of linearized stability (c.f. [18,19]), proving that solutions with initial data that is E µ -close to a normally stable equilibrium will converge exponentially fast to a nearby equilibrium.
As a particular application of our abstract results, we consider the surface diffusion flow, a geometric evolution equation acting on orientable hypersurfaces. Given a fixed reference manifold Σ ⊂ R n , we consider the evolution of surfaces Γ(t) defined in normal direction over Σ via time-dependent height functions h : Σ × R + → R. The governing equation for surface diffusion is then expressed as a fourth-order, parabolic evolution law acting on h = h(t) and we look for solutions in the setting of little-Hölder continuous functions; i.e. E 0 := bc α (Σ) and E 1 := bc 4+α (Σ), for some α ∈ (0, 1).
Considering the setting of Σ = C r ⊂ R 3 , an infinite cylinder of radius r > 0, we work with height functions h(t) : C r → (−r, ∞) which produce so-called axiallydefinable surfaces Γ(h), as in [12]. We show that the resulting surface diffusion flow can be cast as a quasilinear parabolic evolution equation in the form (1.1) with E µ := bc 1+α (Σ) and E β := bc 3+α (Σ), from whence we have µ = 1/4 and β = 3/4 in this setting. Explicitly expressing the singular nonlinearity F 2 , we employ interpolation theory estimates to confirm the necessary singular structure (H2) is satisfied on V µ ∩ E β , where V µ is an appropriately chosen open subset of E µ . The appearance of several critical indices supports the idea that bc 1+α (Σ) is in fact a critical space for surface diffusion flow.
Applying our general results to initial data h 0 ∈ bc 1+α (C r ) we extend wellposedness from [12,Proposition 3.2] to surfaces with only one Hölder continuous derivative. Further, we extend [11, Proposition 2.2, 2.3] by restricting to functions h 0 ∈ bc 1+α symm (C r ) exhibiting azimuthal symmetry around the cylinder C r . Further, enforcing periodicity of h 0 along the central axis of C r , we show stability and instability of cylinders under periodic perturbations with Hölder control on only first order derivatives. In particular, when r > 1, we show that 2π-periodic Γ(h 0 ) surfaces that are bc 1+α -close to C r give rise to global solutions to surface diffusion flow converging to a nearby cylinder exponentially fast. On the other hand, when r < 1, we show that there exist 2π-periodic perturbations which are arbitrarily close to C r in bc 1+α for which solutions escape a neighborhood of the cylinder. We also direct the reader to [4], for additional information concerning the surface diffusion flow for axisymmetric surfaces.
Taking Σ to be an arbitrary compact, connected, immersed manifold, we demonstrate well-posedness of surface diffusion for initial data in bc 1+α (Σ) that are sufficiently close to the manifold Σ, an extension of [6, Theorem 1.1]. Further, in case Σ ⊂ R n is a Euclidean sphere, we apply our generalized stability result to yield stability of the family of spheres under perturbations which require control on only first-order derivatives. This result extends [6,Theorem 1.2], where initial values in bc 2+α (Σ) are considered.
Working also in the setting of surfaces parameterized over a sphere, Escher and Mucha [7] show that small perturbations in the topology of Besov spaces B 5/2−4/p p,2 (Σ) exist globally and converge exponentially fast to a sphere. Although the topologies of these Besov spaces and our little-Hölder spaces are not easily comparable, we note that our stability results hold for any spacial dimension n, while the regularity of perturbations in [7] changes with n. In particular, Escher and Mucha enforce the bound p > 2n+6 3 , which they note only guarantees existence of lower regularity perturbations in C 1+α (Σ) \ C 2 (Σ) when n < 9. (Notice that the authors in [7] consider surfaces in R n+1 .) Regarding a different approach to stability of spheres, we refer to [14,23,24] where the lifespan of solutions, and convergence to equilibria, is controlled via L 2estimates of the second fundamental form. We observe that our assumptions on initial data allow for initial surfaces on which the second fundamental form may not be defined, so that our results are not contained in [14,23,24].
As a final remark on surface diffusion flow, we mention that several authors have considered the flow of surfaces with rough initial data when Γ ⊂ R n+1 is given as the graph of a function over a domain Ω ⊆ R n , see [3,8,9] for instance. In [8], Koch and Lamm prove global existence of solutions to surface diffusion flow with initial surfaces that are merely Lipschitz continuous, and they prove analytic dependence on initial data. However, Koch and Lamm work in the setting of entire graphs (i.e. Ω = R n ) and require a smallness condition on Lipschitz norm which seems to make it difficult to translate their result to more general settings. Working also in the setting of entire graphs, the conclusions of Asai in [3] are closest to our current results, as the author works in spaces of little-Hölder continuous functions. We refer to Remark 2.4 for a detailed account of the results in [3]. In [9], the authors approach surface diffusion flow from the setting of L p maximal regularity on a bounded domain Ω, producing well-posedness for initial data in Besov spaces B 4µ−4/p qp (Ω), for an appropriate choice of µ, p, and q. We briefly outline the current paper. In Section 2, we state and prove our main result, Theorem 2.2. We conclude Section 2 with an extension of well-posedness, giving continuous dependence on initial data in stronger topologies Eμ.
In Section 3, we prove equivalence between measuring stability of equilibria in the space E µ and measuring stability in a smaller space Eμ,μ ∈ [β, 1). We then prove the generalized principle of linearized stability for perturbations in E µ .
In Section 4 we apply our results to various settings for surface diffusion flow. Beginning with axially-definable surfaces parameterized over an infinite cylinder, we conclude well-posedness of surface diffusion flow for general perturbations in bc 1+α . Then we enforce periodicity in the general setting to establish stability / instability of cylinders under perturbations in bc 1+α per (with radius r above / below the threshold r = 1), before producing similar results in the setting of axisymmetric surfaces. We end Section 4 with the setting of surfaces defined over an arbitrary compact reference manifold Σ, and establish well-posedness and stability of spheres with initial data in bc 1+α .

2.
Well-Posedness of (1.1) In this section, we formulate and prove our main result concerning solvability of (1.1). Moreover, we formulate and prove conditions for global existence of solutions. We start with the definition and elementary properties of time-weighted continuous spaces (see [5,Section 2] for more details).
Let E be an arbitrary Banach space and define the spaces of time-weighted continuous functions where J := [0, T ],J := (0, T ], and µ ∈ (0, 1). We also set BC 0 (J, E) := C(J, E), and BC 1 0 (J, E) := C 1 (J, E). Given Banach spaces E 0 and E 1 so that E 1 is densely embedded in E 0 , we define , and respectively. Further, we note that the trace operator γ : E 1,µ (J) → E 0 is welldefined and, assuming M µ (E 1 , E 0 ) = ∅ (as we do throughout), the trace space γE 1,µ (J) coincides with the continuous interpolation space The important inequality (1.3) should be viewed in relation to applications of interpolation we will encounter frequently in the article. In particular, if we set . . , m. Then, given x, y ∈ E 1 and t > 0, it follows that Here the constant c j = c j (α, α j ) is the product of interpolation constants from E β and E βj (c.f. [1, Proposition 2.2.1]), while C 0 = C 0 (α, α 1 , . . . , α m ) is an upper bound for the family of all such constants, j = 1, . . . , m.
(b) In the proof of Theorem 2.2 below, we address both subcritical and critical indices j. The difference in approaches to these two cases can be viewed in context of (2.2). In particular, note that when j is subcritical, the exponent on t is strictly positive, since ρ j α + α j is exactly the left-hand side of (1.3). Meanwhile, when j is critical we have a trivial exponent on t, but it must hold that ρ j > 0 in this case. Thus, when j is critical we focus on the term |t 1−µ x| ρj α E1 which has a positive exponent (a property not necessarily holding in the subcritical case). (a) (Local Solutions) Given any x 0 ∈ V µ , there exist positive constants τ = τ (x 0 ), ε = ε(x 0 ), and σ = σ(x 0 ) such that (1.1) has a unique solution ) and enjoys the regularity (c) (Global Solutions) If the solution u(·, x 0 ) satisfies the conditions: then it holds that t + (x 0 ) = ∞ and so u(·, x 0 ) is a global solution of (1.1). Moreover, if the embedding E 1 → E 0 is compact, then condition (i) may be replaced by the assumption: and some τ ∈ (0, t + (u 0 )).
Before proceeding to the proof of the theorem we add some remarks. We first note that the embedding It thus follows that the map [(t, x) → u(t, x)] defines a locally Lipschitz continuous semiflow on V µ .
Remark 2.4. (a) We recall briefly that local Lipschitz continuity of a semiflow on V µ means that x)] is continuous on D, and for all (t 0 , x 0 ) ∈ D there exists a product neighborhood U × V ⊂ D and c > 0 so that (b) Local well-posedness of (1.1) was also considered by Asai [3] in the presence of a singular right-hand side F : V µ ∩ E 1 → E 0 . In particular, the author assumes that F satisfies Here the author has p and E θ := (E 0 , E 1 ) 0 θ,∞ appropriately chosen, with θ ∈ [µ, 1), so that p + (θ − µ)/(1 − µ) < 1. This setting is similar to our condition (H2) if one allows j = 1, β = 1, ρ j = p, and β j = θ, whereby it follows that Asai only considers subcritical weights. Further, we note that in [3, Theorem 1.1] the author proves Hölder continuous dependence on initial data in V µ , whereas we obtain Lipschitz continuity. No additional geometric properties for solutions are established in [3].
Proof of Theorem 2.2. (a) We follow the structure of related proofs in [9] and [20], where L p -maximal regularity is assumed. We note that sub-critical and critical indices required distinct proofs in [9] and [20], respectively, whereas both cases can be handled in the same setting here. Choose x 0 ∈ V µ and fix ε 0 > 0 so that Applying (H1) and (H2), we obtain constants L = L(ε 0 ) > 0 and C ε0 > 0 so that It follows from (H1) and [10, and The previous inequalities are justified by strong continuity of the semigroup e −tA(x0) in E µ , [5, Lemma 2.2(c)], and [5, Equation (3.7)], respectively. We will construct a contraction mapping on a closed subset of E 1,µ (J T ) given by We proceed by first proving that T x is well-defined (see Claims 1 and 2 below), then we show that T x is in fact a contraction mapping on W x (J T , r) for r, T and x appropriately chosen (see Claims 3 and 4 below).
Claim 1: For r, T, ε chosen sufficiently small and positive, if Applying (2.6), (2.8), and (2.9), we compute Henceforth, we assume x is sufficiently close to x 0 (in E µ ) and r, T, ε are given appropriately small so that Claim 1 holds. It follows that given any . Employing (H1) and the bounds (2.4) and (2.11), we compute From (2.12) and (2.13), we draw the following conclusions. (2.14) As an additional observation, note (2.12) and (2. Lastly, we consider the term F 2 (v), and we first observe that The assertion in (2.15) now follows from the embedding E β → E βj and the observation that |v(t)| E β and |v(s)| E β are bounded for values s, t that are bounded away from 0. The latter statement means that for each In order to show boundedness of , which is feasible by the density of the embedding E 1 → E µ , and write for each t ∈J T . Next, employing (2.2) and (2.16), we have With T, r and x chosen as above, we have now shown that the right hand side of equation (2.10) conclude the proof of the theorem, we must show that T x is a contraction mapping on W x (J T , r) for appropriately chosen r, T, and x.
x, by the property of maximal regularity and the definition of the mapping T x . Thus, it suffices to show that T provided r, T, and ε are chosen sufficiently small.
We begin with the observation using the fact that M 1 ≥ 1. Note that choosing ε sufficiently small, this term can be bounded by r/4 for all x ∈B Eµ (x 0 , ε).
we apply maximal regularity of A(x 0 ) to bound the first term of (2.17) where C T1 > 0 is the constant of maximal regularity for the interval [0, T 1 ]; recalling that T 1 > 0 was introduced before bounds (2.6)-(2.8). The first two terms of (2.18) are bounded as in (2.12) and (2.13), respectively -which are both bounded by r/4C T1 for r, T, and ε sufficiently small. Addressing the last term in (2.18), we first split can be made arbitrarily small by taking T sufficiently small. Meanwhile, we apply (2.5) and Remark 2.1 to bound where M 0 := C ε0 C 0 and M 2 , M 3 are constants chosen as follows. Applying part of (2.11) and Young's inequality, we select M 2 > 0 so that v−u x0 for all T > 0 sufficiently small so that u x0 E1,µ(J T ) ≤ r.
Finally, note that all terms in (2.19) have a linear factor of r and an additional factor that can be made arbitrarily small by restricting the sizes of r, T, and ε. In context of Remark 2.1(b), we note that terms involving subcritical index j are made small with T alone, while critical indices j require restriction on the size of r. We conclude that the last term in (2.18) can be bounded by r/4 and (2.17) can thus be bounded by r for all r, T, and ε chosen sufficiently small. This proves Claim 3.
Continuing with individual terms in (2.23), we apply (2.4) and (2.22) to get and, also applying (2.11), we have Meanwhile, by Young's inequality and (2.22), recalling that M 1 ≥ 1, we have which we apply in combination with (2.5), Remark 2.1, and (2.21) to bound with M 0 := C ε0 C 0 . Combining all terms involving v 1 − v 2 E1,µ(J T ) and likewise terms involving |x 1 − x 2 | Eµ , note that (2.23) takes on the desired structure for the claim. Moreover, every factor multiplying the terms v 1 − v 2 E1,µ(J T ) can be made arbitrarily small by taking either r, T, or ε sufficiently small. Note that the same cannot be said for every factor of |x 1 − x 2 | Eµ , as seen in the last term of (2.23). Regardless, we have thus proved Claim 4.
Finally, fix r, T, and ε small enough so that κ ≤ 1 2 . We thus have the estimate for every v i ∈ W xi (J T , r) and x i ∈B Eµ (x 0 , ε). Let x 1 = x 2 = x ∈ B Eµ (x 0 , ε) be given. Then and so T x is a strict contraction on W x (J T , r). Applying Banach's fixed point theorem, we obtain a unique fixed point which solves (1.1) by construction of the mapping T x . Furthermore, for which completes the proof of the first statement of the theorem.
(b) By a standard argument, we can extend the local solution obtained in part (a) to a maximal solution on some right-open interval [0, t + (x 0 )). To confirm this maximal solution satisfies the stated regularity, we consider a portion of this extension argument. In particular, with x 0 ∈ V µ given, we apply part (a) to produce the solution u 1 (·, x 0 ) ∈ E 1,µ ([0, τ 1 ]) on some interval [0, τ 1 ]. Then, we note that x 1 := u 1 (τ 1 , x 0 ) ∈ E 1 ∩ V µ , and so we may apply part (a) again to produce the solution u 2 (·, x 1 ) ∈ E 1,µ ([0, τ 2 ]) on a second interval [0, τ 2 ]. It follows that satisfies (1.1) with u(0) = x 0 and regularity u ∈ E 1,µ ([0, τ 1 + τ 2 ]). To prove this last claim, it suffices to show that u 2 ∈ C([0, τ 2 ], E 1 ), in particular lim t→0 + u 2 (t) = x 1 . For that purpose, we fix ε > 0 so that the result of part (a) holds for x ∈B Eµ (x 1 , ε) and choose δ ∈ (0, τ 2 ) sufficiently small that We conclude this section on well-posedness with the following extension of (2.3), accounting for the dependence of solutions on initial data residing in smaller spaces Eμ ⊂ E µ . This result will be useful in the following section as we consider long-term dynamics of solutions that start in E µ and instantaneously regularize to spaces Eμ.

Normal Stability
With well-posedness of (1.1) established, we investigate the long-term behavior of solutions that start near equilibria. In particular, in this section we demonstrate that the so-called generalized principle of linearized stability (c.f. [18,19]) continues to hold on E µ , provided the pertinent assumptions are satisfied. As a first step in this direction, we prove that stability of equilibria can be tracked in either the topology of E µ or, equivalently, in the stronger topology of Eμ.
Proposition 3.1. Suppose the assumptions of Theorem 2.2 hold,μ ∈ [β, 1), and suppose u * ∈ V µ ∩ E 1 is an equilibrium for (1.1). Then u * is stable in the topology of E µ ⇐⇒ u * is stable in the topology of Eμ.
Suppose that u * is stable in Eμ. Let ε > 0 be given and set εμ = ε/c µ . By the stability assumption, there exists a number δμ such that every solution of (1.1) with initial value x 0 ∈ B Eμ (u * , δμ) exists globally and satisfies Next, by continuous dependence on initial data, we can choose δ ∈ (0, η) sufficiently small such that for all x 0 ∈ B Eμ (u * , δ). In particular, after a short time, we are simply tracking the solutions u(·, x 0 ) in the stronger topology of Eμ. Hence, by (3.3) and (3.7), u(t, x 0 ) ∈ B Eµ (u * , ε) for any initial value x 0 ∈ B Eµ (u * , δ). This completes the proof of Proposition 3.1.
In addition to (H1)-(H2) we now assume that whereμ ∈ [β, 1) is a fixed number. Here we note that V µ ∩ Eμ ⊂ Eμ is open, and that differentiability is understood with respect to the topology of Eμ.
Let E ⊂ V µ ∩ E 1 denote the set of equilibrium solutions of (1.1), which means that u ∈ E if and only if u ∈ V µ ∩ E 1 and A(u)u = F 1 (u) + F 2 (u).
Given an element u * ∈ E, we assume that u * is contained in an m-dimensional manifold of equilibria. This means that there is an open subset U ⊂ R m , 0 ∈ U , and a C 1 -function Ψ : U → E 1 , such that • Ψ(U ) ⊂ E and Ψ(0) = u * , • the rank of Ψ (0) equals m, and (3.10) We assume furthermore that near u * there are no other equilibria than those given by Ψ(U ), i.e. E ∩ B E1 (u * , r 1 ) = Ψ(U ), for some r 1 > 0.
For u * ∈ E, we define where A , F 1 and F 2 denote the Fréchet derivatives of the respective functions. We denote by N (A 0 ) and R(A 0 ) the kernel and range, respectively, of the operator A 0 .
After these preparations we can state the following result on convergence of solutions starting near u * .
Theorem 3.2. Suppose u * ∈ V µ ∩ E 1 is an equilibrium of (1.1), and suppose that the functions (A, F 1 , F 2 ) satisfy (H1)-(H2) as well as (3.9). Finally, suppose that u * is normally stable, i.e., (i) near u * the set of equilibria E is a C 1 -manifold in E 1 of dimension m ∈ N, (ii) the tangent space for E at u * is given by N (A 0 ), (iii) 0 is a semi-simple eigenvalue of A 0 , i.e., N ( Then u * is stable in E µ . Moreover, there exists a constant δ = δ(μ) > 0 such that each solution u(·, x 0 ) of (1.1) with initial value x 0 ∈ B Eµ (u * , δ) exists globally and converges to some u ∞ ∈ E in Eμ at an exponential rate as t → ∞.
Remark 3.3. Theorem 3.2 yields convergence of u(·, x 0 ) in the stronger norm of Eμ for initial values in E µ . We note that this holds true for anyμ ∈ [β, 1), with δ depending onμ.

Applications to Surface Diffusion Flow
In this section, we apply the theory from the previous sections to extend results regarding the surface diffusion flow in various settings. First, we extend [12, Proposition 3.2] regarding well-posedness of the surface diffusion flow in the setting of so-called axially-definable surfaces. We then prove nonlinear stability of cylinders with radius r > 1 (as equilibria of surface diffusion flow) under a general class of periodic perturbations which only require control of first-order derivatives; this result extends [12,Theorem 4.3] and [11,Theorem 4.9] where control of second-order derivatives was also required. At the conclusion of the section, we establish general well-posedness for surface diffusion flow acting on surfaces parameterized over a compact reference manifold Σ ⊂ R n , and conclude normal stability of Euclidean spheres under bc 1+α perturbations.

4.1.
Axially-Definable Setting: Well-Posedness. We begin with a brief introduction to the axially-definable setting and formulation of the problem; for a more detailed account we direct the reader to [12,Sections 2 and 3].
First, given r > 0, let denote the unbounded cylinder in R 3 of radius r, where T := [0, 2π] denotes the one-dimensional torus, with 0 and 2π identified. Next, we fix a parameter α ∈ (0, 1) and define the Banach spaces E 0 := bc α (C r ) and where bc k+α , k ∈ N, denotes the family of k-times differentiable little-Hölder regular functions. In particular, on an open set U ∈ R n , bc α (U ) is defined as the closure of the bounded smooth functions BC ∞ (U ) in the topology of BC α (U ), the Banach space of all bounded Hölder-continuous functions of exponent α. Then bc k+α (U ) consists of functions having continuous and bounded derivatives of order k, whose k th -order derivatives are in bc α (U ). The space bc k+α (C r ) is defined via an atlas of local charts. Taking µ = 1/4 and β = 3/4, we define the continuous interpolation spaces E µ := (E 0 , E 1 ) 0 µ,∞ and E β := (E 0 , E 1 ) 0 β,∞ . It is well-known that the scale of little-Hölder spaces is closed under continuous interpolation (c.f. [13] and [22]) and so these spaces are likewise identified as and With the spaces E 0 , E µ , E β , E 1 thus set, note that condition (1.3) becomes ρ j 2 + β j ≤ 1, so that we have a critical index j exactly when ρ j /2 + β j = 1. Further, with ε > 0 fixed, we define the family of admissible initial values (which coincides with surfaces that remain bounded away from the central axis of rotation) We say that a surface Γ ⊂ R 3 is axially-definable if it can be parameterized as for some height function h : C r → R satisfying h > −r on C r , where ν denotes the outer unit normal field over C r . In the setting of axially-definable surfaces, the surface diffusion flow is expressed as the following evolution equation for timedependent height functions h = h(t, p) = h(t, x, θ): As shown in [12, Section 2.2], the evolution operator G takes the form We have now confirmed that the mappings A, F 1 and F 2 satisfy properties (H1) and (3.9). Regarding confirmation of the structural conditions (H2), we expand terms of (4.3) to confirm where the functions c η,τ , d η,τ,σ depend only upon h and ∂ 1 h, and are analytic by (4.5). Of particular importance in (4.6), we note that third-order derivatives of h appear linearly in terms with at most linear factors of ∂ 2 h, while lower-order terms include at most cubic factors of ∂ 2 h.
Further, it follows that the map [(t, h 0 ) → h(t, h 0 )] defines a semiflow on V µ which is analytic for t > 0 and Lipschitz continuous for t ≥ 0.  (b) Regarding analyticity of the semiflow [(t, h 0 ) → h(t, h 0 )] for t > 0: for any τ > 0 we note that Thus, analyticity holds for t > τ in V β by [5, Theorem 6.1] and (4.5), and then analyticity also holds in V µ by embedding.
(c) In the setting of surfaces expressed as graphs over R n , existence and uniqueness of solutions with initial data in bc 1+α (R n ) was established in [3,Theorem 4.2]. However, we note that the author requires initial values to be slightly more regular than those considered here, due to the fact that he tracks regularity of solutions in a different topology than that of the space where he takes initial data.

4.2.
Axially-Definable Setting: Stability of Cylinders. Considering the stability of cylinders as equilibria for (4.2), we first introduce the 2π-periodic little-Hölder spaces bc k+α per , k ∈ N, defined as the subspace of functions h ∈ bc k+α (C r ) exhibiting 2π-periodicity along the x-axis; i.e.
As shown in [12,, working in this setting allows access to Fourier series representations for height functions h and guarantees the linearized operator DG(h) has a discrete spectrum.
Regarding well-posedness in the periodic setting, it was shown in [12, Proposition 3.4] that G preserves periodicity, so Theorem 4.1(a) continues to hold verbatim with bc per replacing bc throughout. Meanwhile, we note that global solutions in the periodic setting differ slightly from Theorem 4.1(b) owing to the compactness of the embedding bc 4+α per (C r ) → bc α per (C r ) (c.f. Theorem 2.2(c)). Noting that h * ≡ 0 is always an equilibrium of (4.2) (which coincides with the observation that the cylinder C r is an equilibrium of surface diffusion flow), we consider the stability of h * under perturbations in . Further, we denote by M cyl the family of height functionsh such that Γ(h) defines a cylinder C(ȳ,z,r) -symmetric about axis (·,ȳ,z) in R 3 with radiusr > 0in a neighborhood of C r . With these preparations, we state the following stability result. (ii) |h(t, h 0 )| bc 1+δ ≤ M for all t ∈ [τ, t + (h 0 )), for some τ ∈J(h 0 ) and δ ∈ (α, 1), then t + (h 0 ) = ∞, so that h(·, h 0 ) is a global solution.
(b) (Stability) Fix r > 1 andμ ∈ (0, 1). There exists a positive constant δ > 0 such that, given any admissible periodic perturbation with |h 0 | bc 1+α (Cr) < δ, the solution h(·, h 0 ) exists globally in time and converges to someh ∈ M cyl at an exponential rate, in the topology of Eμ. (c) (Instability) For 0 < r < 1 the function h * ≡ 0 is unstable in the topology of bc 1+α per (C r ). Proof. (a) This result follows from Theorem 2.2(c), noting that bc 4+α per (C r ) → bc α per (C r ) is a compact embedding in this periodic setting. Conditions (i)-(ii) guarantee the solution remains bounded away from the boundary ∂V µ,per .
(b) First note that restricting the domains of (A, F 1 , F 2 ) to periodic little-Hölder spaces will maintain the conditions (H1)-(H2) and (4.5), all confirmed in Section 4.1. From the proof of [12,Theorem 4.3] we know that h * is normally stable when r > 1. The conclusion thus follows from Theorem 3.2.
(c) It follows from [12,Theorem 4.3(b)] that h * is unstable in the topology of bc 3+α per (C r ). Instability in bc 1+α per (C r ) then follows from Proposition 3.1.

Axisymmetric Setting.
We turn now to consider (4.2) acting on the scale of axisymmetric little-Hölder spaces bc k+α sym (C r ), k ∈ N, defined as the subspace of functions h ∈ bc k+α (C r ) exhibiting symmetry around the x-axis; i.e.
These functions naturally coincide with surfaces Γ(h) which are symmetric about the central x-axis, as considered in [11]; although we relax the setting slightly by not enforcing axial-periodicity for our well-posedness result. For all such functions with sufficient regularity, it follows that ∂ θ h ≡ 0 and the application of the evolution operator G to h ∈ bc 4+α sym (C r ) produces the simplified expression (4.10) In fact, a complete expansion of individual terms for the operator G is provided in [11,Equations (2

.1)-(2.3)] from which we deduce
where we can explicitly observe the structure of (4.6).
To apply the results of Section 4.1 to the axisymmetric setting, it suffices to note that the property of axisymmetry is preserved by (4.2). This claim is clear from a purely geometric perspective, since the evolution equation (4.2) is completely determined by the geometry of the surfaces Γ(h(t)), and axisymmetry of the surface imparts the same symmetry onto the geometric structure. However, one can also confirm preservation of axisymmetry analytically by confirming that G commutes with the azimuthal shift operators T φ , for φ ∈ T; defined by for (x, θ) ∈ C r .

Surfaces Near Compact
Hypersurfaces. We conclude the paper by looking at the flow of surfaces parameterized over a fixed reference manifold and extend results in [6]. In particular, let Σ denote a smooth, closed, compact, immersed, oriented hypersurface in R n , and let ν Σ be a unit normal vector field on Σ, compatible with the chosen orientation. It follows that there exists a constant a > 0 and an open atlas {U : ∈ L} for Σ so that is a smooth diffeomorphism onto the range R := im(X ), for ∈ L. We capture the evolution of surfaces that are C 1 -close to Σ via time-dependent height functions h : R + × Σ → (−a, a). In particular, to h(t) := h(t, ·) we associate the surface which is parametrized by the mapping As in the previous settings, we let α ∈ (0, 1) and work in spaces of little-Hölder continuous functions with µ = 1/4 and β = 3/4. Further, we define To express the equations for surface diffusion flow of Γ(h(t)) as an evolution equation acting on the height functions h, we direct the reader to [6,Section 2] and [21,Section 5] where details are given for pulling back the governing equation V Γ = ∆ Γ H Γ , defined on Γ(h(t)), to an equivalent equation on the reference manifold Σ. We thus arrive at an expression h t (t, p) = [G(h(t))](p) for t > 0, p ∈ Σ, h(0) = h 0 on Σ, (4.13) where the evolution operator G takes the form (c.f. [21,Section 5]) (4.14) Utilizing expressions given in [15, or [16, Section 2.2], we expand (4.14) and confirm properties (H1), (H2) and (3.9). For the structure of the Laplace-Beltrami operator in local coordinates, we have (employing the standard summation convention over repeated instances of i, j, k taking values from 1 to (n − 1) -the dimension of the manifold) where ϕ is a scalar function on Γ(h) and ϕ * := Ψ * h ϕ its pull-back to Σ through the parameterization Ψ h . The coefficient functions a ij and b k are expressed as where τ Σ i p denote elements of a basis for the tangent space T p Σ to a point p ∈ Σ, while τ i Σ p make up a corresponding basis for the dual of T p Σ, and · · is the inner product in Euclidean space R n . Further, M 0 (h) := (I − hL Σ ) −1 depends upon h and the Weingarten tensor L Σ on Σ (i.e. no derivatives of h appear in M 0 (h)) and projects onto the tangent space Γ(h). Here the normal vector to Γ(h) is and hence P Γ(h) only depends upon first-order derivatives of h. Therefore, secondorder derivatives of h only appear in b k when the derivative ∂ i acts on P Γ(h) in the first term of the inner product. With no further factors of ∂ 2 h appearing in the expression, it follows that ∂ 2 h only appears linearly in the functions b k (h, ∂ 1 h, ∂ 2 h).
By [16,Section 2.2] or [15,Section 3.5], the mean curvature function has the following structure in local coordinates, (4.16) Thus, applying (4.15) to (4.16), one confirms that G exhibits the quasilinear structure Considering condition (H1), note that, in every local chart U , the principal symbol σ[A(h)] coincides with the expression given for the principal symbolσ[P (h)] in [21,Section 5]. Thus, we have A(h) ∈ M µ (E 1 , E 0 ) for all h ∈ V µ , and we likewise conclude Considering condition (H2), by the argument in Section 4.1, it suffices to show that F 2 exhibits the same structure as (4.6). Thus, applying (4.15) to (4.16) (noting that (4.16) is the pulled back expression of mean curvature), we first consider the four scenarios where third-order derivatives arise in G(h), namely: In all such scenarios, when ∂ 3 h is produced it appears linearly and it multiplies factors of ∂ 2 that appear at most linearly. Next, considering all cases within which second-order derivatives arise in G(h) -without accompanying factors of ∂ 3 h -one likewise confirms that at most cubic factors of ∂ 2 h appear. Therefore, we conclude that conditions (H1), (H2) and (3.9) all hold for (4.13), and we thus produce the following extension of [6, Theorem 2.2] by application of Theorem 2.2. Note that the embeddings bc k+α (Σ) → bc α (Σ) are compact here, since the domain Σ is itself compact. Theorem 4.6 (Well-Posedness). Fix α ∈ (0, 1) and a > 0. Let Σ be a smooth, closed, compact, immersed, oriented hypersurface in R n on which there exists an open atlas {U : ∈ L} where X : (p, r) → p + rν Σ (p) : Σ × (−a, a) → R n is a smooth diffeomorphism onto R := im(X ), for ∈ L.
Further, the map [(t, h 0 ) → h(t, h 0 )] defines a semiflow on V µ which is analytic for t > 0 and Lipschitz continuous for t ≥ 0.
Remark 4.7. (a) We note that the global existence result in Theorem 4.6(b) is limited, as it fails to account for the possibility of updating the reference manifold Σ as Γ(h(t)) is leaving the tubular neighborhood, but this result is sufficient for considerations of stability/instability when Σ is an equilibrium.
(b) Further regularity of the surfaces Γ(h(t)) have been shown in certain settings. In particular, when Σ is additionally assumed to be a smooth embedded surface, it follows from [21,Theorem 5.2], and instantaneous regularization of solutions, that Γ(h(t)) is also smooth for all t ∈J(h 0 ). Likewise, if Σ is real analytic and embedded, then Γ(h(t)) is also real analytic. 4.5. Stability of Euclidean Spheres. In the particular case that Σ is a Euclidean sphere, the function h * ≡ 0 is normally stable in E 1 by [6, Section 3] and we thus conclude the following extension of [6,Theorem 1.2]. Note that our result shows stability of spheres under surface diffusion flow with control on only first derivatives of perturbations.
Theorem 4.8. Fix α ∈ (0, 1),μ ∈ (0, 1). Let Σ be a Euclidean sphere in R n and choose a > 0 so that the mapping [(p, r) → p + rν Σ (p)] is a diffeomorphism on Σ × (−a, a). There exists a constant δ > 0 such that, given any admissible perturbation Γ(h 0 ) for h 0 ∈ V µ with |h 0 | bc 1+α (Σ) < δ, the solution h(·, h 0 ) exists globally in time and converges to someh ∈ M sph at an exponential rate, in the topology of Eμ. Here M sph denotes the family of all spheres which are sufficiently close to Σ in R n .
An immediate consequence of Theorem 4.8 is a relaxation of convexity constraints for stable perturbations of a sphere. In particular, note that every bc 1+αneighborhood of a sphere contains non-convex hypersurfaces. This corollary provides a different approach to the same result in [7], where the authors prove the claim by showing that non-convex perturbations of spheres exist in B 5/2−4/p p,2 (Σ). Corollary 4.9. There exist non-convex hypersurfaces Γ 0 such that the solution h(·, h 0 ) to (4.13), with Γ(h 0 ) = Γ 0 , exists globally in time and converges exponentially fast to a sphere.