Microscopic Coupling

5.2 Microscopic Solvability 83

84 5 SOLVABILITY OF THE TWO SCALE MODEL Consider in the following a possibly reduced time interval I_τ = [0, τ] with 0 < τ ≤ T and set

M_τ,α :=n

φ∈C^α(I_τ, C_per² (Y))

φ(t, y)>0 ∀t ∈I_τ, ∀y ∈Yo . Define

B_R,τ = n

(φ, c_s)∈

C^α(I_τ, C_per² (Y))²

kφkC^α(I_τ,C²(Y))+kcskC^α(I_τ,C²(Y)) ≤R, (φ, c_s)(0,·) = (φi ni, c_{s,i ni})o , for R > max{kφi nik_C²_(Y),kcs,i nik_C²_(Y)}. B_R,τ is a nonempty and closed (with re-spect to the C^α(Iτ, C²(Y))-norm) set. Note, that BR,τ is in general not a subset of M_τ,α×C^α(I_τ, C_per² (Y)), sinceφ(t, y)>0 is not necessarily fulfilled.B_R,τ complies with that only for certain choices of R and τ:

Proposition 5.22 (Well-definedness of S on B_R,τ). Suppose φ_{i ni}(y)>0 for all y ∈Y. For any R >max{kφi nik_C²_(Y),kcs,i nik_C²_(Y)} there exist a time τ₁ >0, depending on R, such that

B_R,τ1 ⊂M_τ1,α×C^α(I_τ1, C_per² (Y)).

Proof. It is to show that (φ, c_s) ∈ B_R,τ1 implies φ(t, y) > 0 for all y ∈ Y and t ≤ τ₁, with a suitable τ1>0.

Consider an arbitrary but fixed R > max{kφi nikC²(Y),kcs,i nikC²(Y)} and suppose (φ, c_s)∈B_R,τ. Then, φ is α–Hölder continuous in time with Hölder constant ≤ R.

If φ_{i ni}(y) > 0 for all y ∈ Y, then there exists d := min_y∈Y φ_{i ni}(y) > 0, because Y is compact. So,

|φ(t, y)−φi ni(y)| ≤Rt^α,

and thus φ(t, y)>0 for all y ∈Y and t ≤τ₁:= _2R^{d 1/α} . Proposition 5.22 ensures that the operator

S: B_R,τ1 →[C_per^1,2+2α(I×Y)]²

is well defined. Furthermore, there is a configuration of R and τ, such that S maps BR,τ into itself:

Proposition 5.23 (Self-mapping). Suppose φ_{i ni}(y) > 0 for all y ∈ Y. There exist postive numbers R₀>0 and τ₂ >0 such that

S: B_R0,τ2 →B_R0,τ2.

R₀ and τ₂ depend on the macroscopic coupling data, the initial data and the boundary data for the Stokes system and the elasticity equation.

5.2 Microscopic Solvability 85 Proof. Suppose 0 < τ ≤ τ₁, with τ₁ from Proposition 5.22, and ( ˜φ,c˜_s)∈B_R,τ. Set (φ, c_s) =S( ˜φ,c˜_s). By construction of S it is (φ, c_s)(0,·) = (φi ni, c_{s,i ni}).

It remains to show thatkφkC^α(I_τ,C²(Y))+kcskC^α(I_τ,C²(Y)) ≤R: The a priori estimates for the single parts of the problem, see Theorems 5.8, 5.14 and 5.19, imply that

kφkC^1,2+2α(I_τ×Y)+kcskC^1,2+2α(I_τ×Y)

≤c(κ)

1 +k˜csk²_C(Iτ,C2(Y))+kC^Vk²_C(Iτ)+k∇xVk²_C(Iτ) +kPk²_C(Iτ)+kbk²

C(I_τ,W_r^2−1/r(Y×{0}))+kφi nikC^2+2α(Y)+kcs,i nikC^2+2α(Y)

, (5.76) where κ is an upper bound for kφk˜ C(I_τ,C²(Y)). For any τ ≤ τ₁, κ can be choosen in-dependently of φ˜ and R: Due to φ˜ ∈ C^α(I_τ, C_per² (Y)) and τ ≤ τ₁ = _2R^{d 1/α}

, with d := min_y∈Y φ_{i ni}(y), it is for 0≤t ≤τ

kφ(t, y)˜ −φ_{i ni}(y)k_C²_(Y) ≤Rt^α ≤ d 2, and thus

sup

t∈Iτ

kφ(t)k˜ C²(Y) ≤ kφi nikC²(Y)+ d 2 =:κ.

So, the constantc in (5.76) can be choosen independently of φ˜and R, and (5.76) can be written as

kφkC^1,2+2α(I_τ×Y)+kcskC^1,2+2α(I_τ×Y)

≤c₁(C^V, V, P, b, φ_{i ni}, c_{s,i ni}) +c₂k˜c_sk²_C(Iτ,C2(Y)), (5.77) where

c₁(C^V, V, P, b, φ_{i ni}, c_{s,i ni}) =c

1 +kC^Vk²_C(I)+k∇xVk²_C(I)+kPk²_C(I)+kφi nikC^2+2α(Y)

+kcs,i nik_C^2+2α_(Y)+kbk²

C(I,W_r^2−1/r(Y×{0}))

. Next, note that

kc˜_skC(Iτ,C²(Y)) = max

t∈I_τ kc˜_s(t)−c_{s,i ni}+c_{s,i ni}kC²(Y) ≤Rτ^α+kcs,i nikC²(Y). Consequently, (5.77) becomes

kφkC^1,2+2α(I_τ×Y)+kcskC^1,2+2α(I_τ×Y) ≤c˜₁(C^V, V, P, b, φ_{i ni}, c_{s,i ni}) + ˜c₂R²τ^2α, (5.78) with

˜

c1(C^V, V, P, b, φi ni, cs,i ni) =c1(C^V, V, P, b, φi ni, cs,i ni) + ˜c2kcs,i nik²_C2(Y).

86 5 SOLVABILITY OF THE TWO SCALE MODEL The continuous embedding

C^1,2+2α(I_τ ×Y),→C^α(I_τ, C²(Y)),

see Lemma 2.12, implies together with (5.78) that

kφkC^α(I_τ,C²(Y))+kcskC^α(I_τ,C²(Y)) ≤c₃ kφkC^1,2+2α(I_τ×Y)+kcskC^1,2+2α(I_τ×Y)

≤c₃c˜₁(C^V, V, P, b, φ_{i ni}, c_{s,i ni}) +c₃c˜₂R²τ^2α. (5.79) Choose now R0 := 2˜c1c3 and τ2 >0 such thatc˜2c3R0τ₂^2α ≤ ¹₂, i.e.τ2≤(2˜c2c3R0)^−2α1 . It follows from (5.79) that

kφkC^α(Iτ2,C²(Y))+kcskC^α(Iτ2,C²(Y)) ≤R0.

Finally, there is a choice of R and τ such that S: B_R,τ →B_R,τ

is a strict contraction:

Proposition 5.24 (Contraction). Consider R₀ from Proposition 5.23. There exists a number τ₃>0 such that the operator

S: B_R0,τ3 →B_R0,τ3 is a strict contraction.

Proof. Suppose 0 < τ ≤ τ₂, with τ₂ from Proposition 5.23, and ( ˜φ_i,c˜_s,i)∈B_R0,τ, i = 1,2. Set(φi, cs,i) = S( ˜φi,c˜s,i). The continuity estimates of Lemmata 5.9, 5.15 and 5.21 (with C₁^V =C₂^V) imply that

kφ1−φ₂kC^1,2+2α(I_τ×Y)+kcs,1−c_s,2kC^1,2+2α(I_τ×Y)

≤c(κ)kuˆ₁+ ˆu₂k_C(I,W2

r( ˆQs)) kφ˜₁−φ˜₂kC(I_τ,C²(Y))+k˜c_s,1−c˜_s,2kC(I_τ,C²(Y))

, where uˆ_i = Sel asti c ◦ SStok es( ˜φ_i,c˜_s,i) and κ ≥ max{kφ˜₁kC(I_τ,C²(Y)),kφ˜₂kC(I_τ,C²(Y))}. As seen in the proof of Proposition 5.23, κ can be choosen independently of φ˜_i ∈B_R0,τ, if τ ≤τ1, with τ1 from Proposition 5.22. This is satisfied here. Furthermore, the a priori estimates of Theorems 5.8 and 5.14 yield together with ( ˜φ_i,c˜_s,i)∈B_R0,τ

kuˆ1+ ˆu2k_C(Iτ,W2

r( ˆQs)) ≤c1(κ, C^V,∇V, P, b)

+c₂(κ) k˜c_s,1kC(I_τ,C²(Y))+k˜c_s,2kC(I_τ,C²(Y))

≤c1(κ, C^V,∇V, P, b) + 2c2(κ)R₀

≤c(φ_{i ni}, c_{s,i ni}, C^V,∇V, P, b, R0).

5.2 Microscopic Solvability 87 This leads to

kφ1−φ₂kC^1,2+2α(I_τ×Y)+kcs,1−c_s,2kC^1,2+2α(I_τ×Y)

≤c kφ˜₁−φ˜₂k_C(Iτ,C²_(Y))+k˜c_s,1−c˜_s,2k_C(Iτ,C²_(Y)) ,

with a constant c only depending on initial, boundary and macroscopic coupling data.

By construction, it is ( ˜φ₁−φ˜₂)(0, y) = (˜c_s,1−c˜_s,2)(0, y) = 0, and since φ˜_i and c˜_s,i belong toC^α(Iτ, C²(Y)), it follows

kφ˜₁−φ˜₂kC(I_τ,C²(Y)) ≤τ^αkφ˜₁−φ˜₂kC^α(I_τ,C²(Y)), kc˜_s,1−c˜_s,2kC(Iτ,C²(Y)) ≤τ^αk˜c_s,1−c˜_s,2kC^α(Iτ,C²(Y)), and consequently

kφ1−φ₂kC^1,2+2α(I_τ×Y)+kcs,1−c_s,2kC^1,2+2α(I_τ×Y)

≤c τ^α kφ˜₁−φ˜₂kC^α(I_τ,C²(Y))+k˜c_s,1−c˜_s,2kC^α(I_τ,C²(Y))

. The continuous embedding

C^1,2+2α(I_τ ×Y),→C^α(I_τ, C²(Y)), see Lemma 2.12, implies

kφ1−φ2kC^α(Iτ,C²(Y))+kcs,1−cs,2kC^α(Iτ,C²(Y))

≤c τ˜ ^α kφ˜1−φ˜2kC^α(Iτ,C²(Y))+k˜cs,1−c˜s,2kC^α(Iτ,C²(Y))

. Choose now τ₃ such that c τ˜ ₃^α <1. This finishes the proof.

So at last, everything is prepared to prove the solvability of the coupled microscopic problem as the most important result in section 5.2. It is formulated in the following theorem:

Theorem 5.25(Existence and uniqueness of a solution of the coupled microscopic prob-lem). SupposeC^V(x ,·),∇xV(x ,·), P(x ,·)∈C(Iτ₃)forx ∈S0, withτ3from Proposition 5.24. Assume furthermore that

b(x ,·,·)∈C(I_τ3, W_r,per^2−1/r(Y × {0})), φ_{i ni}(x ,·), cs,i ni(x ,·)∈C_per^2+2α(Y),

withφ_{i ni}(x , y)>0 for all y ∈Y. Then, there exists a unique solution(φ, c_s, v , p, u)(x) of (3.33) – (3.42) with

φ(x), c_s(x)∈C_per^1,2+2α(I_τ3 ×Y), ˆ

v(x)∈C(Iτ₃,[W_r,per² ( ˆQl K)]³), ˆ

p(x)∈C(I_τ3, W_r,per¹ ( ˆQ_{l K})), ˆ

u(x)∈C(I_τ3,[W_r,per² ( ˆQ_s)]³),

88 5 SOLVABILITY OF THE TWO SCALE MODEL for some 0< α < ¹₂, r > _1−2α⁶ . This solution satisfies the a priori estimate

kφ(x)kC^1,2+2α(I_τ

3×Y)+kcs(x)kC^1,2+2α(I_τ

3×Y)+kˆv(x)k_C(I

τ3,W_r,per² ( ˆQ_{l K}))

+kˆp(x)k_C(Iτ

3,W_r,per¹ ( ˆQ_{l K}))+ku(xˆ )k_C(Iτ

3,W_r,per² ( ˆQs))

≤c

1 +kC^V(x)k²_C(Iτ

3)+k∇xV(x)k²_C(Iτ

3)+kP(x)k²_C(Iτ

3)

+kb(x)k²

C(Iτ3,W_r^2−1/r(Y×{0}))+kφi ni(x)k_C^2+2α_(Y)+kcs,i ni(x)k²_C2+2α(Y)

.

(5.80)

Proof. The assumptions for Banach’s fixed point theorem, see 2.18, on the operator S: B_R0,τ3 →B_R0,τ3

are fulfilled and so, there exists a unique fixed point in B_R0,τ3. Any fixed point(φ, c_s) of S, together with

(ˆv ,p) =ˆ SStokes(φ, c_s), uˆ=Selastic(ˆv ,p, φ)ˆ

solves (3.33) – (3.42). Due to Theorems 5.8, 5.14 and 5.19, it is φ(x), c_s(x) ∈C_per^1,2+2α(I_τ3×Y).

As seen in Proposition 5.23, (φ, c_s) satisfy kφkC^1,2+2α(I_τ3×Y)+kcskC^1,2+2α(I_τ3×Y) ≤c R₀,

and by the definition of R₀ in the proof of Proposition 5.23 and the estimates of Lemmata 5.8 and 5.14 it follows (5.80).

It remains to show that the just found solution is unique not only in B_R0,τ3, but also in M×C(Iτ₃, C_per² (Y)). Suppose therefore that(φ_i, c_s,i)∈M×C(Iτ₃, C_per² (Y)),i = 1,2, are fixed points of S. Note, that(φ_i, c_s,i) also belong toC_per^1,2+2α(I_τ3×Y), due to Theorems 5.8, 5.14 and 5.19, and thus to C^α(I_τ3, C_per² (Y)). Since (φ_i, c_s,i) satisfy the same initial condition, it isk(φ1−φ₂)(0)kC²(Y) =k(cs,1−c_s,2)(0)kC²(Y) = 0. As seen in Proposition 5.24, the estimate

kφ1−φ₂kC^α(I_τ,C²(Y))+kcs,1−c_s,2kC^α(I_τ,C²(Y))

≤c(κ)τ^αkuˆ₁+ ˆu₂k_C(Iτ,W2

r( ˆQs)) kφ1−φ₂kC^α(I_τ,C²(Y))+kcs,1−c_s,2kC^α(I_τ,C²(Y))

, is satisfied for any τ ∈ I_τ3, where κ ≥ max{kφ1kC(I_τ

3,C²(Y)),kφ2kC(I_τ

3,C²(Y))}. It follows that k(φ1−φ₂)(t)k_C²_(Y) =k(cs,1−c_s,2)(t)k_C²_(Y) = 0 for t ∈[0, τ], if

c(κ)τ^αkuˆ₁+ ˆu₂k_C(I

τ,W_r²( ˆQ_s)) <1.

Repeat the argument to show thatk(φ1−φ₂)(t)kC²(Y) =k(cs,1−c_s,2)(t)kC²(Y)= 0 not only on [0, τ]: If k(φ1−φ2)(t0)kC²(Y) = k(cs,1−cs,2)(t0)kC²(Y) = 0 for some t0 ∈Iτ₃,

5.2 Microscopic Solvability 89 then it isk(φ1−φ₂)(t)k_C²_(Y) =k(cs,1−c_s,2)(t)k_C²_(Y) = 0 for allt ∈[t₀, t₀+τ(t₀)]∩I_τ3. This shows that the set

I⁰ :=

t ∈I_τ3

k(φ1−φ₂)(t)kC²(Y) =k(cs,1−c_s,2)(t)kC²(Y) = 0

is an open subset of I_τ3, and it is not empty because 0 ∈ I⁰. But since t 7→ k(φ, cs)(t)k_C²_(Y) is continuous, I⁰ is also closed in I_τ3 and therefore I⁰ = I_τ3. This proves uniqueness of the solution of (3.33) – (3.42).

With the statement of Theorem 5.25, the main goal concerning the analysis for the microscopic problem is reached. Until here, a macroscopic pointx ∈S₀ was fixed. This sections ends with necessary preparations for the micro-macro-coupling: Investigation of the regularity with respect to x ∈ S₀ and continuity with respect to the coupling data. The answers are formulated in the following Lemmata:

Lemma 5.26 (Regularity with respect to x ∈ S₀). Suppose that C^V, ∇xV, P, b, φ_{i ni} andc_{s,i ni} depend continuously onx ∈S₀. Then, the solution of (3.33) – (3.42) depends continuously on x ∈S₀ and

kφk_C(S0,C^1,2+2α_(Iτ

3×Y))+kcsk_C(S0,C^1,2+2α_(Iτ

3×Y))+kˆvk_C(I

τ3×S0,W_r,per² ( ˆQ_{l K}))

+kpkˆ _C(I

τ3×S0,W_r,per¹ ( ˆQ_{l K}))+kukˆ _C(I

τ3×S0,W_r,per² ( ˆQ_s))

≤c

1 +kC^Vk²_C(Iτ

3×S₀)+kbk²

C(I_τ3×S₀,W_r^2−1/r(Y×{0})) +k∇xVk²_C(Iτ

3×S₀)

+kPk²_C(Iτ

3×S0)+kφi nikC(S₀,C^2+2α(Y))+kcs,i nik²_C(S

0,C^2+2α(Y))

.

(5.81)

Proof. Suppose for i = 1,2 points x_i ∈ S₀ and set C_i^V = C^V(x_i), φ_i = φ(x_i) and cs,i =cs(xi). Analogously to the proof of Proposition 5.24 it holds for τ ∈Iτ₃

kφ1−φ₂kC^α(I_τ,C²(Y))+kcs,1−c_s,2kC^α(I_τ,C²(Y))

≤c τ^α kφ1−φ₂kC^α(I_τ,C²(Y))+kcs,1−c_s,2kC^α(I_τ,C²(Y))

+kC₁^V−C₂^VkC(Iτ)

, (5.82) with a constantc >0 depending only on the initial, boundary and macroscopic coupling data. As long as c τ^α <1, it follows

kφ1−φ2kC^α(Iτ,C²(Y))+kcs,1−cs,2kC^α(Iτ,C²(Y)) ≤ckC₁^V −C₂^VkC(Iτ).

Repeating these arguments, starting with arbitrary t₀ ∈ I_τ3 as initial time, leads to an analogous estimate as (5.82) on the time interval [t₀, t₀+ τ] ∩I_τ3, with a constant

˜

c depending on kφi(t0)kC^2+2α(Y) and kcs,i(t0)kC^2+2α(Y) instead of kφi(0)kC^2+2α(Y) and kcs,i(0)kC^2+2α(Y). Thanks to the a priori estimate (5.80), the mentioned constant c˜can in fact be choosen independently oft₀, such that

kφ1−φ₂k_C^α_{([t0,t0+τ],C}²_(Y))+kcs,1−c_s,2k_C^α_{([t0,t0+τ],C}²_(Y)) ≤ckC₁^V −C₂^VkC([t₀,t₀+τ]), as long ast0+τ ≤τ3 and c τ <˜ 1. This proves

kφ(x1)−φ(x2)kC^α(Iτ3,C²(Y))+kcs(x1)−cs(x2)kC^α(Iτ3,C²(Y)) ≤ckC^V(x1)−C^V(x2)kC(Iτ3),

90 5 SOLVABILITY OF THE TWO SCALE MODEL with a constantc >0depending only on the initial, boundary and macroscopic coupling data. The right hand side tends to zero for |x1−x₂| →0, sinceC^V is continuous with respect to x. It follows that φ, c_s and, due to Lemmata 5.9, 5.15, also vˆ, pˆand uˆare continuous with respect to x.

Take the maximum with respect to x ∈S0 on both sides of the a priori estimate (5.80) to prove (5.81).

Lemma 5.27 (Continuity with respect to the coupling data). Suppose C₁^V, C₂^V ∈C(I_τ3×S₀) and denote by φ_i, c_s,i,uˆ_i,vˆ_i and pˆ_i, i = 1,2, the corre-sponding solutions of the microscopic problem (3.33) – (3.42). These solutions depend locally Lipschitz continuous on C₁^V and C₂^V, i.e. if kC_i^VkC(Iτ3×S₀) ≤ R, for some R > 0, then

kφ1−φ₂kC(S₀,C^1,2+2α(I_τ

3×Y))+kcs,1−c_s,2kC(S₀,C^1,2+2α(I_τ

3×Y))

+kˆv₁−vˆ₂k_C(I

τ3×S0,W_r,per² ( ˆQ_{l K}))+kˆp₁−pˆ₂k_C(I

τ3×S0,W_r,per¹ ( ˆQ_{l K}))

+kuˆ₁−uˆ₂k_C(I

τ3×S₀,W_r,per² ( ˆQ_s))

≤ckC₁^V −C₂^VkC(I_τ3×S₀),

(5.83)

with a constant c >0 depending on R.

Proof. Consider first fixedx ∈S₀. Analogously to the proof of Lemma 5.26 it holds kφ1(x)−φ₂(x)kC^α(I_τ3,C²(Y))+kcs,1(x)−cs,2(x)kC^α(I_τ3,C²(Y)) ≤ckC₁^V(x)−C₂^V(x)kC(Iτ3), with a constantc >0depending only on the initial, boundary and macroscopic coupling data. The continuity estimates of Lemmata 5.9 and 5.15 then imply

kφ1(x)−φ₂(x)k_C^1,2+2α_(Iτ

3×Y)+kcs,1(x)−c_s,2(x)k_C^1,2+2α_(Iτ

3×Y)

+kˆv₁(x)−vˆ₂(x)k_C(I

τ3,W_r,per² ( ˆQ_{l K}))+kpˆ₁(x)−pˆ₂(x)k_C(I

τ3,W_r,per¹ ( ˆQ_{l K}))

+kuˆ₁(x)−uˆ₂(x)k_C(I

τ3,W_r,per² ( ˆQ_s))

≤ckC₁^V(x)−C₂^V(x)kC(I_τ

3).

Taking the maximum with respect to x ∈S₀ proves (5.83).

Remark 5.28. The proofs of the last two Lemmata work also with less reg-ularity assumptions on C^V as for example C^V ∈ L₂(S₀, C(I_τ3)) and then lead to φ∈L₂(S₀, C^1,2+2α(I_τ3 ×Y)) etc. But note, that the spaces L₂(S₀, C(I_τ3)) and C(I_τ3, L₂(S₀))do not coincide, and it is not clear, how to prove C^V ∈L₂(S₀, C(I_τ3))as a solution of the macroscopic problem.

The only microscopic quantity, which occurs in the macroscopic problem as coupling datum, is the microscopic mean value c¯_s. Therefore, the following Lemma, which is an obvious consequence of the Lemmata 5.26 and 5.27, is stated explicity:

5.2 Microscopic Solvability 91 Lemma 5.29 (On the microscopic mean value c¯_s). It holds

kc¯skC¹(Iτ3,C(S₀)) ≤c

1 +kC^Vk²_C(Iτ

3×S0)+k∇xVk²_C(Iτ

3×S0)+kPk²_C(Iτ

3×S0)

+kbk²

C(I_τ3×S₀,W_r^2−1/r(Y×{0}))+kφi nikC(S₀,C^2+2α(Y))

+kcs,i nik²_C(S

0,C^2+2α(Y))

, and

kc¯_s,1−c¯_s,2kC¹(I_τ3,C(S₀)) ≤ckC₁^V −C₂^VkC(Iτ3×S₀), with the same constantc >0 as in Lemma 5.27.

Proof. Note that Y does not depend on t and thus

∂tc¯s(x , t) =∂t

Z

Y

cs(x , t, y) dy = Z

Y

∂tcs(x , t, y) dy =∂tcs(x , t).

The statements follow from

|f¯(x , t)|= Z

Y

f(x , t, y) dy

≤ kf(x , t)kC(Y)

Z

Y

1 dy , f ∈ {cs, ∂_tc_s}, and |Y|= 1 and estimates (5.81) and (5.83) respectively.

92 5 SOLVABILITY OF THE TWO SCALE MODEL

Im Dokument A two scale model for liquid phase epitaxy with elasticity (Seite 83-92)