Definition 4.1. Viscosity sub/super/solution for the non-local eikonal equation)
4. Proof of the main theorem
Now we are on the stage to prove the main theorem. Throughout this section, we use the same notations as in the previous sections.
For any T ∈ (0,1], let us set a compact and convex subset X0(T) in C([0, T];L2(Ω)), by putting:
X0(T) :=
⎧⎪
⎪⎨
⎪⎪
⎩
ξ∈W1,2(0, T;L2(Ω))
∩L∞(0, T;H1(Ω))
t 0
|ξt(τ)|2L2(Ω) dτ+κ0|∇ξ(t)|2L2(Ω)n
≤κ0|∇v0|2L2(Ω)n+ 1, and
|ξ(t)|L∞(Ω)≤2M0, for any t∈[0, T]
⎫⎪
⎪⎬
⎪⎪
⎭
;
and for any ˜v∈X0(T), let us consider Cauchy problem (3.2) with the initial value w0∈DN satisfying (3.6). Then, as is seen in the last section, problem (3.2) admits a unique solutionw∈W1,2(0, T;L2(Ω))∩L∞(0, T;H1(Ω)), and we find a positive constantρ0=ρ0(|v0|L∞(Ω),|w0|H1(Ω), R), independent ofT∈(0,1], such that
|w|W1,2(0,T;L2(Ω))+|w|L∞(0,T;H1(Ω)) ≤ρ0.
Also, taking the time-derivative of the both sides of (3.4) and multiplying the both sides of the result by wt(t), we further find a positive constant ρ1 = ρ1(|v0|L∞(Ω),|v0|H1(Ω),|w0|H2(Ω), r, R), independent ofT ∈(0,1], such that
ρ1≥ρ0 and|w|C1([0,T];L2(Ω))≤ρ1.
Here, for anyT ∈(0,1], let us introduce a compact and convex subsetY0(T), by putting:
Y0(T) := η∈C1([0, T];L2(Ω)) r≤η≤Ra.e. inQT, and
|η|C1([0,T];L2(Ω))+|η|L∞(0,T;H1(Ω))≤ρ1
; to define an operatorPT :X0(T)−→Y0(T), which maps any function ˜v ∈X0(T) into the solutionw ∈ Y0(T) of Cauchy problem (3.2). Then, for eachT ∈(0,1], the operatorPT is continuous in the sense that:
wi:=PTv˜i→w:=PT˜v inC([0, T];L2(Ω)) asi→+∞,
if{v˜i} ⊂X0(T), ˜v∈X0(T) and ˜vi →˜vin C([0, T];L2(Ω)) asi→+∞. (4.1)
Solvability for a PDE Model of Regional Economic Trend 411 The continuous dependence (4.1) is checked by applying Gronwall’s lemma to the following differential inequality:
1 2
d
dt|(wi−w)(t)|2L2(Ω)≤σ1
eα0v˜i(t)wi(t)1−α0−eα0v(t)˜ w(t)1−α0,(wi−w)(t)
L2(Ω)
≤σ1e2α0M0
R1−α0+ 1
rα0 |(wi−w)(t)|2L2(Ω)+|(˜vi−˜v)(t)|2L2(Ω)
; for a.e.t∈[0, T], that is obtained in a standard way with use of a similar technique as in the proof of Lemma 3.1.
Next, for anyT ∈(0,1] and any ˜w∈Y0(T), let us consider Cauchy problem (3.3). Then, since:
|(log ˜w)t(t)|2L2(Ω)= w˜t(t)
˜ w(t)
2
L2(Ω)
≤ ρ21
r2 for anyt∈[0, T]; (4.2) the following energy estimate:
t 0
|vt(τ)|2L2(Ω)dτ +κ0|∇v(t)|2L2(Ω)n
≤κ0|∇v0|2L2(Ω)n+ν12ρ21
r2 ãt, for anyt∈[0, T];
(4.3)
is easily obtained by multiplying the both sides of the (heat) equation byvt(t).
Now, for each T ∈ (0,1], let us define a solution operator QT : Y0(T)
−→ C([0, T];L2(Ω)), which maps any ˜w ∈ Y0(T) into the solution v ∈ W1,2(0, T;L2(Ω))∩L∞(0, T;H1(Ω)). Then, on account of the uniqueness of so- lutions and the demi-closedness of the maximal monotone operator, the operator QT shows the following continuity:
vi :=QTw˜i→v:=QTw˜ inC([0, T];L2(Ω)) asi→+∞,
if{w˜i} ⊂Y0(T), ˜w∈Y0(T) and ˜wi→w˜ in C([0, T];L2(Ω)) asi→+∞; (4.4) for anyT ∈(0,1].
Furthermore, on account of (4.2), (4.3) and Lemma 3.3, we find a sufficiently small constantT∗∈(0,1] such that
(QTw)˜ |[0,T∗]∈X0(T∗) for anyT ∈[T∗,1] and any ˜w∈Y0(T),
where the notation “|[0,T∗]” denotes the restriction of functions inC([0, T];L2(Ω)), withT ∈[T∗,1], onto the compact interval [0, T∗]. Therefore, by the compactness of X0(T∗) in C([0, T∗];L2(Ω)), the operatorQT∗ :Y0(T∗)−→X0(T∗) is well defined and continuous in the sense of (4.4) underT =T∗.
Consequently, we figure out that the compositionST∗ :=PT∗◦ QT∗ :Y0(T∗)
−→ Y0(T∗) forms a well-defined and continuous operator in the topology of C([0, T∗];L2(Ω)). SinceY0(T∗) is a compact and convex subset inC([0, T∗];L2(Ω)), Schauder’s fixed point theorem enables us to conclude the existence of a fixed pointw∗∈Y(T∗) for the operator (iteration)ST∗. Here, puttingv∗:=QT∗w∗and u∗ := ev∗, the pair {u∗, w∗} fulfills all conditions (s1)–(s3) under T = T∗. This entails the existence of time-local solutions of our system{(2.1),(2.2)}.
Acknowledgment. The research for this paper is partially supported by a grant from Institute for Advanced Research Hiroshima Shudo University, 2004.
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Modelling, Analysis and Simulations, GAKUTO Internat. Ser. Math. Sci. Appl., Gakk¯otosho.
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Ken Shirakawa
Department of Applied Mathematics, Faculty of Engineering, Kobe University 1-1 Rokkodai, Nada, Kobe, 657-8501, Japan
e-mail:sirakawa@cs.kobe-u.ac.jp Akio Ito
Department of Electronic Engineering and Computer Sciences, School of Engineering Kinki University, 1 Takayaumenobe, Higashihiroshimashi, Hiroshima, 739-2116, Japan e-mail:aito@hiro.kindai.ac.jp
Atsushi Kadoya
Department of Economic Informatics, Faculty of Economic Sciences, Hiroshima Shudo University, 1-1-1 Otsuka, Asaminami-ku
Hiroshimashi, Hiroshima, 731-3195, Japan e-mail:kadoya@shudo-u.ac.jp
International Series of Numerical Mathematics, Vol. 154, 413–423 c 2006 Birkh¨auser Verlag Basel/Switzerland
Surface Energies in Multi-phase Systems with Diffuse Phase Boundaries
Bj¨ orn Stinner
Abstract. A Ginzburg–Landau type functional for a multi-phase system in- volving a diffuse interface description of the phase boundaries is presented with the following calibration property: Prescribed surface energies (possibly anisotropic) of the phase transitions are correctly recovered in the sense of a Γ-limit as the thickness of the diffuse interfaces converges to zero. Possible applications are grain boundary motion and solidification of alloys on which numerical simulations are presented.
Mathematics Subject Classification (2000).82B26, 74N20, 74E10.
Keywords.Phase transitions, phase field model, sharp interface model, surface energy, anisotropy.
1. Introduction
Consider a domain Ω ⊂ Rd, d ∈ {1,2,3}, which is subdivided into several (not necessarily connected) regions Ωα, α ∈ {1, . . . ,M}, M ∈ N, separated by hy- persurfaces Γαβ, 1 ≤α < β ≤ M. The interfaces are supposed to carry energy obtained by integrating surface densities which may depend on the orientation of the hypersurface. The total energy of the system has the form
FSI =
α<β
Γαβ
σαβ(ναβ)dHd−1. (1.1) To avoid wetting effects is is assumed thatσαβ+σβδ> σαδ>0 for each triple of phasesα, β, δ.
Energies as in (1.1) can be approximated by Ginzburg–Landau energies of the form
FP F =
Ω
εa(φ,∇φ) +1 εw(φ)
dLd. (1.2)
Here,φ= (φ1, . . . , φM) : Ω→ΣM with ΣM:=
v∈RM: M α=1
vα= 1 and 0≤vα≤1 for allα= 1, . . . ,M is a set of phase field variables. For each α, φα is assigned to one of the phases (labelledαand corresponding to Ωα) and describes its presence in a point of Ω.
The functiona(φ,∇φ) is a non-negative gradient term, andw(φ) is a multi-well potential withMminima corresponding to the phases. It is well known that such Ginzburg–Landau energies cause transition regions where the phase fields vary from one of the minima ofw, i.e., from one of the phases, to another one. The thickness of the interfacial layers is of the orderε, a small length scale.
Bellettini et al. [2] showed that the Γ-limit of (1.2) asε →0 has the form (1.1), and they proved a relation between the σαβ and the functions a and w.
Using matched asymptotic expansions, Sternberg [10] for the isotropic case and Garcke et al. [5] for the general case found the slightly simpler relation
σαβ(ν) = inf
p 1
−1
w(p)a(p, p⊗ν)dy,
p∈C0,1([−1,1]; ΣM), p(−1) =eα, p(1) =eβ
(1.3) whereeα and eβ are the corners of the simplex ΣM corresponding to the phases αandβ, i.e.,eη= (δηθ)Mθ=1with the Kronecker symbolsδηθ. Using numerical sim- ulations they showed that this formula holds true for a large class of anisotropies.
Vice versa, it is a non-trivial task to construct functions aand w such that the surface energies obtained via (1.3) coincide with given surface energies (which, in applications, may be known from experiments).
A possible solution has been found by the author in collaboration with H.
Garcke and R. Haas (see [7]) and will be presented in the following sections. First some facts on the minimization problem (1.3), afterwards the precise aims are stated. Some additional conditions onaandware imposed:
Definition 1.1. Let TΣM:=
v= (v1, . . . , vM)∈RM: M α=1
dα= 0
. The functiona: ΣM×(TΣM)d→Ris an admissible gradient term if
a(φ, X)≥0, a(φ, ηX) =η2a(φ, X) ∀φ∈ΣM, X∈(TΣM)d, η∈R, (1.4) aαβ(ν) :=a(seβ+ (1−s)eα,(eβ−eα)⊗ν) (1.5) does not depend ons∈[0,1] ∀α, β∈ {1, . . . ,M}.
Surface Energies in Multi-phase Systems 415 The functionw: ΣM→Ris an admissible multi-well potential if
w(φ)≥0∀φ∈ΣM, w(φ) = 0 ⇔ φ∈ {e1, . . . , eN}, (1.6) w(seβ+ (1−s)eα) =wαβs2(1−s)2, ∀s∈[0,1], α, β∈ {1, . . . ,M}. (1.7) A minimizer of the problem in (1.3) fulfills after appropriate rescaling to some functionφ:R→ΣM (cf. [5] for this procedure)
w,φ(φ) +a,φ(φ, ∂zφ⊗ν)− d dz
a,X(φ, ∂zφ⊗ν)ν
=λ,
zlim→∞φ(z) =eβ, lim
z→−∞φ(z) =eα,
(1.8) whereλ=M1 M
i=1w,φi(φ) +a,φi(φ, ∂zφ⊗ν)−dzd
a,Xi(φ, ∂zφ⊗ν)ν
is a Lagrange multiplier andλ=λ(1, . . . ,1)∈RM. In particular, the minimization problem in (1.3) reads now
σαβ(ν) = inf
φ R
a(φ, ∂zφ⊗ν) +w(φ)
dz, φ∈C0,1(R; ΣM), lim
z→−∞φ(z) =eα, lim
z→∞φ(z) =eβ
. (1.9) Lemma 1.2. Consider a function of the form
φ(z) =χ(z)eβ+ (1−χ(z))eα. (1.10) Thenφsolves (1.8) for admissible functionsaandwif
χ(z) =1 2
1 + tanh3 wαβ
aαβ(ν)
z 2
and if (1.11)
λ=w,φi+|χ|2a,φi− d dz
χa,Xiν
(1.12) for someλindependent ofi∈ {1, . . . ,M}wherew,φi is evaluated atφ(z) and the derivatives ofaat (φ(z),(eβ−eα)⊗ν).
Moreover, ifφis a solution to (1.9) then the surface energy is σαβ(ν) =1
3 3
aαβ(ν)wαβ (1.13)
Proof. Given φ as in (1.10) obviously φ(z) → eβ ⇔ χ(z) → 1 ⇔ z → ∞ and φ(z) → eα ⇔ χ(z) → 0 ⇔ z → −∞ whence the second line of (1.8) follows.
By assumption (1.4) a,φ is two-homogeneous and a,X is one-homogeneous with respect to the second variable. Since∂zφ(z)⊗ν =χ(z)(eβ−eα)
a,φ(φ(z), ∂zφ(z)⊗ν) =|χ(z)|2a,φ(φ(z),(eβ−eα)⊗ν), a,X(φ(z), ∂zφ(z)⊗ν) =χ(z)a,X(φ(z),(eβ−eα)⊗ν).
The first line of (1.8) then follows directly from (1.12).
A straightforward calculation shows the identities
|χ|2=wαβ
aαβχ2(1−χ)2, χ=wαβ
aαβχ(1−χ)(1−2χ). (1.14)
By assumption (1.5)a(φ(z),(eβ−eα)⊗ν) =aαβ(ν). Therefore, ifφsolves (1.9) then the surface energy is (substituting z2
3aαβ(ν) wαβ =t) σαβ(ν) =
R
a(φ, ∂zφ⊗ν) +w(φ)
dz
=
R
|χ(z)|2aαβ(ν) +wαβχ(z)2(1−χ(z))2
dz
=
R
2wαβ(1 + tanh(t))2(1−tanh(t))2 .
aαβ(ν) wαβ dt
=1 3
3
aαβ(ν)wαβ.
Now it is possible to state the aim precisely:
Task. Construct admissible functionsaandwsuch that 1. the values 13
aαβ(ν)wαβ coincide with given surface energiesσαβ(ν), 2. the function φ(z) = χ(z)eβ + (1−χ(z))eα with χ given by (1.11) fulfills
condition(1.12), and
3. this functionφ(z)solves (1.9).