Spectral analysis for linearizations of the Allen-Cahn equation around rescaled stationary solutions with triple-junction [Elektronische Ressource] / vorgelegt von Tobias Kusche

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• Mathematics

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• Publié par: universitat_regensburg
• Publié le: 01 janvier 2006
• Nombre de lectures: 9
• Langue: English

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Spectral analysis for linearizations of
the Allen-Cahn equation around
rescaled stationary solutions with
triple-junction
Dissertation
zur Erlangung des Doktorgrades der Naturwissenschaften
(Dr. rer. nat.) der Fakult¨at fur¨ Mathematik
der Universit¨at Regensburg
vorgelegt von
Tobias Kusche
aus
Regensburg
2006Promotionsgesuch eingereicht am: 10.01.2006
Die Arbeit wurde angeleitet von: Prof. Dr. Harald Garcke
Prufungsaussc¨ huß: Vorsitzender: Prof. Dr. Uwe Jannsen
1. Gutachter: Prof. Dr. Harald Garcke
2. Gutachter: Prof. Dr. Stanislaus Maier-Paape
weiterer Prufe¨ r: Prof. Dr. Wolfgang HackenbrochContents
List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1 Vector-valued Sturm-Liouville operators 19
21.1 Exponential L -bounds . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2 Pointwise exponential bounds . . . . . . . . . . . . . . . . . . . . 25
1.3 The range of Sturm-Liouville sytems . . . . . . . . . . . . . . . . 29
1.4 Convergence of the spectrum . . . . . . . . . . . . . . . . . . . . . 32
2 Spectral analysis for a two-phase transition 35
2.1 Standing wave solutions . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Linearizations around standing waves . . . . . . . . . . . . . . . . 41
2.2.1 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.2 Convergence of the ground state . . . . . . . . . . . . . . . 46
3 Spectral analysis at the triple-junction 51
3.1 Sobolev spaces with symmetry . . . . . . . . . . . . . . . . . . . . 51
3.2 Rescaled stationary solutions . . . . . . . . . . . . . . . . . . . . . 56
3.3 Linearizations around rescaled stationary solutions . . . . . . . . 59
3.3.1 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . 59
3.3.2 Exponential decay of eigenfunctions . . . . . . . . . . . . . 69
3.3.3 The range space . . . . . . . . . . . . . . . . . . . . . . . . 83
3.3.4 Convergence of the ground state . . . . . . . . . . . . . . . 85
3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4 Discussion 97
A Measure theory 101
B Operator theory 103
C Sesquilinear forms 109
12List of symbols
B (x) p. 14 Im(z) p. 13r
kC (Ω,B) p. 16 ker(f) p. 14
kC (Ω,B) p. 16 K p. 13
k 2C (Ω,B) p. 16 L (Ω,K) p. 16
b
∞ 2C (Ω,B) p. 16 L (Ω,K) p. 16J
∞ ∞C (Ω,B) p. 16 L (Ω,K) p. 160
∞ 2C (Ω) p. 53 L (I ) p. 420,G odd
det(A) p. 15 L(H) p. 15
diam(U) p. 13 L(H ,H ) p. 151 2
dim U p. 14 lin M p. 14
div as operator p. 107 M(m,K) p. 15
div u p. 17 N p. 13
αD u α multiindex p. 17 N p. 130
kD u k∈N p. 17 O(m) p. 15
D T operator or p. 15 O(g()),→ 0 g function p. 14T
Gsesquilinear form p. 109 P p. 52Ω
Gl(m,K) p. 15 P(X) p. 13
kH (Ω,K) p. 16 r() p. 60
k,∞H (Ω,K) p. 16 R p. 13+
k,∞
H (Ω,K) p. 16 R p. 13−loc
◦
∞k
R p. 13H (Ω,K) p. 17 +
k Re(z) p. 13H (I ) p. 42odd
k sign(z) p. 13H (Ω) p. 53G
◦
k supp(u) p. 14H (Ω) p. 53G
mS(R ) p. 15I n p. 15
K
T p. 60I p. 41
∗T T operator p. 15im(f) p. 13
34 as operator p. 107 [A,B] p. 25
4u p. 17 T ⊂S T, S operators p. 15
%() p. 60 ⊂⊂ p. 13
nρ(T) p. 103 ⊗ T T operator p. 15i ii=1
∂σ(T) p. 103 p. 18
∂ν
kσ (T) p. 103 ∂U ∈C p. 18d
σ (T) p. 103 ∇ as operator p. 107e
σ (T) p. 103 ∇u p. 17p
t
≥ for operators p. 111 u →u t sesquilinear form p. 109n
◦
|.| p. 13 p. 13U
ck.k p. 14 U p. 13X
k.k k p. 16C (Ω)
k.k k p. 17H (Ω)
k.k k,∞ p. 17H (Ω)
k.k 2 p. 16L
J
k.k ∞ p. 16L (Ω)
k.k 2 p. 16L (Ω)J
k.k p. 15L(B ,B )1 2
k.k p. 15tr
h.,.i p. 13
h.,.i p. 162L
h.,.i p. 162L
J
h.,.i p. 15
tr
h.,.i p. 14X
w
−→ p. 14
[T] T operator p. 17
[V] V function or p. 105 f.
real number p. 13
4Introduction
Phase-field models
Via formal calculations, Bronsard and Reitich, [BR], studied the asymptotic be-
havior of the vector-valued Ginzburg-Landau equation
∂ T 2 u = 2 4u −(DW(u )) (1)
∂t
∂ u| = 0 or u (x,t)| =h(x) (2)∂Ω ∂Ω
∂ν
u (x,0) =g(x) (3)
nas → 0. We consider this equation on an open domain Ω ⊂ R and for
m mu : Ω×R → R , where m,n ≥ 2. The potential W : R → R is smooth+
and attains its minimum value zero at exactly three distinct pointsa,b, andc, so
as to model a three-phase physical system. Instead of equation (1), we can also
consider the vector-valued Allen-Cahn equation
T∂2 2 ˆ v = 4v − DW(v ) . (4)
∂t
Equation (1) equals (4) via
1ˆW(x) := W(x),
2
and
1 v (x,t) :=u x, t .
22
The question is how the solution u of (1), (2), and (3) behaves as → 0. The
phase-field parameter > 0 represents the thickness of the transition layer be-
tweendifferentphases. Therefore, weexpectthatu approachesasharpinterface
model as → 0. One such sharp interface model is the mean-curvature flow.
nRoughly speaking, this is a family (Γ ) of smooth manifolds inR such thatt t∈[0,t]
the signed distance function d(.,t) of Γ fulfillst
∂
4d(x,t) = d(x,t), t∈ [0,T], x∈ Γ.t
∂t
5A precise definition is given in [AS]. For m = 1, i.e. the scalar Allen-Cahn
equation, de Mottoni and Schatzman, [deMS], proved that there exist initial
data for the Allen-Cahn equation such that the corresponding solutions converge
to the minima ofW uniformly outside each tubular neighborhood of Γ as→ 0.t
Essentially, the proof is a rigorous justification of formal asymptotic expansion,
i.e. it is supposed that in a tubular neighborhood of (Γ ) the solution u ist t∈[0,T]
approximately given by the asymptotic expansion
N
X d (x,t) iu (x,t) = u ,x,t , t∈ [0,T], x∈ Γ (δ).i tA
i=0
Note that Γ (δ) := {x ∈ Ω : dist(x,Γ ) < δ}. The function d is the modifiedt t
distance function, i.e.
NX
id (x,t) =d(x,t)+ d (x,t), t∈ [0,T], x∈ Γ (δ). i t
i=1
If one puts u into the Allen-Cahn equation, expands the term DW (u ) viaA A
Taylor expansion, and arranges the terms according to their -power, the results
are equations for the u of the formi
L u =R (d ), (5)0 i i−1 i−1
whereR (d ) depends only on known quantities and the functiond whichi−1 i−1 i−1
2is not determined so far. The operator L has domain H (R,C) and is given by0
00 2L u =−u +D W(θ )u.0 0
The function θ is the unique increasing solution of0
00−θ +DW(θ) = 0, θ(0) = 0,
that connects the two distinct minima of W. Equation (5) has a solution if and
only if
⊥R (d )∈ ker(L ) .i−1 i−1 0
This determines d , as dim ker(L ) = 1. As the solutions of (5) decay at ani−1 0
exponential rate, the approximate solution u can be extended to Ω. The resultA
is a family of approximate solutions (u ) such thatA ∈(0,1)

d(x,t)
2u (x,t) =θ +O( ), x∈ Γ (δ),0 tA
and
∂2 2 k u − 4u +DW (u ) =O , → 0.A A A∂t
6The integerk∈N grows with the lengthof the asymptotic expansion. Important
fortheproofoftheconvergenceu →u istoanalyzethebehaviorofthesmallestA
eigenvalue λ of the operator1
2d 2L =− +D W(θ ) (6) 02dz

1 12that is equipped with Neumann boundary conditions in L − , ,C . This

deliversthe[deMS]-estimatefortheAllen-Cahnoperator, i.e. thesmallesteigen-
value of
2 2 − 4+D W (u )A
2behaveslikeO( ),→ 0. Theoperatorthatisgivenbythedifferentialexpression
2 2 − 4+D W (u )A
is called Allen-Cahn operator. It represents the linearization of the Allen-Cahn
equation around the approximate solution u .A
Concerning the vector valued Allen-Cahn equation, Bronsard and Reitich proved
shorttimeexistencefortheproblemofthreecurvesΓ movingbymeancurvaturei
suchthatthethreecurvesmeetatatriple-junctionm(t), andtheotherendpoint
of each curve lies on the boundary of Ω - cf. figure 1.
G
1
G
3
G
2
Figure 1: Three-phase boundary motion.
Via formal asymptotic expansion, Bronsard and Reitich obtained the evolu-
tion laws of three-phase boundary motion derived by material scientists. At the
triple junction m(t), they used the expansion
N
X x−m(t) iu (x,t)≈ u ,t .i

i=0
For the function u , the expansion leads to the equation0
T
−4u +(DW(u )) = 0.0 0
7Moreover, in directions tangentially to the interfaces, one expects that u ap-0
proaches the standing wave solution that connects two minima of W. The ex-
istence of such an u was rigorously proved in the work of [BGS], details given0
in chapter 3. This is the first step in the proof of rigorous convergence to the
limiting flow. If one pursues the formal calculation to determine the u ’s, he isi
led to equations of the form
L u =R . (7)0 i i−1
The function R depends only on known quantities, and L is given by thei−1 0
differential expression
2−4+D W(u ).0
The operator L was introduced in [BGS]. It’s domain is given by the set of0
22 2all elements in (H (R ,C)) that are equivariant with respect to the symmetry
group G of the equilateral triangle. A byproduct of the proofs in [BGS] is that
L is self-adjoint and positive semidefinite.0
Target of the endevours
Now, we consider the case m = 2. In [BGS], they proved the existence of a
solution θ of0
00 t−θ +(DW(θ )) = 000
which connects two distinct global minima