The Cahn-Hilliard equation and the Biharmonic heat kernel on edge manifolds

0.27 MB·English

by  Boris Vertman

THE CAHN-HILLIARD EQUATION AND THE BIHARMONIC
HEAT KERNEL ON EDGE MANIFOLDS
BORIS VERTMAN
Abstract. We construct the biharmonic heat kernel of a suitable closed self-
adjoint extension of the bi-Laplacian on a manifold with incomplete edge singu-
larities. We establish mapping properties of the biharmonic heat operator and
derive short time existence for certain semi-linear equations of fourth order. We
3 apply the analysis to a class of semi-linear partial differential equations in partic-
1 ular the Cahn-Hilliard equation and obtain asymptotics of the solutions near the
0 edge singularity.
2

n
a 1. Introduction
J

0 The Cahn-Hilliard equation was proposed by Cahn and Hilliard in [Cah61],
3
[CaHi58] as a simple model of the phase separation process, where at a fixed tem-

] perature the two components of a binary fluid spontaneously separate and form
P
domains that are pure in each component. Let ∆ denote a self-adjoint extension of
S
a Laplace operator. Then the Cahn-Hilliard equation may be stated in the following
.
h
form
t
a ∂ u(t)+∆2u(t)+∆(u(t) u(t)3)) = 0, u(0) = u , t (0,T).
m t 0
− ∈
Global existence for solutions to the Cahn-Hilliard equation has been established
[

by Elliott and Songmu [ElSo86], and Caffarelli and Muler [CaMu95]. In the setup
1
v of singular manifolds however, there is still a question of asymptotics of solutions
9 at the singular strata. This aspect has been studied by the recent work of Roidos
9
and Schrohe [RoSc12] in the context of manifolds with isolated conical singularities,
2
7 which has motivated the present discussion here. Using the notion of maximal regu-
.
1 larity, they establish short time existence of solutions to the Cahn-Hilliard equation
0 in certain weighted Mellin-Sobolev spaces which then yields regularity and asymp-
3
totics of solutions near the conical point.
1
:
v Inthis paper we study theCahn-Hilliardequation inthegeometric setup ofspaces
i
X with incomplete edges, which generalizes the notion of isolated conical singularities.
r Our method is different from [RoSc12] and uses the microlocal construction of the
a
heat kernel for the bi-Laplacian. Precise analysis of the asymptotic behaviour of the
heatkernel allowsforaderivationofthemappingpropertiesoftheheatoperatorand
a subsequent argument on the short time existence for certain semi-linear equations
of fourth order, in analogy to [JL03] and [BDV11]. Existence and asymptotics
for solutions to the Cahn-Hilliard equation comes as an application of the general
results.
The microlocal analysis of the biharmonic heat kernel on edge spaces also allows
for derivation of Schauder estimates and ultimately leads to short time existence
results for fourth order PDE’s, including the Bock flow [BaDy11] on singular spaces.
Date: January 31, 2013.
2000 Mathematics Subject Classification. 53C44; 58J35;35K08.
1
2 BORISVERTMAN
Thiswillbethesubject offorthcominganalysis. Wealsopointoutthatourapproach
is not limited to squares of Laplacians on functions, but yields similar results for
general powers of Hodge Laplacians on differential forms along the same lines.
Our main result may be summarized briefly as follows.
Theorem 1.1. Let (M,g) be a Riemannian manifold with an admissible Riemann-
ian incomplete edge metric g. Then the biharmonic heat kernel H for the Friedrichs
self-adjoint extension of the Laplace-Beltrami operator ∆ lifts to a polyhomogeneous
g
distribution on a blowup of R+ M2 and defines a biharmonic heat operator with
×
various mapping properties, such that certain semi-linear fourth order equations,
among then the Cahn-Hilliard equation, admit a short time solution of a certain
regularity.
This paper is organized as follows. We first introduce the basic geometry of
incomplete edge spaces in 2. We then provide a microlocal construction of the heat
§
kernel for a certain self-adjoint extension of the bi-Laplacian in 3. We proceed
§
with with establishing mapping properties of the biharmonic heat operator in 4.
§
We conclude with a short time existence result for certain semi-linear equations of
fourth order and apply the analysis to the particular example of the Cahn-Hilliard
equation in 5.
§
Acknowledgements
The author would like to thank Elmar Schrohe, Rafe Mazzeo and Eric Bahuaud
for insightful discussions and gratefull acknowledges the support by the Hausdorff
Research Institute at the University of Bonn.
Contents
1. Introduction 1
Acknowledgements 2
2. Spaces with incomplete edge singularities 2
3. Microlocal heat kernel construction 4
4. Mapping properties of the biharmonic heat operator 13
5. Short time existence of semi-linear equations of fourth order 18
References 19
2. Spaces with incomplete edge singularities
We introduce the fundamental geometric aspects of spaces with incomplete edge
singularities, described in detail in [Maz91], compare also [MaVe12]. Let M be
◦
a compact stratified space with a single top-dimensional stratum M and a single
lower dimensional stratum B By the definition of stratified spaces, B is a smooth
closed manifold. Moreover there is an open neighbourhood U M of B and a radial
function x : U M R, such that U M is a smooth fibr⊂e bundle over B with
fibre C(F) = (0∩,1) →F, a finite open con∩e over a compact smooth manifold F. The
×
restriction of x to each fibre is the radial function of that cone.
THE CAHN-HILLIARD EQUATION AND THE BIHARMONIC HEAT KERNEL ON EDGES 3
The singular stratum B in M may be resolved and defines a compact manifold M
with boundary ∂M, where ∂M is the total space of a fibration φ : ∂M B with
→
the fibre F. The resolution process is described in detail for instance in [Maz91f].
The neighborhood U lifts to a collar neighborhood U of the boundary, which is a
smooth fibration of cylinders [0,1) F over B with the radial function x.
×
Definition 2.1. A Riemannian manifold (M ∂M,g) := (M,g) has an incomplete
edge singularity at B if over U the metric g e\quals g = g +h, where h = O(x)
0 | |g0
as x 0 and f
→
g ↾ U ∂M = dx2 +x2gF +φ∗gB,
0
\
with gB being a Riemannian metric on the closed manifold B, and gF a symmetric
2-tensor on the fibration ∂M which restricts to a Riemannian metric on each fibre
F.
We set m = dimM,b = dimB and f = dimF. Clearly, m = 1+b+f. We assume
henceforth f = dimF 1. Otherwise the incomplete edge manifold reduces to a
≥
compactmanifoldwithboundary, whereourdiscussionbelowisnolongerapplicable.
Similarly to other discussions in the singular edge setup, see [Alb07],
[BDV11],[BaVe11] and [MaVe12], we additionally require φ : (∂M,gF +φ∗gB)
→
(B,gB) to be a Riemannian submersion. If p ∂M, then the tangent bundle T ∂M
p
∈
splits into vertical and horizontal subspaces as TV∂M TH∂M, where TV∂M is the
p ⊕ p p
tangent space to the fibre of φ through p and TH∂M is the orthogonal complement
p
of this subspace. φ is said to be a Riemannian submersion if the restriction of the
tensor gF to TH∂M vanishes. We say that an incomplete edge metric g is feasible
p
if φ is a Riemannian submersion and in addition the Laplacians associated to gF at
each y B are isospectral.
∈
The reasons behind the feasibility assumptions are as follows. Let y = (y ,...,y )
1 b
be the local coordinates on B lifted to ∂M and then extended inwards. Let z =
(z ,...,z ) restrict to local coordinates on F along each fibre. Then (x,y,z) are the
1 f
local coordinates on M near the boundary. Consider the Laplace Beltrami operator
∆ associated to (M,g) and its normal operator N(x2∆) , defined as the limiting
y0
operator with respect to the local family of dilatations (x,y,z) (λx,λ(y y ),z)
0
and acting on functions on the model edge R+ F Rb wi→th incomple−te edge
s × × u
metric g = ds2+s2gF + du 2. Under the first admissibility assumption, N(x2∆)
ie | | y0
is naturally identified with s2 times the Laplacian for that model metric.
The second condition on isospectrality is imposed to ensure polyhomogeneity of
the associated heat kernels when lifted to the corresponding parabolicblowup space.
More precisely we only need that the eigenvalues of the Laplacians on fibres are
constant in a fixed range [0,1], though we still make the stronger assumption for a
clear and convenient representation.
Definition 2.2. Let (M,g) be a Riemannian manifold with a feasible edge metric.
This metric g = g +h is said to be admissible if in addition to feasibility
0
(i) the lowest non-zero eigenvalue λ > 0 of the Laplace-Beltrami operator ∆
0 F,y
associated to (F,gF φ−1(y)) for any y B, satisfies λ0 > dimF.
| ∈
(ii) h vanishes to second order at x = 0, i.e. h = O(x2) as x 0.
| |g0 →
4 BORISVERTMAN
The reasonsbehindtheadmissibility assumptions areoftechnical ratherthangeo-
metric nature and somewhat less straightforward to explain. However we point out
that condition (i) is satisfied by a rescaling of gF which geometrically corresponds
to decreasing the cone angle uniformly along the edge. Condition (ii) in particular
holds for even metrics which depend on x2 instead of x. The admissibility assump-
tions yield particular information on the heat kernel expansion, which is then used
in Proposition 3.5.
An important ingredient in the analysis of singular edge spaces is the vector space
of edge vector fields smooth in the interior of M and tangent at the boundary ∂M
e
V
to the fibres of the fibration. This space is closed under the ordinary Lie bracket
e
V
of vector fields, hence defines a Lie algebra. Its dfescription in local coordinates is as
follows. Consider the local coordinates (x,y,z) on M near the boundary. Then the
edge vector fields are locally generated by
e
V
x∂ ,x∂ ,...,x∂ ,∂ ,...,∂ .
x y1 yb z1 zf
We may now define th(cid:8)e Banach space of continuous s(cid:9)ections C0(M), continuous
ie
on M up to the boundary and fibrewise constant at x = 0. This is precisely the
space of continuous sections with respect to the topology on M induced by the
Riemfannian metric g. Banach spaces of higher order are defined as follows, compare
[BDV11].
Definition 2.3. Let (M,g)beaRiemannianmanifoldwithanincompleteedgemet-
ric. Let denote a collection of ∆ and a choice of derivatives in x−1 2,x−1 , ,
which wDill be specified later. Then for each k N we define a B{anachVespaceVe Ve}
∈
C2k(M, ) := u C2k(M) C0(M) X ∆ju C0(M),X , j = 0,..,k 1 ,
ie D { ∈ ∩ ie | ◦ ∈ ie ∈ D − }
k
with the norm u := u + X ∆ju .
2k ∞ ∞
k k k k k ◦ k
j=0 X∈D
X X
3. Microlocal heat kernel construction
3.1. Self-adjoint extension of the bi-Laplacian. Let ∆ denote the Laplace
Beltrami operator acting on functions on an incomplete edge space (M,g) with a
feasible incomplete edge metric g. As a first step we fix a self-adjoint extension of
the bi-Laplacian ∆2. Consider the space of square-integrable forms L2(M,g), with
respect to g. The maximal and minimal closed extensions of ∆ are defined by the
domains
(∆) := u L2(M,g) ∆u L2(M,g) ,
max
D { ∈ | ∈ }
(∆) := u (∆) u C∞(M) such that (3.1)
Dmin { ∈ Dmax | ∃ j ∈ 0
u u and ∆u ∆u both in L2(M,g) .
j j
→ → }
where ∆u L2 is a priori understood in the distributional sense. Under the unitary
∈
rescaling transformation in the singular edge neighborhood
Φ : L2(U ,dvol(g)) L2(U ,x−fdvol(g)), u xf/2u, (3.2)
→ 7→
THE CAHN-HILLIARD EQUATION AND THE BIHARMONIC HEAT KERNEL ON EDGES 5
the Laplacian ∆ takes the form
∂2 1 f 1 2 1
∆Φ := Φ ∆ Φ−1 = + ∆ + − .
◦ ◦ −∂x2 x2 F,y 2 − 4
!
(cid:18) (cid:19)
Lemma 3.1 ([MaVe12]). Let (M,g) be an incomplete edge space with a feasible
edge metric. Considerthe increasingsequence of eigenvalues (σ )p of ∆ , counted
j j=1 F,y
with their multiplicities, such that ν2 := σ + (f 1)2/4 [0,1). The associated
j j − ∈
indicial roots are given by γ± = ν +1/2. Then any u (∆) expands under
j ± j ∈ Dmax
the rescaling (3.2) as x 0
→
p
Φu c+[u]ψ+(x,z;y)+c−[u]ψ−(x,z;y) +Φu˜, u˜ (∆),
∼ j j j j ∈ Dmin
j=1
X(cid:0) (cid:1)
where the leading order term of each ψ± is the corresponding solution of the indicial
j
operator. More precisely, let φ denote the normalized σ -eigenfunctions of of ∆ .
j j F,y
Then
√x(logx)φ (z;y), ν = 0,
ψ+(x,z;y) = xγj+φ (z;y), ψ−(x,z;y) = j j
j j j (xγj−(1+ajx)φj(z;y), νj > 0,
with a R uniquely determined by ∆. The coefficients c±[u] are of negative regu-
j ∈ j
larity in y and the asymptotic expansion holds only in a weak sense, i.e. there is an
expansion of the pairing u(x,y,z)χ(y)dy for any test function χ C∞(B).
B ∈
The Friedrichs self-adjoRint extension of ∆ has been identified in [MaVe12] as
(∆ ) = u (∆) : c−[u] = 0 . (3.3)
D F { ∈ Dmax | ∀j=1,..,p j }
Note that the sequence of eigenvalues (σ )p of ∆ starts with σ = 0 for j =
j j=1 F,y j
1,..,dimH0(F). The corresponding indicial roots compute to γ = 1/2 (f 1)/2
j
± −
and the coefficient are constant in z F. Consequently, in view of the explicit form
∈
of the rescaling transformation in (3.2), the Friedrichs domain contains precisely
those elements in the maximal domain whose leading term in the weak expansion
as x 0 is given by x0 with fibrewise constant coefficients. In particular
→
(∆) C0(M) (∆ ). (3.4)
Dmax ∩ ie ⊂ D F
We fix a self-adjoint extension of the bi-Laplacian as the square of ∆
F
(∆2 ) = u ∆ ∆u ∆ . (3.5)
F F F
D { ∈ | ∈ }
3.2. Biharmonic heat kernel on a model edge. Inthesecondstepweconstruct
the heat kernel for ∆2 . We begin with studying the homogeneity properties of the
F
heat kernel for the bi-Laplacian in the model case of an exact edge ( = Rb
C(F),dy2 +g) where (C(F) = (0, ) F,g = ds2 +s2gF) is an exact uEnbounde×d
∞ ×
cone over a closed Riemannian manifold (F,gF). The Laplacian ∆ on the exact
E
edge is then a sum of the Laplacian on (C(F),g) and the Euclidean Laplacian on
Rb. Consider the scaling operation (λ > 0)
Ψ : C∞(R+ 2) C∞(R+ 2),
λ
×E → ×E
(Ψ u)(t,(s,y,z),(s,y,z)) = u(λ4t,(λs,λ(y y),z),(λs,λy,z)).
λ
−
e e e e e e e
6 BORISVERTMAN
Under the scaling operation we find
(∂ +∆2)Ψ u = λ4Ψ (∂ +∆2)u. (3.6)
t E λ λ t E
Consequently, given the heat kernel H for the Friedrichs extension of ∆2 (or at
E E
that stage any other self-adjoint extension), any multiple of Ψ H still solves the
λ E
heat equation and also maps into the domain of ∆2. For the initial condition we
E
obtain substituting Y = λy,S = λs
lim (Ψ H )(t, s, y, z, s, y, z)u(s, y, z)sf dsdydz
λ E e e e e
t→0
ZE
=limλ−1−b−f H (λ4t, λse, λeye Ye, ze, Se, Ye, ze)ue(S/eλ, Y/λ, z)Sf dSdYdz
E
t→0 −
ZE
=λ−1−b−fu(λs/λ, λy/λ, z) = λ−1−b−fu(s, y, z).
e e e e e e e e e e e
By uniqueness of the heat kernel we obtain
Ψ H = λ1−b−fH . (3.7)
λ E E
In addition to the homogeneity properties of H , we also require a full asymptotic
E
expansion of the biharmonic heat kernel as (s,s) 0. We accomplish this by
→
establishing an explicit integral representation of H . Under the unitary rescaling
E
(3.2) and a spectral decomposition of L2(F,gF) into σ2-eigenspaces of ∆ , we may
e F
write for the rescaled model edge Laplacian
2
f 1 1
∆Φ = ∂2 +s−2 ∆ + − +∆
E − s F 2 − 4 Rb
!
(cid:18) (cid:19)
2
f 1 1
= ∂2 +s−2 σ2 + − +∆ =: l +∆ ,
− s 2 − 4 Rb ν(σ) Rb
!
σ (cid:18) (cid:19) σ
M M
where ν(σ) := σ2 +(f 1)2/4 and l is defined on C∞(0, ). The Friedrichs
− ν(σ) 0 ∞
extension of ∆ is compatible with the decomposition, compare a similar discussion
E
p
in ([Ver08], Proposition 4.9). As a special case of Lemma 3.1, l has unique self-
ν
adjoint extension L in L2(R+) for ν 1, and in case of ν [0,1), solutions
ν
u D(l ) admit a partial asymptotic≥expansion as s 0 ∈
ν,max
∈ →
s−ν+1/2, ν (0,1),
u(s) = u+c+[u]sν+1/2 +c−[u] ∈ u D(l ).
ν,min
(√slog(s), ν = 0, ∈
Then the Frieedrichs extension L of l is defined, similar toe(3.3), by requiring
ν ν
c−[u] = 0, and moreover, identifying ∆ with its unique self-adjoint extension in
Rb
L2(Rb), we may write
∆Φ = L +∆ . (3.8)
E,F ν(σ) Rb
σ
M
Consequently, it suffices to construct the biharmonic heat kernel for L + ∆ in
ν Rb
L2(R+ Rb). Denote by J the ν-th Bessel function of first kind and consider the
ν
×
Hankel transform of order ν 0
≥
∞
(H u)(s) := √ss′J (ss′)u(s′)ds′, u C∞(0, ). (3.9)
ν ν ∈ 0 ∞
Z0
THE CAHN-HILLIARD EQUATION AND THE BIHARMONIC HEAT KERNEL ON EDGES 7
By ([Co59], Chapter III) and also by ([Les97], Proposition 2.3.4), the Hankel
transform extends to a self-adjoint isometry on L2(R+). We denote by
(Fu)(ξ) := (2π)−b/2 u(y)e−iy·ξdy, u C∞(Rb),
∈ 0
Rb
Z
theFourier transformonRb, which extends toanisometric automorphismof L2(Rb).
Consequently, G := H F defines an isometric automorphism of L2(R+ Rb)
ν ν
such that G−1 = H F−◦1. Applying ([Les97], Proposition 2.3.5), we arrive a×t the
ν ν◦
following
Proposition 3.2. The isometric automorphism G diagonalizes L + ∆ . More
ν ν Rb
precisely,
D(L +∆ ) = u L2(R+ Rb) (S2 + Ξ 2)G u L2(R+ Rb) ,
ν Rb ν
{ ∈ × | | | ∈ × }
G (L +∆ )G−1 = S2 + Ξ 2,
ν ν Rb ν | |
where X,Ξ denote multiplication operators by x R+ and ξ Rb, respectively.
∈ ∈
Similary, the isometry G diagonalizes the squared operator (L +∆ )2, identify-
ν ν Rb
ing its action with (S2 + Ξ 2)2. Consequently we may express the biharmonic heat
kernelof(L +∆ )2 asan|in|tegralintermsofBessel functions. Foru C∞(R+ Rb)
ν Rb ∈ 0 ×
we find
e−t(Lν+∆Rb)2u (s,y) = G e−t(S2+|Ξ|2)2G−1u (s,y)
ν ν
(cid:16) (cid:17) ∞ (cid:16) ∞ (cid:17)
= (2π)−b/2 ei(y−ye)ξ√ssJ (sρ)J (sρ)ρe−t(ρ2+|ξ|2)2dρdξ
ν ν
ZRbZ0 (cid:18)ZRbZ0 (cid:19)
u(s,y)dsdy.
e e
Denotebyφ thenormalizedσ2-eigenfunctionof∆ , wherewecount theeigenvalues
σ F
σ2 Spece(∆e )ewiteh their multiplicities. Then, as a consequence of (3.8), we finally
F
∈
obtain for the Φ-rescaled biharmonic heat kernel on a model edge
∞
HΦ = (2π)−b/2 ei(y−ye)ξ√ssJ (sρ)J (sρ)ρe−t(ρ2+|ξ|2)2dρdξ
E ν(σ) ν(σ)
σ ZRbZ0
M
φ (z) φ (z).
e e σ σ
· ⊗
The ν-th Bessel function of first kind admits an asymptotic expansion for small
arguments J (ζ) ∞ a ζν+2j, as ζ 0. This yields an asymptotic expaension of
ν ∼ j=0 j →
HΦ as (s,s) 0 and consequently, rescaling back, we obtain as s 0
E → P →
H (t, s, y, z, s, y, z) a (t,s,y,y,z,z)sγ+2j, (3.10)
E ν,j
e ∼
γ
X
where the summation is over aell eγ =e (f 1)/2 e+ eσ2 +e(f 1)2/4 with σ2
− − − ∈
Spec∆ , counted with multiplicity, and each coefficient a lies in the corresponding
F ν,j
p
σ2-eigenspace. We summarize the properties of H , established above, in a single
E
proposition for later reference.
Proposition 3.3. Considerthe model edge ( = Rb C(F),dy2+g), where (C(F) =
E ×
(0, ) F,g = ds2 +s2gF) is an exact unbounded cone over a closed Riemannian
∞ ×
manifold (Ff,gF). Fix the Friedrichs self-adjoint extension of the associated Laplace
8 BORISVERTMAN
Beltrami operator ∆ . Then the biharmonic heat kernel H of ∆2 is homogeneous
E E E
of order ( 1 b f) under the scaling operation (λ > 0)
− − −
Ψ : C∞(R+ 2) C∞(R+ 2),
λ
×E → ×E
(Ψ u)(t,(s,y,z),(s,y,z)) = u(λ4t,(λs,λ(y y),z),(λs,λy,z)).
λ
−
Moreover, H admits an asymptotic expansion as (s,s) 0 with the index set given
E
by E +2N , where e e e →e e e e
0
e
(f 1) (f 1)2
E = γ 0 γ = − + − +σ2, σ2 Spec∆ ,
F
{ ≥ | − 2 4 ∈ }
r
uniformly in other variables and with coefficients taking value in the corresponding
σ2-eigenspace.
3.3. Construction of the biharmonic heat kernel. We can now proceed from
the analysis of the heat kernel on the model edge to the construction of the heat
kernel H for the bi-Laplacian on a space (M,g) with an incomplete feasible edge
metric. The heat kernel construction here follows ad verbatim the discussion in
[MaVe12] for the edge Laplacian, with the only difference that for the bi-Laplacian
now rather t1/4 instead of √t is treated as a smooth variable.
The heat kernel is a function on M2 = R+ M2. Let (x,y,z) and (x,y,z) be the
h ×
coordinates on the two copies of M near the edge. Then the local coordinates near
thecornerinM2 aregivenby(t,(x,y,z),(x,y,z)f). Thekernel H(t,(x,y,z),(x,y,z))
h e e e
has a non-uniform behaviour at the submanifolds
A = (t = 0,(0,y,z),(0,y,z)) eR+e e∂M2 y = y , e e e
{ ∈ × | }
D = (t = 0,(x,y,z),(x,y,z)) R+ M2 x = x, y = y, z = z .
{ e e ∈ × | e }
Exactly as in thecase of theHodge Laplacianon edges, see [MaVe12], we introduce
e e e f e e e
anappropriateblowupoftheheatspaceM2, suchthatthecorrespondingheatkernel
h
lifts to a polyhomogeneous distribution in the sense of the definition below. This
procedure has been introduced by Melrose in [Mel93]. For self-containment of the
paper we repeat the definition of polyhomogeneity as well as the blowup process
here.
Definition 3.4. Let W be a manifold with corners, with all boundary faces embed-
ded, and (H ,ρ ) N anenumeration of its boundaries andthe corresponding defin-
{ i i }i=1
ing functions. For any multi-index b = (b ,...,b ) CN we write ρb = ρb1...ρbN.
Denote by (W) the space of smooth ve1ctor fieNlds∈on W which lie tange1nt toNall
b
boundary faVces. A distribution ω on W is said to be conormal, if ω ρbL∞(W) for
some b CN and V ...V ω ρbL∞(W) for all V (W) and for e∈very ℓ 0. An
1 ℓ j b
index se∈t E = (γ,p) C∈ N satisfies the follow∈inVg hypotheses: ≥
i
{ } ⊂ ×
(i) Re(γ) accumulates only at plus infinity,
(ii) For each γ there is P N , such that (γ,p) E if and only if p P ,
γ 0 i γ
(iii) If (γ,p) E , then (γ +∈j,p′) E for all j ∈N and 0 p′ p. ≤
i i
∈ ∈ ∈ ≤ ≤
An index family E = (E ,...,E ) is an N-tuple of index sets. Finally, we say that
1 N
a conormal distribution ω is polyhomogeneous on W with index family E, we write
THE CAHN-HILLIARD EQUATION AND THE BIHARMONIC HEAT KERNEL ON EDGES 9
ω AE (W), if ω is conormal and if in addition, near each H ,
∈ phg i
ω a ργ(logρ )p, as ρ 0,
∼ γ,p i i i →
(γX,p)∈Ei
withcoefficientsa conormalonH , polyhomogeneouswithindexE atanyH H .
γ,p i j i j
∩
The homogeneity property (3.7) contains the information how precisely the sub-
manifolds A,D M2 need to be blown up such that the heat kernel becomes
⊂ h
polyhomogeneous. To get the correct blowup of M2 we first bi-parabolically (t1/4 is
h
viewed as a coordinate function) blow up the submanifold
A = (t,(0,y,z),(0,y,z)) R+ ∂M2 : t = 0,y = y M2.
{ ∈ × } ⊂ h
The resulting heat-space [M2,A] is defined as the union of M2 A with the interior
spherical normalbundleofAinhMe e2. Theblowup[M2,A]isendohew\ed withtheunique
h h
minimal differential structure with respect to which smooth functions in the interior
of M2 and polar coordinates on M2 around A are smooth. As in [MaVe12], this
h h
blowup introduces four newboundary hypersurfaces; we denotethese by ff (thefront
face), rf (the right face), lf (the left face) and tf (the temporal face).
The actual heat-space blowup M2 is obtained by a bi-parabolicblowup of [M2,A]
h h
along the diagonal D, lifted to a submanifold of [M2,A]. The resulting blowup M2
h h
isdefined asbeforeby cutting out the submanifold andreplacing it with itsspherical
normal bundle. It is a manifold with boundaries and corners, visualized in Figure
below.
t1/4
β
lf rf
−→
ff
x x
M2 M2
h tf tf h
td e
Figure 1. Heat-space Blowup M2.
h
The projective coordinates on M2 are then given as follows. Near the top corner
h
of the front face ff, the projective coordinates are given by
x x y y
ρ = t1/4, ξ = , ξ = , u = − , z, y, z, (3.11)
ρ ρ ρ
e e
where in these coordinates ρ,ξ,ξ areethe defining functions of the boundary faces ff,
e e
rf and lf respectively. For the bottom corner of the front face near the right hand
side projective coordinates are geiven by
t x y y
τ = , s = , u = − , z, x, y, z, (3.12)
x4 x x
whereinthesecoordinatesτ,s,xarethedefiningefunctionsoftf,rfandffrespectively.
e e e
For the bottomcorner of thee front faece near thee left hand side projective coordinates
e
10 BORISVERTMAN
are obtained by interchanging the roles of x and x. Projective coordinates on M2
h
near temporal diagonal are given by
t1/4 (x x) y y e x(z z)
η = , S = − , U = − , Z = − , x, y, z. (3.13)
x t1/4 t1/4 t1/4
In these coordinates tf is the facee in the limite(S,U,Ze) e , ff and td are defined
by x,η, respectiveely. The blowdown map β |: M2 | →M2∞is ien leocael coordinates
h → h
simply the coordinate change back to (t,(x,y,z),(x,y,z)).
Ine case the edge manifold is an exact edge ( = Rb C(F),dy2 + g) where
(C(F) = (0, ) Ff,g = ds2 + s2gF), PropEoesiteione 3×.3 implies that H lifts
E
∞ ×
to a polyhomogeneous conormal distribution on the heat space blowup, of order
( m),m = 1 + b + f, at the front and the temporal diagonal faces, vanishing to
in−finite order at tf, and with the index set at rf and lf given by E +2N , where
0
(f 1) (f 1)2
E = γ 0 γ = − + − +σ2, σ2 Spec∆ .
F
{ ≥ | − 2 4 ∈ }
r
In the general case of a feasible edge space (M,g), H is only an initial parametrix
E
and solves the heat equation only to first order. Repeating almost ad verbatim the
heat kernel construction in case of the edge Laplacian in [MaVe12], we arrive at
the following
Proposition 3.5. Let (Mm,g) be an incomplete edge space with a feasible edge
metric g. Then the lift β∗H is polyhomogeneous on M2 of order ( dimM) at ff
h −
and td, vanishing ot infinite order at tf, and with the index set at rf and lf given by
E +N where
0
(f 1) (f 1)2
E = γ 0 γ = − + − +σ2, λ Spec∆ .
F,y
{ ≥ | − 2 4 ∈ }
r
More precisely, if s denotes the boundary defining function of rf, we obtain
∞ ∞
β∗H sγ+2ja (β∗H)+ sγ+2+ja′ (β∗H) as s 0,
∼ γ,j γ,j →
!
γ∈E j=0 j=0
X X X
where the coefficients a (H) are of order ( m) at the front face and lie in their cor-
γ,j
−
responding ∆ eigenspaces. The higher coefficients a′ (β∗H) are of order ( m+1)
F,y γ,j −
at ff.
Proof. Recall the heat kernel construction in [MaVe12], which we basically follow
here. Denote by ∆ the Laplace Beltrami operator on (M,g). We write := ∂ +∆2
t
L
for the heat operator. The restriction of the lift β∗(t ) to ff is called the normal
L
operator N (t ) at the front face (at the fibre over y B) and is given in
ff L y0 0 ∈
projective coordinates (3.12) explicitly as follows
2
N (t ) = τ ∂ + ∂2 fs−1∂ +s−2∆ +∆Rb
ff L y0 τ − s − s F,y0 u
(cid:18) (cid:19)
(cid:16) (cid:17)
2
=: τ ∂ + ∆C(F) +∆Rb .
τ s,y0 u
(cid:18) (cid:19)
(cid:16) (cid:17)
N (t ) does not involve derivatives with respect to (y ,x,y,z) and hence acts tan-
ff 0
L
gentially to the fibres of the front face. Consequently in our choice of an initial
e e e