Analysis and optimal boundary control of a nonstandard system of phase field equations

0.32 MB·English

by  Pierluigi Colli

Analysis and Optimal Boundary
Control of a Nonstandard System
of Phase Field Equations
Pierluigi Colli1, Gianni Gilardi1,
2 and Ju¨rgen Sprekels2
1
0
2

n
Abstract. We investigate a nonstandard phase field model of Cahn-Hilliard type. The
a
J model, which was introduced in [16], describes two-species phase segregation and consists

0 of a system of two highly nonlinearly coupled PDEs. It has been studied recently in [5], [6]
3
for the case of homogeneous Neumann boundary conditions. In this paper, we investigate

] the case that the boundary condition for one of the unknowns of the system is of third
P
kind and nonhomogeneous. For the resulting system, we show well-posedness, and we
A
study optimal boundary control problems. Existence of optimal controls is shown, and
.
h
the first-order necessary optimality conditions are derived. Owing to the strong nonlinear
t
a
couplings in the PDE system, standard arguments of optimal control theory do not apply
m
directly, although the control constraints and the cost functional will be of standard type.
[

2
v
1 Introduction
1
4
5
Let Ω IR3 denoteanopenandboundeddomainwhosesmoothboundary Γ hasoutward
5
⊂
. unit normal n, let T > 0 be a given final time, and let Q := Ω (0,T), Σ := Γ (0,T).
1
× ×
0 In this paper, we study the following initial-boundary value problem:
2
1
(ε+2ρ)µ +µρ ∆µ = 0 a.e. in Q, (1.1)
: t t
v −
i δρ ∆ρ+f′(ρ) = µ a.e. in Q, (1.2)
X t −
∂ρ ∂µ
r = 0, = α(u µ) a.e. on Σ, (1.3)
a
∂n ∂n −
ρ(x,0) = ρ (x), µ(x,0) = µ (x), for a.e. x Ω. (1.4)
0 0
∈
The PDE system (1.1)–(1.2) constitutes a phase field model of Cahn-Hilliard type that
describes phase segregation of two species (atoms and vacancies, say) on a lattice in the
1
Dipartimento di Matematica “F. Casorati”, Universit`a di Pavia, Via Ferrata, 1, 27100 Pavia, Italy,
e-mail: [email protected], [email protected]
2
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany,
e-mail: [email protected]
Key words: Nonlinear phase field systems, Cahn–Hilliard systems, parabolic systems, optimal boundary
control, first-order necessary optimality conditions.
AMS (MOS) Subject Classification: 74A15, 35K55, 49K20.
2 Optimal Boundary Control of a Nonstandard Phase Field System
presence of diffusion. It has been introduced recently in [16] and [5]; for the general physi-
calbackground, wereferthereaderto[16]. Theunknownvariablesaretheorderparameter
ρ, interpreted asavolumetric density, andthechemical potential µ. Forphysical reasons,
we must have 0 ρ 1 and µ > 0 almost everywhere in Q. The boundary (control)
≤ ≤
function u on the right-hand side of (1.3) plays the role of a microenergy source. More-
2
over, ε and δ are positive constants, and the nonlinearity f is a double-well potential
defined in (0,1), whose derivative f′ is singular at the endpoints ρ = 0 and ρ = 1; a typ-
ical example is f = f +f , with f smooth and f (ρ) = c(ρ log(ρ)+(1 ρ) log(1 ρ)),
1 2 2 1
− −
where c is a positive constant.
The PDE system (1.1)–(1.4) is singular, with highly nonlinear and nonstandard coupling.
In particular, unpleasant nonlinear couplings involving time derivatives occur in (1.1),
and the expression f′(ρ) in (1.2) may become singular. In the recent papers [5], [6], well-
posedness and asymptotic behavior for t and ε 0 of the system (1.1)–(1.4) were
→ ∞ ց
established for the case when the second boundary condition in (1.3) is replaced by the
homogeneous Neumann boundary condition ∂µ/∂n = 0; a distributed optimal control
problem for this situation was analyzed in [7]. We also refer to the papers [3] and [4],
where the corresponding Allen-Cahn model was discussed.
The paper is organized as follows: in Section 2, we state the general assumptions and
prove the existence of a strong solution to the problem. Section 3 is concerned with the
issues of uniqueness and stability. Section 4 then brings the study of a boundary control
problem for the system (1.1)–(1.4). We show existence of a solution to the optimal control
problem and derive the first-order necessary optimality conditions, as usual given in terms
of the adjoint system and a variational inequality.
Throughout the paper, we make repeated use of H¨older’s inequality, of the elementary
Young inequality
1
ab γa2 + b2, for every a,b 0 and γ > 0, (1.5)
≤ 4γ ≥
of the interpolation inequality
v v θ v 1−θ v Lp(Ω) Lq(Ω),
k kLr(Ω) ≤ k kLp(Ω)k kLq(Ω) ∀ ∈ ∩
1 θ 1 θ
where p,q,r [1,+ ], θ [0,1], and = + − , (1.6)
∈ ∞ ∈ r p q
and, since dimΩ 3, of the continuity of the embeddings H1(Ω) Lq(Ω) for 1 q 6,
≤ ˆ ⊂ ≤ ≤
where, with constants C > 0 depending only on Ω,
q
v Cˆ v v H1(Ω), 1 q 6, (1.7)
Lq(Ω) q H1(Ω)
k k ≤ k k ∀ ∈ ≤ ≤
and where the embeddings are compact for 1 q < 6. We also use the Sobolev spaces
≤
Hs(Ω) of real order s > 0 and recall the compact embeddings Hs(Ω) H1(Ω) and
⊂
Hs(Ω) C(Ω) for s > 1 and s > 3/2, respectively, and, e.g., the estimate, with a
constan⊂t Cˆ > 0 depending only on Ω,
∞
v Cˆ v v H2(Ω). (1.8)
k kC(Ω) ≤ ∞k kH2(Ω) ∀ ∈
Colli — Gilardi — Sprekels 3
2 Problem statement and existence
Consider the initial-boundary value problem (1.1)–(1.4). For convenience, we introduce
the abbreviated notation
H = L2(Ω), V = H1(Ω), W = w H2(Ω) : ∂w/∂n = 0 on Γ .
∈
(cid:8) (cid:9)
We endow these spaces with their standard norms, for which we use self-explaining nota-
tion like ; for simplicity, we also write for the norm in the space H H H.
V H
k·k k·k × ×
Recall that the embeddings W V H are compact. Moreover, since V is dense
⊂ ⊂
in H, we can identify H with a subspace of V∗ in the usual way, i.e., by setting
u,v V∗,V = (u,v)H for all u H and v V , where , V∗,V denotes the duality
h i ∈ ∈ h· ·i
pairing between V∗ and V . Then also the embedding H V∗ is compact.
⊂
We make the following assumptions on the data:
(A1) f = f +f , where f C2(0,1) is convex, f C2[0,1], and
1 2 1 2
∈ ∈
limf′(r) = , limf′(r) = + . (2.1)
rց0 1 −∞ rր1 1 ∞
(A2) ρ W, f′(ρ ) H, µ V , and
0 0 0
∈ ∈ ∈
0 < ρ (x) < 1 x Ω, µ 0 a.e. in Ω. (2.2)
0 0
∀ ∈ ≥
(A3) u H1(0,T;L2(Γ)), and u 0 a.e. on Σ.
∈ ≥
(A4) α L∞(Γ), and α(x) α > 0 for almost every x Γ.
0
∈ ≥ ∈
Notice that (A2) implies that ρ C(Ω) and, thanks to the convexity of f , also that
0 1
∈
f(ρ ) H.
0
∈
The following existence result resembles that of Theorem 2.1 in [5].
Theorem 2.1 Suppose that the hypotheses (A1)–(A4) are satisfied. Then the system
(1.1)–(1.4) has a solution (ρ,µ) such that
ρ W1,∞(0,T;H) H1(0,T;V) L∞(0,T;W), (2.3)
∈ ∩ ∩
µ H1(0,T;H) C0([0,T];V) L2(0,T;H3/2(Ω)), (2.4)
∈ ∩ ∩
f′(ρ) L∞(0,T;H), (2.5)
∈
0 < ρ < 1 a.e. in Q, µ 0 a.e. in Q. (2.6)
≥
Remark 2.2 The H3/2 space regularity for µ is optimal due to the L2 space regularity
of u given by (A3). Nevertheless, both equation (1.1) and the boundary condition for
µ contained in (1.3) can be understood a.e. in Q and a.e. on Σ, respectively, and the
standard integration by parts is correct, as we briefly explain (so that we can both refer to
that formulation and use integration by parts). In principle, one can replace the equation
and the boundary condition by the usual variational formulation, namely
[(ε+2ρ)µ +µρ ] vdx+ µ vdx+ α(µ u)vdσ = 0
t t
Z Z ∇ ·∇ Z −
Ω Ω Γ
4 Optimal Boundary Control of a Nonstandard Phase Field System
(where dσ standsforthesurfacemeasure)forevery v V ,a.e.in (0,T),oranintegrated-
∈
in-time version of it. This implies that (1.1) is satisfied in the sense of distributions,
whence ∆µ belongs to L2(Q) by comparison, and the equation can be understood a.e.
in Q, a posteriori. The last regularity (2.4) of µ and the condition ∆µ L2(0,T;L2(Ω))
∈
just observed also ensure that the trace ∂µ has a meaning in the space L2(0,T;L2(Γ))
∂n|Σ
due to the trace theorem [15, Thm. 7.3] (we just observe that the space Ξ−1/2(Ω) that
enters such a result is larger than L2(Ω)), so that the boundary condition can be read
a.e. on Σ.
Proof of Theorem 2.1. The proof follows closely the lines of the proof of Theorem 2.1 in
[5], where a homogeneous Neumann boundary condition for µ was investigated.
Step 1: Approximation. We employ an approximation scheme based on a time delay in
the right-hand side of (1.2). To this end, we introduce for τ > 0 the translation operator
: L1(0,T;H) L1(0,T;H), which for v L1(0,T;H) and almost every t (0,T) is
τ
T → ∈ ∈
defined by
( )(t) := v(t τ) if t > τ, and ( )(t) := µ if t τ . (2.7)
τ τ 0
T − T ≤
Now, let N IN be arbitrary, and τ := T/N. We seek functions (ρτ,µτ) satisfying
∈
(2.3)–(2.6) (with (ρ,µ) replaced by (ρτ,µτ)), which solve the system
(ε+2ρτ)µτ +µτρτ ∆µτ = 0 a.e. in Q, (2.8)
t t −
δρτ ∆ρτ +f′(ρτ) = µτ a.e. in Q, (2.9)
t − Tτ
∂ρτ ∂µτ
= 0, = α(u µτ) a.e. on Σ, (2.10)
∂n ∂n −
ρτ(x,0) = ρ (x), µτ(x,0) = µ (x), for a.e. x Ω. (2.11)
0 0
∈
We note that Remark 2.2 also applies to the approximating problem. To prove the
existence of a solution, we put t := nτ, I := [0,t ], 1 n N, and consider for
n n n
≤ ≤
1 n N the problem
≤ ≤
(ε+2ρn)µn +µnρn ∆µn = 0 a.e. in Ω I , (2.12)
t t − × n
∂µn
µn(0) = µ a.e. in Ω, = α(u µn) a.e. on Γ I , (2.13)
0 n
∂n − ×
δρn ∆ρn +f′(ρn) = µn−1 a.e. in Ω I , (2.14)
t − Tτ × n
∂ρn
ρn(0) = ρ a.e. in Ω, = 0, a.e. on Γ I . (2.15)
0 n
∂n ×
Notice that the operator acts on functions that are not defined on the entire interval
τ
T
(0,T). However, its meaning is still given by (2.7) if n > 1, and for n = 1 we simply put
µn−1 = µ .
τ 0
T
Clearly, wehave (ρτ,µτ) = (ρN,µN) if (ρN,µN) exists. Weclaimthatthesystems (2.12)–
(2.15) can be uniquely solved by induction for n = 1,...,N, where, for 1 n N,
≤ ≤
ρn W1,∞(I ;H) H1(I ;V) L∞(I ;W), (2.16)
n n n
∈ ∩ ∩
µn H1(I ;H) C0(I ;V) L2(I ;H3/2(Ω)), (2.17)
n n n
∈ ∩ ∩
0 < ρn < 1 a.e. in Ω I , µn 0 a.e. in Ω I . (2.18)
n n
× ≥ ×
Colli — Gilardi — Sprekels 5
To prove the claim, suppose that for some n 1,...,N the problem (2.12)–(2.15)
∈ { }
has a unique solution satisfying (2.16)–(2.18), where the index n is replaced by n 1.
−
Then it follows with exactly the same argument as in the proof of Theorem 2.1 in [5] that
the initial-boundary value problem (2.14), (2.15) has a unique solution ρn that satisfies
(2.16) and the first inequality in (2.18). Substituting ρn in (2.12), we infer that the linear
initial-boundary value problem (2.12), (2.13) has a unique solution µn satisfying (2.17).
Notice here that the regularity of µn follows from the fact that u H1(0,T;L2(Γ)).
∈
It remains to show that µn is nonnegative almost everywhere. To this end, we test (2.12)
by (µn)−, where (µn)− denotes the negative part of µn. Using integration by parts
−
and the boundary condition in (2.13), we obtain the identity
1 t d t
(ε+2ρn) (µn)− 2 dxds+ (µn)− 2 dxds
2 Z0ZΩ dt(cid:16) (cid:12) (cid:12) (cid:17) Z0ZΩ(cid:12)∇ (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
t t
+ α (µn)− 2 dσds + αu(µn)−dσds = 0.
Z Z Z Z
0 Γ (cid:12) (cid:12) 0 Γ
(cid:12) (cid:12)
From the fact that ρn, ρ , µ , α, u are all nonnegative, we infer that
0 0
ε (µn)−(t) 2 dx (ε+2ρn(t)) (µn)−(t) 2 dx
Z ≤ Z
Ω(cid:12) (cid:12) Ω (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(ε+2ρ ) µ− 2 dx = 0.
≤ Z 0 0
Ω (cid:12) (cid:12)
(cid:12) (cid:12)
Hence, (µn)− = 0, i.e., µn 0 a.e. in Ω I , and the claim is proved.
n
≥ ×
Step 2: A priori estimates. Now that the well-posedness of the problem
(2.8)–(2.11) is established, we perform a number of a priori estimates for its solution.
For the sake of a better readability, we will omit the index τ in the calculations. In
what follows, we denote by C > 0 positive constants that may depend on the data of the
system but not on τ. The meaning of C may change from line to line and even in the
same chain of inequalities.
First estimate. Since ∂ (ε/2)µ2 + ρµ2 = (ε + 2ρ)µ + µρ µ, testing of (2.8) by µ
t t t
yields, for every t [0,T](cid:0), (cid:1) (cid:0) (cid:1)
∈
ε t t
µ2 +ρµ2 (t)dx + µ 2dxds + αµ2dσds
ZΩ(cid:16)2 (cid:17) Z0ZΩ|∇ | Z0ZΓ
ε t
= µ2 +ρ µ2 (t)dx + αuµdσds,
ZΩ(cid:16)2 0 0 0(cid:17) Z0ZΓ
whence, using Young’s inequality and (A2)–(A4), we can conclude that
µ L∞(0,T;H)∩L2(0,T;V) C. (2.19)
k k ≤
Second estimate. Next, we test (2.9) by ρ . Applying (2.19), recalling the fact that
t
f(ρ ) H, and invoking Young’s inequality, we easily see that
0
∈
6 Optimal Boundary Control of a Nonstandard Phase Field System
ρ H1(0,T;H)∩L∞(0,T;V) + f(ρ) L∞(0,T;L1(Ω)) C. (2.20)
k k k k ≤
Third estimate. We rewrite Eq. (2.9) in the form
∆ρ + f′(ρ) = δρ f′(ρ) + µ
− 1 − t − 2 Tτ
and observe that the right-hand side is bounded in L2(Q). Hence, applying a standard
procedure (e.g., testing by f′(ρ)), and invoking elliptic regularity, we find that
1
ρ + f′(ρ) C. (2.21)
k kL2(0,T;W) k 1 kL2(Q) ≤
Fourth estimate. We differentiate Eq. (2.9) formally with respect to t and test the
resulting equation with ρ (this argument can be made rigorous, see [5]). Since, owing to
t
the convexity of f , f′′(ρ) is nonnegative almost everywhere, we find the estimate
1 1
δ t δ
ρ (t) 2 + ρ 2 dxds ∆ρ f′(ρ )+µ 2
2 k t k Z Z |∇ t| ≤ 2 k 0 − 1 0 0kH
0 Ω
t t
+ max f′′(ρ) ρ 2 dxds + (∂ µ) ρ dxds
0≤ρ≤1 | 2 |Z Z | t| Z Z tTτ t
0 Ω 0 Ω
t−τ
C + µ (s)ρ (s+τ)dxds. (2.22)
t t
≤ Z Z
0 Ω
In order to estimate the last integral, we substitute for µ , using Eq. (2.8). It follows,
t
using integration by parts:
t−τ t−τ 1
µ ρ ( +τ)dxds = (∆µ µρ )ρ ( +τ)dxds
t t t t
Z Z · Z Z ε+2ρ − ·
0 Ω 0 Ω
t−τ µ 2ρ ( +τ)
t
= ∇ ρ ( +τ) + · µ ρ
Z Z (cid:20)− ε+2ρ ·∇ t · (ε+2ρ)2 ∇ · ∇
0 Ω
1
ρ µρ ( +τ) dxds
t t
− ε+2ρ · (cid:21)
t−τ α
(u µ)ρ ( +τ)dσds. (2.23)
t
− Z Z ε+2ρ − ·
0 Γ
Exactly as in the proof of Theorem 2.1 in [5], the domain integral in the second and third
lines of (2.23) can be estimated from above by an expression of the form
1 t t
ρ 2 dxds + C 1 + µ(s) 2 ρ (s) 2 dxds . (2.24)
2 Z0ZΩ|∇ t| (cid:16) Z0 k kV k t kH (cid:17)
Observe that, owing to the inequality (2.19), the mapping s µ(s) 2 belongs to
7→ k kV
L1(0,T).
Colli — Gilardi — Sprekels 7
Finally, we estimate the boundary term in the last line of Eq. (2.23). To this end,
recall that by the trace theorem there is a constant c > 0 , independent of τ, such that
Ω
v c v for all v V . Moreover, we have ρ 0 and α L∞(Γ). Therefore,
L2(Γ) Ω V
k k ≤ k k ∈ ≥ ∈
we obtain that
t−τ α
(u µ)ρ ( +τ)dσds
t
(cid:12)Z0 ZΓ ε+2ρ − · (cid:12)
(cid:12) (cid:12)
(cid:12) t−τ (cid:12)
C ρ (s+τ) u(s) + µ(s) ds
t L2(Γ) L2(Γ) L2(Γ)
≤ Z k k k k k k
0 (cid:0) (cid:1)
t−τ
C ρ (s+τ) u(s) + µ(s) ds
t V L2(Γ) V
≤ Z k k k k k k
0 (cid:0) (cid:1)
1 t
ρ (s) 2 ds + C. (2.25)
≤ 4 Z k t kV
0
Now we may combine the estimates (2.22)–(2.25) and employ Gronwall’s inequality to
conclude that
ρt L∞(0,T;H)∩L2(0,T;V) C. (2.26)
k k ≤
The same argument as in the derivation of (2.22) then shows that also
kρtkL∞(0,T;W) + kf1′(ρ)kL∞(0,T;H) ≤ C. (2.27)
Fifth estimate. We test equation (2.8) by µ . Formal integration by parts (this can be
t
made rigorous), using (A3), the trace theorem and Young’s inequality, yields:
t 1 α
ε µ 2 dxds + µ(t) 2 + µ(t) 2dσ
Z Z | t| 2 k∇ kH Z 2 | |
0 Ω Γ
t t
C + αuµ dσ + µρ µ dxds
t t t
≤ Z Z Z Z | |
0 Γ 0 Ω
t t
C + αu(t)µ(t)dσ αu µdσds + µρ µ dxds
t t t
≤ Z − Z Z Z Z | |
Γ 0 Γ 0 Ω
C t t
+ γ µ(t) 2 + µ(s) 2 ds + µ ρ µ dxds
≤ γ k kV Z k kV Z Z | || t|| t|
0 0 Ω
C t ε t
+γ µ(t) 2 + µ(s) 2 ds + µ (s) 2 ds
≤ γ k kV Z k kV 2Z k t kH
0 0
t
+ C ρ (s) 2 µ(s) 2 ds
Z k t kL4(Ω)k kL4(Ω)
0
C ε t
+γ µ(t) 2 + µ (s) 2 ds
≤ γ k kV 2Z k t kH
0
t
+ C 1+ ρ (s) 2 µ(s) 2 ds. (2.28)
Z k t kV k kV
0 (cid:0) (cid:1)
8 Optimal Boundary Control of a Nonstandard Phase Field System
Hence, using (2.26), choosing γ > 0 sufficiently small, and invoking Gronwall’s lemma,
we can conclude that
µ H1(0,T;H)∩L∞(0,T;V) C. (2.29)
k k ≤
Sixth estimate. Since 0 < ρ < 1 a.e. in Q, and using (2.26), (2.29) and the continuity
of the embedding V L4(Ω), we can estimate as follows:
⊂
(ε+2ρ)µt +µρt L2(Q) C µt L2(Q) + µ L∞(0,T;L4(Ω)) ρt L2(0,T;L4(Ω))
k k ≤ k k k k k k
C µt L2(Q) + µ L∞(0,T;V) ρt L2(0,T;V) C. (2.30)
≤ k k k k k k ≤
(cid:0) (cid:1)
Comparison in (2.8) then shows the boundedness of ∆µ in L2(Q), and it follows from
(2.8), (A3) and standard elliptic estimates that also
µ C. (2.31)
k kL2(0,T;H3/2(Ω)) ≤
Step 3: Conclusion of the proof. Collecting all the above estimates, it turns out that
there is some sequence τ 0 such that
k
ց
µτk µ weakly star in
→
H1(0,T;H) L∞(0,T;V) L2(0,T;H3/2(Ω)),
∩ ∩
ρτk ρ weakly star in W1,∞(0,T;H) H1(0,T;V) L∞(0,T;W),
→ ∩ ∩
f′(ρτk) ξ weakly star in L∞(0,T;H).
1 →
Thanks to the Aubin-Lions lemma (cf., [14, Thm. 5.1, p. 58]) and similar results to be
found in [17, Sect. 8, Cor. 4], we also deduce (recall that even H3/2(Ω) is compactly
embedded into V ) the strong convergences
µτk µ strongly in C0([0,T];H) L2(0,T;V),
→ ∩
ρτk ρ strongly in C0([0,T];V)
→
and the Cauchy conditions (1.4) as a consequence. In particular, employing a standard
monotonicity argument (cf., e.g., [1, Lemma 1.3, p. 42]), we conclude that 0 < ρ <
1 and ξ = f′(ρ) a.e. in Q. The strong convergence shown above also entails that
1
f′(ρτk) f′(ρ) strongly in C0([0,T];H) (because f′ is Lipschitz continuous), and that
2 → 2 2
µτk µ strongly in L2(Q).
Tτk →
Now notice that the above convergences imply, in particular, that
ρτk ρ strongly in C0([0,T];L6(Ω)),
→
ρτk ρ weakly in L2(0,T;L4(Ω)),
t → t
µτk µ strongly in L2(0,T;L4(Ω)),
→
µτk µ weakly in L2(Q).
t → t
Colli — Gilardi — Sprekels 9
From this, it is easily verified that
µτk ρτk µρ weakly in L1(0,T;H),
t → t
ρτk µτk ρµ weakly in L2(0,T;L3/2(Ω)).
t → t
Now, we are ready to take the limit as k in (2.8)–(2.10) (written for τ = τ ).
k
→ ∞
Precisely, we can do that as far as ρ is concerned, while it is easier to take the limit in
the variational formulation of (2.8) that accounts for the boundary condition (the same
as mentioned in Remark 2.2), or in the following integrated-in-time version of it
T T
[(ε+2ρτ)µτ +µτρτ] vdxdt+ µτ vdxdt
Z Z t t Z Z ∇ ·∇
0 Ω 0 Ω
T
+ α(µτ u)vdσdt = 0 for every v L∞(0,T;V).
Z Z − ∈
0 Γ
Then, we obtain the analogue for µ, which implies (1.1) and (1.3) .
2
3 Boundedness, uniqueness, and stability
In this section, we derive results concerning boundedness, uniqueness and stability of
the solutions to system (1.1)–(1.4). With respect to boundedness, we have the following
result, which resembles Theorem 2.3 in [5].
Theorem 3.1 Suppose that (A1)–(A4) are fulfilled, and suppose that the following
conditions are satisfied:
(A5) µ L∞(Ω), inf ρ (x) > 0, sup ρ (x) < 1.
0 0 0
∈ x∈Ω x∈Ω
(A6) u L∞(Σ).
∈
Then any solution (ρ,µ) of (1.1)–(1.4) fulfilling (2.3)–(2.6) also satisfies
µ µ∗, ρ ρ , and ρ ρ∗ a.e. in Q (3.1)
∗
≤ ≥ ≤
for some constants µ∗ > 0 and ρ ,ρ∗ (0,1) that depend on the structure of the system
∗
∈
and T , on the initial data, and on an upper bound for the L∞ norm of u, only.
Proof. Let usjustshow theboundedness of µ andthefirst estimate(3.1); theresults for
ρ then follow in exactly the same manner as in the proof of Theorem 2.3 in [5]. Also the
result for µ follows – up to some changes that are necessary due to the different boundary
condition for µ – bythe samechain ofarguments asintheproofofTheorem 2.3 in[5]; but
since this proof does not seem to be standard, we provide it for the reader’s convenience.
So let (ρ,µ) be any solution to the system (1.1)–(1.4), (2.3)–(2.6). We set
Φ0 := max 1, µ0 L∞(Ω), u L∞(Σ) ,
{ k k k k }
choose any k IR such that k Φ , and introduce the auxiliary function χ L∞(Q)
0 k
∈ ≥ ∈
by putting, for almost every (x,t) Q,
∈
χ (x,t) = 1 if µ(x,t) > k, and χ (x,t) = 0 otherwise.
k k
10 Optimal Boundary Control of a Nonstandard Phase Field System
Then, we test (1.1) by (µ k)+. We obtain, for any t [0,T],
− ∈
ε t
+ρ(t) (µ(t) k)+ 2 + (µ k)+ 2dxds
ZΩ(cid:16)2 (cid:17)| − | Z0ZΩ|∇ − |
t
+ α(µ u)(µ k)+dσds
Z Z − −
0 Γ
t t
= ρ (µ k)+ 2dxds ρ µ(µ k)+dxds
t t
Z Z | − | − Z Z −
0 Ω 0 Ω
t
= k ρ (µ k)+dxds.
t
− Z Z −
0 Ω
Now observe that α and ρ are nonnegative and that, by definition of k,
α(µ u)(µ k)+ = α (µ k)+ 2 + (k u)(µ k)+ 0 a.e. in Q.
− − | − | − − ≥
(cid:0) (cid:1)
Hence, using H¨older’s inequality, we obtain from the above equality the estimate
ε t
(µ(t) k)+ 2 + (µ k)+ 2dxds
2 k − kH Z Z |∇ − |
0 Ω
t
k χ (s) ρ (s) (µ k)+(s) ds,
≤ Z k k kL7/2(Ω)k t kL14/3(Ω)k − kL2(Ω)
0
whence, using the Gronwall-Bellman lemma as in [2, Lemma A.4, p. 156],
T 1/2
ε (µ k)+ 2 + (µ k)+ 2dxdt
(cid:16) k − kC0([0,T];H) Z0 ZΩ|∇ − | (cid:17)
k T
χ (t) ρ (t) dt
≤ √ε Z k k kL7/2(Ω)k t kL14/3(Ω)
0
k
ρ χ . (3.2)
≤ √ε k tkL7/3(0,T;L14/3(Ω))k kkL7/4(0,T;L7/2(Ω))
Next, we apply the continuity of the embedding V L6(Ω) and the interpolation in-
⊂
equality (1.6) with p = 2, q = 6, r = 14/3, and θ = 1/7. It follows that
T 3/7
ρ ρ (t) 1/3 ρ (t) 2 dt
k tkL7/3(0,T;L14/3(Ω)) ≤ (cid:16)Z0 k t kL2(Ω)k t kL6(Ω) (cid:17)
T 3/7
ρ 1/7 ρ (t) 2 dt C ρ 6/7 D ,
≤ k tkL∞(0,T;H)(cid:16)Z0 k t kL6(Ω) (cid:17) ≤ k tkL2(0,T;V) ≤ 0
where D is a positive constant depending only on the data of the problem. Moreover,
0
we have
T 1/2 4/7
χ = χ (x,t) 7/2dx dt
k kkL7/4(0,T;L7/2(Ω)) hZ0 (cid:16)ZΩ| k | (cid:17) i
T 1/2 1·8
= χ (x,t) 4dx dt 2 7 = χ 8/7 .
hZ0 (cid:16)ZΩ| k | (cid:17) i k kkL2(0,T;L4(Ω))