Continuous maximal regularity on singular manifolds and its applications

In this article, we set up the continuous maximal regularity theory for a class of linear differential operators on manifolds with singularities. These operators exhibit degenerate or singular behaviors while approaching the singular ends. Particular examples of such operators include differential operators defined on domains, which degenerate fast enough toward the boundary. Applications of the theory established herein are shown to the Yamabe flow, the porous medium equation, the parabolic $p$-Laplacian equation and the thin film equation. Some comments about the boundary blow-up problem, and waiting time phenomena for singular or degenerate parabolic equations can also be found in this paper.


Introduction
The main objective of this article is to establish the continuous maximal regularity for a family of degenerate or singular elliptic operators on a class of manifolds with singularities, called singular manifolds. These results generalize the work in the previous paper [36]. The notation of singular manifolds used in this paper was first introduced by H. Amann in [3]. Roughly speaking, a manifold (M, g) is singular iff it is conformal to one whose local patches are of comparable sizes, and all transition maps and curvatures have uniformly bounded derivatives, i.e., (M, g/ρ 2 ) has the aforementioned properties for some ρ ∈ C ∞ (M, (0, ∞)). In [?], it is shown that the class of all such (M, g/ρ 2 ), called uniformly regular Riemannian manifolds, coincides with the family of complete manifolds with bounded geometry if we restrict ourselves to manifolds without boundary.
In [5], the author built up the L p -maximal regularity for a family of second-order elliptic operators satisfying a certain ellipticity condition, called uniformly strongly ρ-elliptic. By this, the author means that the principal part −div(C( a, gradu)) of a differential operator fulfils (C( a, ξ)|ξ) g * ∼ ρ 2 |ξ| 2 g * , for any cotangent field ξ. Here a is a symmetric (1, 1)-tensor field on (M, g), and C(·, ·) denotes complete contraction. See Section 3 for the precise definition. If two real-valued functions f and g are equivalent in the sense that f /c ≤ g ≤ cf for some c ≥ 1, then we write f ∼ g. These operators, as we can immediately observe from the above relationship, can exhibit degenerate or singular behaviors while approaching the singular ends. In [5], H. Amann also looked at manifolds with boundary. We generalize this concept of uniformly strong ρ-ellipticity to elliptic operators of arbitrary even order acting on tensor bundles. A linear operator A := 2l r=0 C(a r , ∇ r ·) of order 2l, where a r is a (σ + τ + r, τ + σ)-tensor field, is said to be uniformly strongly ρ-elliptic if its principal part C(a 2l , ∇ 2l ·) satisfies that for each cotangent field ξ and every (σ, τ )-tensor field η, it holds (C(a 2l , η ⊗ (−iξ) ⊗2l )|η) g ∼ ρ 2l |η| 2 g |ξ| 2l g * .
(1.1) Moreover, in Section 3.1, we show that this ellipticity condition can actually be replaced by a weaker one, called normal ρ-ellipticity. But for the sake of simplicity, we still stay with uniformly strong ρ-ellipticity stated above in the introduction.
By imposing some mild regularity condition on the coefficients a r of A, called sregularity, we are able to prove the following result.
Here u ∈ bc s,ϑ (M, V σ τ ) iff ρ ϑ u is a (σ, τ )-tensor field with little Hölder continuity of order s. The precise definition of weighted little Hölder spaces will be presented in Section 2.2. An operator A is said to belong to the class H(E 1 , E 0 ) for some densely embedded Banach couple E 1 d ֒→ E 0 , if −A generates a strongly continuous analytic semigroup on E 0 with dom(−A) = E 1 . By means of a well-known result of G. Da Prato, P. Grisvard [25] and S. Angenent [7], this theorem yields the continuous maximal regularity property of A. Theorem 1.1 generalizes the results in [36] in the sense that taking ρ ∼ 1 M , it agrees with the continuous maximal regularity theory in [36] on uniformly regular Riemannian manifolds. Note that the theory established therein can somehow be considered as a generalization of the work on manifolds with cylindrical ends by R.B. Melrose [28,29] and his collaborators.
The proof of the main theorem follows the ideas in [36,Section 3]. The cornerstone of this proof is a properly defined retraction and coretraction system between weighted function spaces over manifolds and in Euclidean spaces, see Section 2.3 for the precise definition. This system enables us to apply the well-studied elliptic and parabolic theory in Euclidean spaces and translate it into the manifold framework.
One important feature of this paper is that it is application-oriented. We apply Theorem 1.1 to several well-known evolution equations. These results are stated in Section 4. As an example in geometric analysis, we show the well-posedness of the Yamabe flow on singular manifolds. The Yamabe flow arises as an alternative approach to the famous Yamabe problem. It was introduced by R. Hamilton shortly after the Ricci flow, and studied extensively by many authors afterwards. The reader may consult [36,Section 5] for a brief historical account of this problem.
In addition to its application to geometric analysis, we also apply the main theorem to two well-known relatives of the heat equation, namely, the porous medium equation and the parabolic p-Laplacian equation, on a singular manifold (M, g). J.L. Vázquez [39,40] proved existence and uniqueness of non-negative weak solutions of Dirichlet problems for the porous medium equation. In a landmark article [14], P. Daskalopoulos and R. Hamilton showed existence and uniqueness of smooth solutions for the porous medium equation, and the smoothness of the free boundary, namely, the boundary of the support of the solution, under mild assumptions on the initial data. In the past decade, there has been rising interest in investigating the porous medium equation on Riemannian manifolds. See [11,15,23,30,42,43] for example. To the best of the author's knowledge, research in this direction is all restricted to the case of complete, or even compact, manifolds. The result that we state in Section 4.1 seems to be the first one concerning existence and uniqueness of solutions to the porous medium equation on manifolds with singularities. Following the same strategy, we study the p-Laplacian equation, a nonlinear counterpart of the Laplacian equation, which is probably one of the best known examples of degenerate or singular equations in divergence form. In Section 4.3, we explore the parabolic p-Laplacian equation on a singular manifold (M, g): Here p > 1 with p = 2. This problem has been studied extensively on Euclidean spaces. The two books [17,18] contain a detailed analysis and a brief historical description of this problem. There are several generalizations of the elliptic p-Laplacian equation on Riemannian manifolds. But fewer have been achieved for its parabolic version above. See [15] for instance. The study of these nonlinear heat equations also produces intriguing applications for degenerate boundary value problems or boundary blow-up problems. In Section 3.2, it is shown that any smooth domain (Ω, g m ) in R m with compact boundary can be realized as a singular manifold, where g m denotes the Euclidean metric. Then we can prove the local existence and uniqueness of solutions to the following boundary blow-up problem for 1 < p < 2 in little Hölder spaces.
as long as the initial data u 0 belongs to a properly chosen open subset in some Hölder space.
Another application of the continuous maximal regularity theory established in this paper concerns parabolic equations with higher order degeneracy on domains with compact boundary. The order of the degeneracy is measured by the rate of decay in the ellipticity condition while approaching the boundary. See Theorem 3.8 for a precise description. This result extends the work in [26,41] to unbounded domains and to higher order elliptic operators. In the last subsection, we prove a local existence and uniqueness theorem for a generalized multidimensional thin film 3) if the initial data decays sufficiently fast to the boundary of its support. Here α 1 , α 2 are two constants, n > 0, and Ω ⊂ R m is a sufficiently smooth domain. This generalized model was first investigated by J.R. King in [22] in the one dimensional case. Later, a multidimensional counterpart has been studied with periodic boundary condition on cubes in [9]. An interesting waiting-time phenomena can be observed from our approach. The mathematical investigation of the thin film equation was initiated by F. Bernis and A. Friedman in [8]. An intriguing feature of free boundary problems associated with degenerate parabolic equations is the waiting-time phenomena of the supports of the solutions. This phenomena has been widely observed and studied by many mathematicians. See [13,16,20,21,37] for example. The waiting-time phenomena for the case α 1 , α 2 = 0, the original thin film equation, has been explored in several of the papers listed above. Our result extends the results in the above literature for the generalized system (1.3).
It is worthwhile mentioning that sometimes to establish the theory for nonlinear parabolic equations, in some sense, is easier than that for linear equations. This surprising phenomena can be observed from the heat equation ∂ t u − ∆ g u = 0. Note that ∆ g = C(g * , ∇ 2 ·). In this case, the principal symbol of ∆ g can be computed as The power of the weight function ρ is different from (1.1) in this case. This breaks the uniform ellipticity conditions of the local expressions of the corresponding differential operators as we can observe from the discussion in Section 3 below. Linear differential operators with degeneracy other than ρ 2 have been investigated by many authors, including B.-W. Schulze [32,33] and his collaborators. But these results depend heavily on the specific geometric structure near the singular ends. In a subsequent paper [35], we treat second order differential operators with a different order of degeneracy from that in (1.1). In the nonlinear case, the nonlinearities sometimes give rise to the right power of ρ in (1.1), as is shown by the examples in Section 4.
This paper is organized as follows. In the rest of this introductory section, we give the precise definitions of uniformly regular Riemannian manifolds and singular manifolds. Section 2 is the stepstone to the theory of differential operators, where we define the weighted Hölder and little Hölder spaces on singular manifolds and introduce some of their properties, following the work of H. Amann in [3,4]. In Section 3, we establish continuous maximal regularity for differential operators satisfying the conditions in Theorem 1.1. In the last section, several applications of continuous maximal regularity theory are presented.
Assumptions on manifolds: Following H. Amann in [3,4], let (M, g) be a C ∞ -Riemannian manifold of dimension m with or without boundary endowed with g as its Riemannian metric such that its underlying topological space is separable. An where H m is the closed half spaceR + × R m−1 and Q m is the unit cube at the origin in R m . We put Q m κ := ϕ κ (O κ ) and ψ κ := ϕ −1 κ . The atlas A is said to have finite multiplicity if there exists K ∈ N such that any intersection of more than K coordinate patches is empty. Put The finite multiplicity of A and the separability of M imply that A is countable.
An atlas A is said to fulfil the uniformly shrinkable condition, if it is normalized and there exists r ∈ (0, 1) such that {ψ κ (rQ m κ ) : κ ∈ K} is a cover for M. Following H. Amann [3,4], we say that (M, g) is a uniformly regular Riemannian manifold if it admits an atlas A such that (R1) A is uniformly shrinkable and has finite multiplicity. If M is oriented, then A is orientation preserving.
Here g m denotes the Euclidean metric on R m and ψ * κ g denotes the pull-back metric of g by ψ κ .
(S2) A is a uniformly regular atlas.
(E3) ϕκ • ψ κ k,∞ ≤ c(k), κ ∈ K,κ ∈K and k ∈ N 0 We write the equivalence relationship as (ρ, K) ∼ (ρ,K). (S1) and (E1) imply that and to say that (M, g; ρ) is a singular manifold. A singular manifold is a uniformly regular Riemannian manifold iff ρ ∼ 1 M . We refer to [5,6] for examples of uniformly regular Riemannian manifolds and singular manifolds. A singular manifold M with an uniformly regular atlas A admits a localization system subordinate to A, by which we mean a family (π κ , ζ κ ) κ∈K satisfying: The reader may refer to [3, Lemma 3.2] for a proof.
Notations: Let K ∈ {R, C}. N 0 is the set of all natural numbers including 0. For any two Banach spaces X, Y , X . = Y means that they are equal in the sense of equivalent norms. The notation Lis(X, Y ) stands for the set of all bounded linear isomorphisms from X to Y . For any Banach space E, we abbreviate F(R m , E) to F(E) with F stands for any function space defined in this article. The precise definitions for these function spaces will be presented in Section 2. Let · ∞ , · s,∞ , · p and · s,p denote the usual norm of the Banach spaces BC(E)(L ∞ (E)), BC s (E), L p (E) and W s p (E) respectively. We denote K-valued function spaces with domain U ∈ {M, Ω} by F(U ) if Ω ⊂ R m with Ω = R m .

Preliminaries
In this section, we define the weighted function spaces on singular manifolds, following the work of H. Amann in [3,4]. Let A be a countable index set. Suppose E α is for each α ∈ A a locally convex space. We endow α E α with the product topology, that is, the coarsest topology for which all projections pr β : α E α → E β , (e α ) α → e β are continuous. By α E α we mean the vector subspace of α E α consisting of all finitely supported elements, equipped with the inductive limit topology, that is, the finest locally convex topology for which all injections E β → α E α are continuous.

Then it induces a conjugate linear bijection
Throughout the rest of this paper, unless stated otherwise, we always assume that is a singular manifold.
In [3, Lemma 3.1], it is shown that M satisfies the following properties: Then, given κ ∈ K, endowed with the Euclidean metric g m .

Weighted function spaces.
For the sake of brevity, we set L 1,loc (X, E) := κ L 1,loc (X κ , E). Then we introduce two linear maps for κ ∈ K: Here and in the following it is understood that a partially defined and compactly supported tensor field is automatically extended over the whole base manifold by identifying it to be zero outside its original domain. We define In the rest of this subsection we assume that k ∈ N 0 . In the first place, we list some prerequisites for the Hölder and little Hölder spaces on X ∈ {R m , H m } from [4, Section 11]. Given any Banach space F , the Banach space BC k (X, F ) is defined by The closed linear subspace BUC k (X, F ) of BC k (X, F ) consists of all functions u ∈ BC k (X, F ) such that ∂ α u is uniformly continuous for all |α| ≤ k. Moreover, It is a Fréchet space equipped with the natural projective topology.
The little Hölder space of order s ≥ 0 is defined by By [4, formula (11.13), Corollary 11.2, Theorem 11.3], we have and for k < s < k + 1 Now we are ready to introduce the weighted Hölder and little Hölder spaces on singular manifolds. Define endowed with the conventional projective topology. Then Here (·, ·) θ,∞ is the real interpolation method, see [  3. Basic properties. In the following context, assume that E κ is a sequence of Banach spaces for κ ∈ K. Then E := κ E κ . We denote by l ϑ ∞ (E) := l ϑ ∞ (E; ρ) the linear subspace of E consisting of all u = (u κ ) such that Similarly, for any k < s < k + 1, we denote by uniformly with respect to κ ∈ K.
In the sequel, we always assume F ∈ {bc, BC}, unless stated otherwise. Define where b = "∞, unif" for F = bc, and b = ∞ for F = BC. Then we have the following proposition.
Proof. This follows immediately from the definition of weighted l b spaces.
Proof. In the indicated references, a different retraction and coretraction system between F s,ϑ (M, V ) and l b (F s ) is defined as follows.
Proof. The assertion with weight ϑ = 0 follows from [ Let ⊕ be the Whitney sum. By bundle multiplication from for v i ∈ Γ(M, V i ) and p ∈ M. We still denote it by m. We can also prove the following point-wise multiplier theorems for function spaces over singular manifolds.
is a bilinear and continuous map.
Proposition 2.7. For any σ, τ ∈ N 0 and ϑ ∈ R, Proof. The case s ∈ N is immediate from the definition of the weighted function spaces. The non-integer case follows from [4, Theorem 16.1].

Continuous maximal regularity
3.1. Continuous maximal regularity on singular manifolds. Throughout the rest of this paper, we always assume that (M, g; ρ) is a singular manifold without boundary.
The other indices are defined in a similar way. [4,Lemma 14.2] implies that C is a bundle multiplication. Making use of [3, formula (3.18)], one can check that for any l-th order linear differential operator so defined, in every local chart (O κ , ϕ κ ) there exists some linear differential operator called the local representation of A in (O κ , ϕ κ ), such that for any u ∈ C ∞ (M, V ) Proof. The assertion is a direct consequence of Propositions 2.5 and 2.7.
Given any angle φ ∈ [0, π], set A linear operator A := A(a) of order l is said to be normally ρ-elliptic if there exists some constant C e > 0 such that for every pair (p, ξ) ∈ M × Γ(M, T * M ) with |ξ(p)| g * (p) = 0 for all p ∈ M, the principal symbol and (ρ l (p)|ξ(p)| l g * (p) + |µ|) (µ +σA π (p, ξ(p))) −1 The constant C e is called the ρ-ellipticity constant of A. To the best of the author's knowledge, this ellipticity condition is the first one formulated for degenerate or singular elliptic operators acting on tensor fields.
We can also introduce a stronger version of the ellipticity condition for A. A is called uniformly strongly ρ-elliptic if there exists some constant C e > 0 such that . HereσA π (p, ξ(p))(η(p)) := (C(a l , η ⊗ (−iξ) ⊗l )(p)|η(p)) g(p) . In [5], H. Amann have used the uniformly strong ρ-ellipticity condition to establish the L p -maximal regularity theory for second order differential operators acting on scalar functions.
We readily check that a uniformly strongly ρ-elliptic operator must be normally ρ-elliptic If A is of odd order, then by replacing ξ with −ξ in (3.3), it is easy to see that ρ(σA π (p, ξ(p))) = C. This is a contradiction. Therefore, every normally ρ-elliptic operator is of even order.
The "if" part follows by a similar argument. Proposition 3.3. Let s ∈ R + \ N and ϑ ∈ R. Suppose that A = A(a) is a 2l-th order linear differential operator, which is normally ρ-elliptic and s-regular with bounds C e and C a defined in (3.4) and (3.6). Then there exist ω = ω(C e , C a ), φ = φ(C e , C a ) > π/2 and E = E(C e , C a ) such that S = ω + Σ φ ⊂ ρ(−A) and Proof. To economize notations, we set It is not hard to check with the assistance of (3.7) that the coefficients (ā κ α ) κ satisfy (ā κ α ) κ ∈ l ∞,unif (bc s (L(E))), |α| ≤ 2l and by Proposition 3.2 thatĀ κ are all normally elliptic with a uniform ellipticity constant for all κ ∈ K. In virtue of [2, Theorems 4.1, 4.2 and Remark 4.6], these two conditions imply the existence of some constants ω 0 = ω 0 (C e , C a ), φ = φ(C e , C a ) > π/2 and E = E(C e , C a ) such that and

3.2.
Domains with compact boundary as singular manifolds. Suppose that Ω ⊂ R m is a C k -domain with compact boundary for k > 2. Then Ω satisfies a uniform exterior and interior ball condition, i.e., there is some r > 0 such that for every x ∈ ∂Ω there are balls B(x i , r) ⊂ Ω and B(x e , r) ⊂ R m \ Ω such that For a ≤ r, we denote the a-tubular neighborhood of ∂Ω by T a . Let i.e., the distance function to the boundary. We define d : in Ω \ T a otherwise. (3.19) Then we have the following proposition.
Proposition 3.7. Let β ≥ 1. Suppose that Ω ⊂ R m is a C k -domain with compact boundary and k > 2. Then (Ω, g m ; d β ) is a C k−1 -singular manifold.
Proof. The case of k = ∞ is a direct consequence of [6, Theorem 1.6]. When k < ∞, one notices that, to parameterize T a , we need to use the outward pointing unit normal of ∂Ω, which is C k−1 -continuous. By a similar argument to [6, Theorem 1.6], we can then prove the asserted statement.
Given any finite dimensional Banach space X, by identifying the singular manifold (M, g; ρ) with (Ω, g m ; d β ), we denote the weighted little Hölder spaces defined on Ω by b s,ϑ β (Ω, X), i.e., b s,ϑ β (Ω, X) = bc s,ϑ (M, X). In view of Remark 3.6, we have the following continuous maximal regularity theorem for elliptic operators with higher order degeneracy on domains.
The above theorem generalizes the results of [26,41] to unbounded domains and elliptic operators with order higher than two.
(b) In Theorem 3.8, taking X to be any infinite dimensional Banach space is also admissible.

The porous medium equation.
We consider the porous medium equation on a singular manifold (M, g; ρ), which reads as follows.

Y. SHAO
On account of the expression ∆ g v = C(g * , ∇ 2 v), it is then a direct consequence of Proposition 3.1 and [10, Proposition 1] that In the above, ∇ := ∇ g , where ∇ g is Levi-Civita connection of g. Given any ϑ ′ ∈ R, by Proposition 2.7 and [35, Proposition 2.6], one obtains A density argument as in the proof for Proposition 2.6 yields ).
For each v ∈ U 1+s , we can check that P (v) is normally elliptic in the sense of [36,Section 3]. Applying the parameter-dependent diffeomorphism technique in [34], Before concluding this subsection, we comment on the Cauchy problem for the porous medium equation and its waiting-time phenomena. Since our conclusion for the porous medium equation, to some extend, can be viewed as a simpler version of the corresponding theory of the thin film equation in Section 4.4, we will only state our results without providing proofs. More details can be found in Section 4.4. (Ω) : inf d ϑ u > 0} with 0 < s < 1, ϑ = −2/(n − 1). We learn from Proposition 3.7 that (Ω, g m ; d) is a C 3 -singular manifold, where d is defined in (3.19). Then by Theorems 3.8 and 4.1, for every f ∈ bc s,ϑ 1 (Ω), the equation ∂ t u + ∆u n = f on Ω T ; with Ω T := Ω × (0, T ), has a unique solution where R g is the scalar curvature with respect to the metric g. g 0 is in the conformal class of the background metric g 0 of M.
We seek solutions to the Yamabe flow (4.11) in the conformal class of the metric g 0 . Let c(m) := m−2 4(m−1) , and define the conformal Laplacian operator L g with respect to the metric g as: L g u := ∆ g u − c(m)R g u.
Let g = u 4 m−2 g 0 for some u > 0. It is well known that by rescaling the time variable equation (4.11) is equivalent to where L 0 := L g0 and u 0 is a positive function. See [27, formula (7)]. It is equivalent to solving the following equation: (4.12) The well-known formula of scalar curvature in local coordinates yields R g = 1 2 g ki g lj (g jk,li + g il,kj − g jl,ki − g ik,lj ).
Remark 4.5. The initial metric g 0 = u 4 m−2 0 g 0 in the above theorem can have unbounded scalar curvature. To keep this already long paper not any longer, we will give more details on this observation elsewhere.
We infer from (4.2) and (4.7) that The principal symbol can be computed as in Section 4.1.
In particular, the above inequality shows that, for 1 < p < 2, as x → ∂Ω This validates the assertion about equation (1.2) in Section 1.

4.4.
The thin film equation on domains. Suppose that Ω ⊂ R m is a C 6 -domain with compact boundary. Then by the discussion in Section 3.2, (Ω, g m ; d β ) with β ≥ 1 is a singular manifold, where d is defined in (3.19). We consider the following thin film equation with n > 0 and degenerate boundary condition. Physically, the power exponent is determined by the flow condition at the liquid-solid interface, and is usually constrained to n ∈ (0, 3]. Since the other choices of n makes no difference in our theory, n ∈ [3, ∞) is also included herein.
For any 0 < s < 1, take ϑ = −4/n and v ∈ E 1 , we define P (u)v :=u n ∆ 2 v + (n + α 1 )u n−1 (Du|D∆v) gm + α 1 u n−1 ∆u∆v It follows from a similar argument as in Section 3.1 that and for every u ∈ U 2+s ϑ , the principal symbol of P (u) can be computed aŝ Thus P (u) is normally ρ-elliptic.
Proof. The proof is essentially the same as that for Theorem 4.1 except that we use Theorem 3.8 instead of Theorem 3.5.
In the case α 1 = 0, we can admit lower regularity for the initial data. In some literature, a more general form of the thin film equation is considered with u n replaced by Ψ(u) = u n + δu 3 with δ ≥ 0 and n ∈ (0, 3]. The term δu 3 is sometimes omitted because it is relatively small compared to u n for n < 3 near the free boundary supp[u(t, ·)].
By identifyingû, f, u 0 ≡ 0 on R m \ Ω,û is nothing but a weak solution to the Cauchy problem ∂ t u + div(u n D∆u + α 1 u n−1 ∆uDu + α 2 u n−2 |Du| 2 Du) = f on R m T ; u(0) = u 0 on R m belonging to the class C 1/2 (J; W 2 1 (R m )) for β ∈ [1, n/(2n − 4)] when n ∈ (2, 3], or for all β ≥ 1 while n ∈ (0, 2] in the sense that J R m {u∂ t φ − ∆udiv(u n Dφ) + α 1 u n−1 ∆u(Dφ|Du) gm + α 2 u n−2 |Du| 2 (Dφ|Du) gm } dx dt = − J R m f φ dx dt for all φ ∈ C 0 (J ; W 2 ∞ (R m )). To prove this statement, one first observes that, by the uniform exterior and interior ball condition, for some sufficiently small a > 0 there is some a-tubular neighborhood of ∂Ω, denoted by T a , such that T a can be parameterized by Λ : (−a, a) × ∂Ω → T a : (r, p) → p + rν p , where ν p is the inward pointing unit normal of ∂Ω at p. By the implicit function theorem, there exists some C 5 -function Θ such that where d is defined in (3.19), and Θ(x) is the closest point on ∂Ω to x.
See [21,Theorem 4.1] for more details. A domain Ω is said to enjoy the external cone property if it satisfies this property at every y ∈ ∂Ω. Note that any u 0 ∈ U 2+s ϑ fulfils the flatness condition of the initial data in [21,Theorem 4.1].