The Microscopic Phase Field Model

72 5 SOLVABILITY OF THE TWO SCALE MODEL

5.2 Microscopic Solvability 73 Proposition 5.16. Suppose f(φ) =−cos(2πφ) and q is given by (3.43). Then f⁰ and ˆ

q satisfy the growth condition

|f⁰(φ)|+|q(φ, cˆ _s,uˆ)| ≤c(κ) 1 +|∇u|ˆ²+|cs|+|φ|

, (5.37)

and the Lipschitz condition

|f⁰(φ₁)−f⁰(φ₂)|+|q(φˆ ₁, c_s,1,uˆ₁)−q(φˆ ₂, c_s,2,uˆ₂)|

≤c(κ) (|φ1−φ₂|+|cs,1−c_s,2|+|∇( ˆu₁+ ˆu₂)| |∇( ˆu₁−uˆ₂)|). (5.38) Proof. The growth condition follows directly from the definition of f andq and (5.35).

f⁰ and qˆare Lipschitz continuous with respect toφ and c_s, and due to (5.35) it is

|f⁰(φ₁)−f⁰(φ₂)|+|q(φˆ ₁, c_s,1,uˆ₁)−q(φˆ ₂, c_s,2,uˆ₂)|

≤c(κ) (|φ1−φ₂|+|cs,1−c_s,2|+|e( ˆu₁) :e( ˆu₁)−e( ˆu₂) :e( ˆu₂)|), where c > 0 is independent of φ_i, c_s,i and u_i, but depends, as the constant in (5.37), onκ. Furthermore, it is due to the third binomial formula

|e( ˆu₁) :e( ˆu₁)−e( ˆu₂) :e( ˆu₂)|=|(e( ˆu₁) +e( ˆu₂)) : (e( ˆu₁)−e( ˆu₂))|

=|e( ˆu₁+ ˆu₂) :e( ˆu₁−uˆ₂)|

≤ |∇( ˆu₁+ ˆu₂)| |∇( ˆu₁−uˆ₂)|.

Consequently, qˆ is at least locally Lipschitz continuous with respect to uˆ and (5.38) follows.

In fact, the following proofs only use the abstract conditions (5.37) and (5.38) and not the exact definitions off and q. Thus, all of the following results hold for any functions f and q, which satisfy (5.37) and (5.38).

Consider test functions w₁, w₂ ∈ L₂(I;H_per¹ (Y)), multiply equations (3.40) and (3.41) withw₁ andw₂respectively, integrate by parts and get the followingweak formulation:

Problem 5.17. Find c_s, φ ∈L₂(I;H_per¹ (Y)) with ∂_tc_s, ∂_tφ ∈L₂(I;H_per¹ (Y)⁰) such that the initial conditions (3.42) are satisfied and for every w₁, w₂ ∈ L₂(I;H¹_per(Y)) the following equations hold true:

Z

I

τ ξ²h∂tφ, w₁i+ Z

Y

ξ²∇φ· ∇w1+ (f⁰(φ) + ˆq(φ, c_s,u))ˆ w₁ dy

dt = 0. (5.39) Z

I

h∂tc_s, w₂i+%_sh∂tφ, w₂i +

Z

Y

D_s∇cs · ∇w2+ c_s

τ_s − C^V τ^V

w₂

dy

dt = 0,

(5.40)

Here,h·,·idenotes the dual pairing on H_per¹ (Y). There exists a unique solution with the following properties:

74 5 SOLVABILITY OF THE TWO SCALE MODEL Theorem 5.18 (Existence and uniqueness of a weak solution of the phase field model).

Assume c_{s,i ni} ∈ L₂(Y), φ_{i ni} ∈ L₂(Y) and uˆ ∈ L₂(I,[W_r,per² ( ˆQ_s)]³), for some r ≥ 3.

Furthermore, suppose that the constantsD_s, τ,ξ,h_A,%_s,τ^V andτ_s are positive. Then, the microscopic problem 5.17 at any fixed pointx ∈S0with givenC^V =C^V(·, x)∈L2(I) has a unique solution (φ, c_s). It satisfies

kφ(x)kL∞(I,L₂(Y))+kφ(x)k_L2(I,H¹_(Y))+kcs(x)kL∞(I,L₂(Y))+kcs(x)k_L2(I,H¹_(Y))

≤c(κ)

1 +ku(xˆ )k²_L

2(I,W_r²( ˆQs))+kC^V(x)kL₂(I)

+kcs,i ni(x)kL₂(Y)+kφi ni(x)kL₂(Y)

,

(5.41)

with κ from (5.1).

Proof. Investigate first the regularity of the coupling term. Due to the growth condition (5.37), the crucial coupling term to study is |∇u|ˆ², which has to be understood in the trace sense as explained above. Suppose uˆ∈[W_r,per² ( ˆQ_s)]³, for fixed t, then :

ˆ

u ∈[W_r²( ˆQ_s)]³ ⇒ |∇u|ˆ² ∈ W_{r /2}¹ ( ˆQ_s)

⇒ tr|∇u|ˆ² ∈ W_{r /2}^1−2/r(Y)

⇒ tr|∇u|ˆ² ∈ L₂(Y), ifr ≥3.

For the norms there holds

k|∇u|ˆ²kL₂(Y) ≤ck|∇ˆu|²k_W^1−2/r

r /2 (Y) ≤ck|∇u|ˆ²k_W1 r /2( ˆQ_s)

≤ck∇ˆuk²_W1

r( ˆQs) ≤ckuˆk²_W2 r( ˆQs),

(5.42)

which follows from the continuity of the trace operator W_{r /2}¹ (Q_s) → W_{r /2}^1−2/r(Y) and the continuity of the embedding W_{r /2}^1−2/r(Y),→L₂(Y), with r ≥3.

The proof of the theorem will be performed in several steps:

Step 1: Solve a linearized problem.

For fixed c˜_s,φ˜∈L₂(I×Y), replacef⁰(φ) + ˆq(φ, c_s,uˆ) in (5.39) by F( ˜c_s,φ) :=˜ f⁰( ˜φ) + ˆq( ˜φ,c˜_s,u).ˆ

This leads to the following linearized problem:

Find c_s, φ∈L₂(I;H_per¹ (Y)) with∂_tc_s, ∂_tφ∈L₂(I;H_per¹ (Y)⁰) such that the initial condi-tions (3.42) are satisfied and for every w1, w2 ∈L2(I;H_per¹ (Y))the following equations

5.2 Microscopic Solvability 75 hold true:

Z

I

τ ξ²h∂tφ, w₁i+

Z

Y

ξ²∇φ· ∇w1dy dt

=− Z

I×Y

F(˜cs,φ)w˜ 1dydt,

(5.43)

Z

I

h∂tc_s, w₂i+

Z

Y

D_s∇cs · ∇w2+ _τ¹

sc_sw₂ dy

dt

= Z

I

Z

Y

C^V

τ^Vw₂dy −%_sh∂tφ, w₂i

dt.

(5.44)

Note, thatF is Lipschitz continuous with respect to c_s and φ, see (5.38), and satisfies the growth condition

|F(c_s, φ)| ≤c(κ) 1 +|∇uˆ|²+|cs|+|φ|

,

see (5.37). Therefore, c˜_s,φ˜∈L₂(I×Y) implies F(˜c_s,φ)˜ ∈L₂(I×Y). Equation (5.43) decouples from (5.44). It is, for given c˜s and φ, a weak formulation of a linear heat˜ equation forφ, independent ofc_s. There exists a unique solutionφ∈L₂(I, H¹_per(Y))of (5.43) with∂_tφ∈L₂(I, H_per¹ (Y)⁰), see [43], Theorem 11.3, p.382.

By the same reference, there is a unique solution c_s ∈ L₂(I, H_per¹ (Y)) with

∂_tc_s ∈L₂(I, H_per¹ (Y)⁰) of (5.44), with the just found ∂_tφ on the righthand side.

Step 2: Estimates for the linearized problem.

Suppose c˜s,i,φ˜i ∈L2(I×Y), i = 1,2, and let cs,i, φi be the corresponding solutions of (5.43), (5.44). Then the functions c˘_s :=c_s,1−c_s,2 and φ˘:=φ₁−φ₂ are solutions of

Z

I

τ ξ²

h∂tφ, w˘ ₁i+

Z

Y

ξ²∇φ˘· ∇w1)dy

dt

=− Z

I×Y

F(˜c_s,1,φ˜₁)−F(˜c_s,2,φ˜₂)

w₁dy dt,

(5.45)

Z

I

h∂tc˘_s, w₂i+

Z

Y

D_s∇˘c_s · ∇w2+ _τ¹

sc˘_sw₂

dy

dt

=− Z

I

%_sh∂tφ, w˘ ₂idt,

(5.46)

withφ(0) = ˘˘ c_s(0) = 0. For z ∈ {˘c_s,φ}, it is˘ ∂_tkz(t)k²_L

2(Y) = 2h∂tz , zi(t) and thus Z t₀

0

h∂tz , zidt = 1

2 kz(t₀)k²_L2(Y)− kz(0)k²_L2(Y)

= 1

2kz(t0)k²_L2(Y), (5.47) for 0 < t₀ ≤ T. Set I_t0 = [0, t₀]. Taking w₁ = χ_It

0(φ₁−φ₂) in (5.45) and using the Lipschitz continuity of F, equation (5.47) and Young’s inequality (2.1) yield

k(φ1−φ2)(t0)k²_L2(Y)+k∇(φ1−φ2)k²_L2(It

0×Y)

≤c

kφ1−φ₂k²_L

2(It0×Y)+kc˜_s,1−c˜_s,2k²_L

2(It0×Y)+kφ˜₁−φ˜₂k²_L

2(It0×Y)

. (5.48)

76 5 SOLVABILITY OF THE TWO SCALE MODEL This estimate also holds, if the gradient term on the left-hand side is neglected. Gron-wall’s inequality (2.4) then implies

kφ1−φ₂kL∞(I,L₂(Y)) ≤c k˜c_s,1−c˜_s,2kL₂(I×Y)+kφ˜₁−φ˜₂kL₂(I×Y)

. (5.49)

Due to the continuous embedding L_∞(I, L₂(Y)) ,→ L₂(I×Y), it follows from (5.48) and (5.49)

kφ1−φ₂k_L2(I,H¹_(Y)) ≤c k˜c_s,1−c˜_s,2kL₂(I×Y)+kφ˜₁−φ˜₂kL₂(I×Y)

, and with (5.45)

k∂t(φ₁−φ₂)k_L2(I,H¹_(Y)⁰₎ ≤c k˜c_s,1−c˜_s,2kL₂(I×Y)+kφ˜₁−φ˜₂kL₂(I×Y)

. (5.50) Setting w2 = χI_t

0(cs,1 −cs,2) in (5.46), it follows again with Young’s inequality with ε >0 (2.2)

k(cs,1−c_s,2)(t₀)k²_L2(Y)+kcs,1−c_s,2k²_L

2(I_t0,H¹(Y))

≤ Z

It0

h∂t(φ₁−φ₂), c_s,1−c_s,2idt

≤c(ε)k∂t(φ1−φ2)k²_L2(It

0,H¹(Y)⁰)+εkcs,1−cs,2k²_L2(It

0,H¹(Y)), which implies together with (5.50) and for ε >0 small enough

kcs,1−c_s,2kL∞(I,L₂(Y))+kcs,1−c_s,2kL₂(I,H¹(Y))

≤c k˜c_s,1−c˜_s,2kL₂(I×Y)+kφ˜₁−φ˜₂kL₂(I×Y)

. (5.51)

An obvious consequence of the estimates (5.49) and (5.51) is kφ1−φ₂kL∞(I,L₂(Y))+kcs,1−c_s,2kL∞(I,L₂(Y))

≤c k˜c_s,1−c˜_s,2kL₂(I×Y)+kφ˜₁−φ˜₂kL₂(I×Y)

. (5.52)

Step 3: Solve the original semi–linear problem using a fixed point argument.

Define the solution operator

F: [L_∞(I, L₂(Y))]²→[L_∞(I, L₂(Y))]²: (˜c_s,φ)˜ 7→(c_s, φ),

which maps given (˜c_s,φ)˜ to the corresponding solutions of (5.43), (5.44). Note, that every function w ∈L_∞(I, L₂(Y))satisfies

kwkL₂(I,L₂(Y)) ≤T^1/2kwkL∞(I,L₂(Y)). This implies, together with estimate (5.52):

kφ1−φ₂kL∞(I,L2(Y))+kcs,1−c_s,2kL∞(I,L2(Y))

≤c T^1/2 kφ˜₁−φ˜₂kL∞(I,L₂(Y))+k˜c_s,1−c˜_s,2kL∞(I,L₂(Y))

.

5.2 Microscopic Solvability 77 Choose 0 < τ₁ ≤ T small enough, such that c τ₁^1/2 < 1. Then, restricted to the time intervalI_τ1 := [0, τ₁], the operator

F: [L_∞(I_τ1, L₂(Y))]² →[L_∞(I_τ1, L₂(Y))]²

is a contraction. Banach’s fixed point Theorem proves the existence of a unique solution (c_s, φ) of (5.39), (5.40) on the possibly reduced time interval [0, τ₁]. Since the choice of τ₁ is independent of the solution (c_s, φ) and its intial data (c_{s,i ni}, φ_{i ni}), finitely many repetitions of this arguments, with (c_s, φ)(τ₁) replacing the initial data, prove the existence of a solution on the whole time interval [0, T].

Step 4: A priori estimate.

Proceed analogously to Step 2: Set w₁ = χ_It

0φ in (5.39). Use the growth condition (5.37) onf⁰ and qˆ

|f⁰(φ)|+|q(φ, cˆ _s,uˆ)| ≤c(κ) 1 +|∇u|ˆ²+|cs|+|φ|

.

Most of the following constants depend onκ, but for readability, this will be omitted in the notation for the moment. It follows

kφ(t0)k²_L2(Y)+k∇φk²_L2(It

0×Y) ≤c

1 +kφk²_L2(It

0×Y)+kcsk²_L2(It

0×Y)

+k|∇u|ˆ²k²_L2(It

0×Y)+kφi nik²_L2(Y) .

(5.53) Gronwall’s inequality implies

kφkL∞(It0,L₂(Y)) ≤c 1 +kcskL₂(It0×Y)+k|∇ˆu|²kL₂(It0×Y)+kφi nikL₂(Y)

. (5.54) Estimate (5.53) leads first to

kφk_L2(It

0,H¹(Y)) ≤c 1 +kφkL₂(I_t

0×Y)+kcskL₂(I_t

0×Y)+k|∇u|ˆ²kL₂(I_t

0×Y)+kφi nikL₂(Y)

, (5.55) next with the continuous embedding L_∞(I_t0, L₂(Y)),→L₂(I_t0×Y) and (5.54) to

kφk_L2(It

0,H¹(Y)) ≤c 1 +kcskL₂(I_t

0×Y)+k|∇u|ˆ²kL₂(I_t

0×Y)+kφi nikL₂(Y)

, (5.56) and finally with (5.39) to

k∂tφkL₂(It0,H¹(Y)⁰) ≤c 1 +kcskL2(It0×Y)+k|∇u|ˆ²kL2(It0×Y)+kφi nikL2(Y)

. (5.57) Set now w₂=χ_It

0c_s in (5.40) and use Young’s inequality withε >0 (2.2) to get kcs(t₀)k²_L

2(Y)+kcsk²_L

2(I_t

0,H¹(Y))

≤c

k∂tφk_L2(It

0,H¹(Y)⁰)kcsk_L2(It

0,H¹(Y)) +kcsk²_L

2(I_t0×Y)

+kC^Vk²_L2(It

0)+kcs,i nik²_L2(Y)

≤c

c(ε)k∂tφk²_L

2(It0,H¹(Y)⁰)+εkcsk²_L

2(It0,H¹(Y))+kcsk²_L2(It

0×Y)

+kC^Vk²_L

2(It0)+kcs,i nik²_L

2(Y)

.

(5.58)

78 5 SOLVABILITY OF THE TWO SCALE MODEL Choosing ε small enough allows to cancel the kcsk_L2(It

0,H¹(Y))–term on the right-hand side of (5.58):

kcs(t₀)k²_L2(Y)+kcsk²_L

2(It0,H¹(Y))

≤c

k∂tφk²_L2(It

0,H¹(Y)⁰)+kcsk²_L2(It

0×Y)+kC^Vk²_L2(It

0)+kcs,i nik²_L2(Y) . Estimate (5.57) then yields

kcs(t₀)k²_L2(Y)+kcsk²_L

2(I_t0,H¹(Y))

≤c

1 +kcsk²_L2(It

0×Y)+k|∇ˆu|²k²_L2(It

0×Y)

+kC^Vk²_L

2(It0)+kcs,i nik²_L

2(Y)+kφi nik²_L

2(Y)

,

(5.59)

and thus with Gronwall’s inequality (2.4)

kcskL∞(It0,L₂(Y)) ≤c 1 +k|∇u|ˆ²kL₂(It0×Y)+kC^VkL₂(It0)+kcs,i nikL₂(Y)+kφi nikL₂(Y)

. (5.60) Combining (5.54), (5.56), (5.59) and (5.60) proves

kφkL∞(It0,L2(Y)) +kφkL₂(It0,H¹(Y))+kcskL∞(It0,L2(Y))+kcskL₂(It0,H¹(Y))

≤c

1+kcskL₂(It0×Y)+k|∇u|ˆ²kL₂(It0×Y)

+kC^VkL₂(I_t0)+kcs,i nikL₂(Y)+kφi nikL₂(Y)

.

(5.61)

The embedding estimate

kcskL₂(It0×Y) ≤ckcskL∞(It0,L₂(Y))

together with (5.42), (5.60) and (5.61) finally implies kφkL∞(I_t0,L₂(Y))+kφkL₂(I_t

0,H¹(Y)) +kcskL∞(I_t0,L₂(Y))+kcskL₂(I_t

0,H¹(Y))

≤c

1 +kukˆ ²_L

2(I_t0,W_r²( ˆQ_s))+kC^VkL₂(I_t

0)+kcs,i nikL₂(Y)+kφi nikL₂(Y)

, for any 0 < t₀ ≤T. Keep in mind, that c depends on κ from (5.37).

Theorem 5.19 (Regularity). Suppose 0 < α < ¹₂, r > _1−2α⁶ and φi ni, cs,i ni ∈C_per^2+2α(Y), and consider given C^V ∈ C(I) and u withuˆ∈C(I,[W_r,per² ( ˆQs)]³). A solution (φ, cs) of (3.40), (3.41), (3.42) belongs to [C_per^1,2+2α(I ×Y)]² with

kφ(x)k_C^1,2+2α_(I×Y)+kcs(x)k_C^1,2+2α_(I×Y)

≤c(κ)

1 +kC^V(x)kC(I)+ku(x)kˆ ²_C(I,W2 r( ˆQs))

+kφi ni(x)k_C^2+2α_(Y)+kcs,i ni(x)k_C^2+2α_(Y) ,

(5.62)

with κ from (5.1).

5.2 Microscopic Solvability 79 Proof. As in the proof of Theorem 5.18, start again with analogous considerations on the coupling term:

Suppose uˆ∈[W_r,per² ( ˆQ_s)]³, for fixedt, then:

ˆ

u ∈[W_r²(Q_s)]³ ⇒ |∇u|ˆ² ∈ W_{r /2}¹ (Q_s)

⇒ tr|∇u|ˆ² ∈ W_{r /2}^1−2/r(Y)

⇒ tr|∇u|ˆ² ∈ C^2α(Y), if r > _1−2α⁶ , for some0 < α < ¹₂. For the norms there holds

k|∇u|ˆ²kC^2α(Y) ≤ck|∇u|ˆ²k_W^1−2/r

r /2 (Y) ≤ck|∇ˆu|²k_W1

r /2( ˆQ_s) ≤ck∇ukˆ ²_W1

r( ˆQs) ≤ckukˆ ²_W2 r( ˆQs). The key idea of the following proof is to use regularity results for the linear heat equation with homogeneous Dirichlet boundary conditions, namely Theorem 9.1 of Ch.IV in [35]

and Theorem 5.1.13 in [38], and perform a bootstrap procedure:

Let Ω⊂R² be a bounded domain such that Y ⊂Ω withC^2+2α–smooth boundary ∂Ω.

Let χ ∈ C₀^∞(Ω) be a cut–off function with χ|Y = 1 and 0 ≤ χ(y) ≤ 1 for all y ∈ Ω.

The functionsφandc_s areY-periodic inH¹(Y)which implies that they can be extended periodically to Ω with φ, c_s ∈ H¹(Ω). In the following, consider the functions χφ and χc_s. Ifφ and c_s solve (5.39) and (5.40) onI×Y, thenχφ and χc_s are weak solutions of

τ ξ²∂_t(χφ)−ξ²∆(χφ) =−χ(f⁰(φ) + ˆq(c_s,u, φ))ˆ −ξ²(φ∆χ+ 2∇χ∇φ), (5.63)

∂_t(χc_s)−D_s∆(χc_s) =χ C^V

τ^V − c_s

τ_s −%_s∂_tφ

−D_s(c_s∆χ+ 2∇χ∇cs) (5.64) onI×Ω with homogeneous Dirichlet conditions on I×∂Ωand initial conditions

χcs(0, y) =χcs,i ni(y), χφ(0, y) =χφi ni(y),

wherec_{s,i ni}, φ_{i ni} are also extended periodically toΩ. The weak formulation of (5.63) and (5.64) is analogous to that in (5.39) and (5.40). Fromc_s, φ∈L₂(I, H¹(Ω)) it follows, that the righthand side of (5.63) is in L2(I ×Ω), due to the growth condition (5.37) (This is also true ifL₂ is replaced by any L_µ, 1≤µ≤ ∞). Omit again the dependency onκ in the following notation.

The application of Theorem 9.1 of Ch.IV in [35] yields χφ∈W₂^1,2(I×Ω),

with

kχφk_W^1,2

2 (I×Ω)≤c

1 +kχφkL₂(I,H¹(Ω))+kχcskL₂(I×Ω)

+kχ|∇u|ˆ²kL₂(I×Ω)+kχφi nikC^2+2α(Ω)

.

80 5 SOLVABILITY OF THE TWO SCALE MODEL From definition of χ and the Y–periodicity of the involved functions it follows φ∈W₂^1,2(I ×Y) with

kφk_W^1,2

2 (I×Y) ≤c

1 +kφkL₂(I,H¹(Y))+kcskL₂(I×Y)+kukˆ ²_C(I,W2

r( ˆQs))+kφi nikC^2+2α(Y)

. (5.65) The norms foruˆandφ_{i ni} are not optimal at that point, but will be needed later anyway.

Note, that (5.65) implies∂tφ∈L2(I×Y), so the righthand side of (5.64) is inL2(I×Ω).

Theorem 9.1 of Ch.IV in [35] can now be applied to equation (5.64) and this yields c_s ∈W₂^1,2(I×Y),

with kcsk_W^1,2

2 (I×Y)≤c kcskL₂(I,H¹(Y))+k∂tφkL₂(I×Y)+kC^VkC(I)+kcs,i nikC^2+2α(Y)

. (5.66) For 0< λ <1, there is the interpolatory inclusion

W₂^1,2(I×Y),→W₂^λ(I, W₂^2(1−λ)(Y))

with continuous embedding, see Lemma 2.14. Furthermore, the embeddings W₂^λ(I, W₂^2(1−λ)(Y)),→L_µ(I, W₂^2(1−λ)(Y)), for λ−1

2 ≥ −1 µ, W₂^2(1−λ)(Y),→W_µ¹(Y), for 2(1−λ)−2

2 ≥1− 2 µ, exist and are continuous, see Theorem 2.9, and therefore

W₂^1,2(I×Y),→W₂^λ(I, W₂^2(1−λ)(Y)),→W₄^0,1(I ×Y)

with continuous embedding. It follows that c_s, φ∈W₄^0,1(I×Y) with kφk_W^0,1

4 (I×Y)+kcsk_W^0,1

4 (I×Y)

≤c

1 +kφk_L2(I,H¹_(Y))+kcsk_L2(I,H¹_(Y))+kukˆ ²_C(I,W2

r( ˆQs))

+kC^VkC(I)+kφi nikC^2+2α(Y)+kcs,i nikC^2+2α(Y)

.

(5.67)

Repetition of the same argument for both equations in L₄(I×Ω) instead ofL₂(I×Ω) implies c_s, φ∈W₄^1,2(I×Y), and thusc_s, φ∈W_µ^0,1(I×Y), for all1 ≤µ < ∞, due to the continuous embeddings W₄^1,2(I×Y) ,→ W₄^λ(I, W₄^2(1−λ)(Y)) ,→W_µ^0,1(I×Y). Together with estimate (5.67) it follows

kφk_W^0,1

µ (I×Y)+kcsk_W^0,1

µ (I×Y)

≤c

1 +kφkL₂(I,H¹(Y))+kcskL₂(I,H¹(Y))+kukˆ ²_C(I,W2 r( ˆQ_s))

+kC^VkC(I)+kφi nikC^2+2α(Y)+kcs,i nikC^2+2α(Y)

.

(5.68)

5.2 Microscopic Solvability 81 Another application of Theorem 9.1 of Ch.IV in [35] yieldsc_s, φ∈W_µ^1,2(I×Y) for any 1≤µ <∞, with

kφk_W^1,2

µ (I×Y)+kcsk_W^1,2

µ (I×Y)

≤c

1 +kφkL₂(I,H¹(Y))+kcskL₂(I,H¹(Y)) +kˆuk²_C(I,W2 r( ˆQ_s))

+kC^VkC(I)+kφi nikC^2+2α(Y)+kcs,i nikC^2+2α(Y)

.

(5.69)

Use again the interpolatory inclusion

W_µ^1,2(I×Y),→W_µ^λ(I, W_µ^2(1−λ)(Y)), 0 < λ <1, (5.70) with continuous embedding, see Lemma 2.14. The embeddings

W_µ^λ(I, W_µ^2(1−λ)(Y)),→C(I, W_µ^2(1−λ)(Y)), W_µ^2(1−λ)(Y),→C^1+2α(Y)

exist and are continuous for λ− _µ¹ > 0 and for 2(1−λ)− _µ² >1 + 2α, see Theorem 2.9. It follows that

W_µ^λ(I, W_µ^2(1−λ)(Y)),→C(I, C^1+2α(Y)), (5.71) forµ > _1−2α⁴ ,0 < α < ¹₂, with continuous embedding. So the right-hand side of (5.63) belongs toC^0,2α(I×Ω)and vanishes on the boundary∂Ω, due toχ∈C₀^∞(Ω). Theorem 5.1.13 in [38] yields that χφ∈C^1,2+2α(I×Ω) with

kχφk_C^1,2+2α_(I×Ω)≤c

1+kχφk_C^0,1+2α_(I×Ω)+kχcsk_C^0,2α_(I×Ω) +kχukˆ ²_C(I,W2

r( ˆQ_s))+kχφi nikC^2+2α(Ω)

.

(5.72) Due to the just achieved regularity forφ, the right-hand side of (5.64) is also an element of C^0,2α(I×Ω) and vanishes on the boundary ∂Ω. Consequently, combining Theorem 5.1.13 in [38] with (5.72), it isχc_s ∈C^1,2+2α(I×Ω) with

kχcskC^1,2+2α(I×Ω)≤c

1 +kχφkC^0,1+2α(I×Ω)+kχcskC^0,1+2α(I×Ω)

+kχukˆ ²_C(I,W2

r( ˆQs)) +kC^VkC(I)

+kχφi nik_C^2+2α_(Ω)+kχcs,i nik_C^2+2α_(Ω) .

(5.73)

The estimates (5.72) and (5.73), together with (5.41), (5.69), (5.70), (5.71) and the Y–periodicity of the involved functions imply, that

φ∈C^1,2+2α(I×Y) and c_s ∈C^1,2+2α(I×Y), with

kφkC^1,2+2α(I×Y)+kcskC^1,2+2α(I×Y)

≤c

1 +kC^VkC(I)+kukˆ ²_C(I,W2

r( ˆQ_s))+kφi nik_C^2+2α_(Y)+kcs,i nik_C^2+2α_(Y)

. (5.74) The constant c depends on κ, since the constant in the growth condition (5.37) does, which was used in the proof.

82 5 SOLVABILITY OF THE TWO SCALE MODEL Remark 5.20. Theorem 5.1.13 in [38], used in the previous proof, is an optimal regu-larity result for linear parabolic equations with Dirichlet boundary conditions, where the right-hand side is Hölder-continuous only in space. This is only an interior regularity re-sult as counterexamples show, see [49]. For the boundary regularity, it is only true under the rather restrictive assumption, that the right-hand side vanishes on the boundary for every t ∈I. Fortunately, it is satisfied here due to the cut–off by χ.

Lemma 5.21 (Continuity with respect to the coupling data). Suppose ˆ

u₁,uˆ₂ ∈C(I, W_r,per² ( ˆQ_s)) and C₁^V, C₂^V ∈ C(I). Denote by φ₁, φ₂ and c_s,1, c_s,2 the corresponding solutions of (3.40)- (3.42). The continuity estimate

k(φ1−φ₂)(x)kC^1,2+2α(I×Y)+k(cs,1−c_s,2)(x)kC^1,2+2α(I×Y)

≤c(κ)

k( ˆu₁+ ˆu₂)(x)k_C(I,W2

r( ˆQ_s))k( ˆu₁−uˆ₂)(x)k_C(I,W2 r( ˆQ_s))

+k(C₁^V −C₂^V)(x)kC(I)

,

(5.75)

holds true, with κ from sections 5.2.1 and 5.2.2.

Proof. The proof for the continuity estimate (5.75) is analogous to that for the a priori estimates (5.41) and (5.62) with the following adaptions: If φ₁, φ₂ and c_s,1, c_s,2 solve (3.40)- (3.42) with corresponding uˆ₁,uˆ₂ and C₁^V, C₂^V, then

φ˜:=φ₁−φ₂ and c˜_s := c_s,1−c_s,2 solve

τ ξ²∂_tφ˜−ξ²∆ ˜φ+f⁰(φ₁)−f⁰(φ₂) + ˆq(c_s,1,uˆ₁, φ₁)−q(cˆ _s,2,uˆ₂, φ₂) = 0,

∂_tc˜_s +%_s∂_tφ˜−D_s∆˜c_s +c˜_s

τ_s −C₁^V −C₂^V τ^V = 0, with initial conditions

˜

c_s(0, y) = 0, φ(0, y˜ ) = 0,

In order to imitate the proofs for estimates (5.41) and (5.62), the growth condition (5.37) forf⁰ andq needs to be replaced by the Lipschitz condition (5.38). Furthermore, it is for r > _1−2α⁶

k|∇( ˆu₁+ ˆu₂)| |∇( ˆu₁−uˆ₂)|kC^2α(Y) ≤ck|∇( ˆu₁+ ˆu₂)| |∇( ˆu₁−uˆ₂)|k_W^1−2/r

r /2 (Y)

≤ck|∇( ˆu₁+ ˆu₂)| |∇( ˆu₁−uˆ₂)|k_W1 r /2( ˆQs)

≤ckuˆ₁+ ˆu₂k_W2

r( ˆQ_s)kuˆ₁−uˆ₂k_W2 r( ˆQ_s).

Proceeding as in the proofs for (5.41) and (5.62), using (5.38) instead of (5.37), finishes the proof.

5.2 Microscopic Solvability 83

Im Dokument A two scale model for liquid phase epitaxy with elasticity (Seite 72-83)