pη(ϕ) Dv
2
L2(0,T ;L2)+ kpkL4/3(0,T ;L2)
+ kΨϕ(ϕ)kL4(0,T ;L2)∩L2(0,T ;L6)+ k∇Nσ(ϕ, σ)kL2(0,T ;L2)
+ kdiv(ϕ ⊗ v)kL4/3(0,T ;L3/2)+ kdiv(σ ⊗ v)kL1(0,T ;L1)≤ CAP. (4.61) We next intend to establish the additional regularities and estimates. In particular, this means that the second regularity in (3.35) is already established. Furthermore, in Step 4 have already shown that
ϕ ∈ H1 0, T ; (H1)0 ∩ L2(0, T ; H3) ⊂ H1 0, T ; H−1 ∩ L2(0, T ; H3).
Invoking a result from interpolation theory [5, Thm. 4.10.2] as well as Lemma3.1, we conclude that ϕ ∈ C
[0, T ]; (H−1, H3)1 2,2
= C [0, T ]; H1 which proves the first regularity in (3.35).
5 “Darcy limit” and existence of weak solutions to the (MCHD) system
This section is devoted to the construction of a weak solution to the multiphase Cahn–Hilliard–Darcy system (1.1) in the sense of Definition3.6. This is achieved by an asymptotic technique, where the positive viscosity functions in the system (MCHB) are sent to zero.
Proof of Theorem 3.7. For every n ∈ N, let ηn and λn be viscosity functions as described in Theorem3.7, and let (ϕn, µn, σn, vn, pn) denote a weak solution of the Cahn–Hilliard–Brinkman system (1.1) obtained from Theorem 3.5 with the choices η = ηn and λ = λn. We point out that by this explicit choice, we do not require the axiom of choice, even though the uniqueness of the weak solutions is unknown. We recall that, owing to Theorem3.5, the solutions (ϕn, µn, σn, vn, pn) satisfy the weak formulation (3.34a)–(3.34d) (written for η = ηn and λ = λn) and exhibit the regularities
ϕn∈ H1(0, T ; (H1)0) ∩ C([0, T ]; H1) ∩ L2(0, T ; H3),
σn ∈ W1,43(0, T ; (H1)0) ∩ C([0, T ]; (H1)0) ∩ L∞(0, T ; L2) ∩ L2(0, T ; H1), ϕn|Γ∈ C([0, T ]; L2Γ), σn|Γ∈ L4(0, T ; L2Γ),
µn ∈ L2(0, T ; H1), vn∈ L2(0, T ; H1), pn∈ L43(0, T ; L2), div(ϕn⊗ vn) ∈ L2(0, T ; L32), div(σn⊗ vn) ∈ L1(0, T ; L32).
(5.1)
Since for any fixed n ∈ N, the viscosities ηn and λnare compatible withA3, there exist constants 0 < η0,n<
η1,n and λ∗,n > 0 such that (3.6) is fulfilled. In view of (3.39), we assume (without loss of generality) that η1,n= λ∗,n= 1 for all n ∈ N and we further fix
η0,n:= inf
p∈RL
ηn(p).
To investigate the convergence of the sequence {(ϕn, µn, σn, vn, pn)}n∈N, we first need to derive suitable bounds that are uniform in n.
Step 1: Uniform estimates. In the following, the letter C denotes generic positive constants that do not depend on n. They may still depend on the initial data and the other constants from Section3.3, except for η0,n. We already know from Theorem3.5that
kϕnkL∞(0,T ;H1)∩L2(0,T ;H3)+ kσnkL∞(0,T ;L2)∩L2(0,T ;H1)+ kσnkL4(0,T ;L2Γ)
+ kµnkL2(0,T ;H1)+ kvnkL2(0,T ;L2)+
pηn(ϕn) Dvn
2
L2(0,T ;L2)+ kpnkL4/3(0,T ;L2)
+ kΨϕ(ϕn)kL4(0,T ;L2)∩L2(0,T ;L6)+ k∇Nσ(ϕn, σn)kL2(0,T ;L2)
+ kdiv(ϕn⊗ vn)kL4/3(0,T ;L3/2)+ kdiv(σn⊗ vn)kL1(0,T ;L1)≤ C. (5.2) We still have to derive additional uniform estimates for the time derivatives of ϕn and σn. To this end, we recall the identities
div(ϕn⊗ vn) = (∇ϕn)vn+ ϕnSv(ϕn, σn) a.e. in Q, (5.3) div(σn⊗ vn) = (∇σn)vn+ ϕnSv(ϕn, σn) a.e. in Q. (5.4) Let now ζ ∈ L83(0, T ; H1) be an arbitrary test function. Using the continuous embedding H32 ,→ L3as well as Lemma3.1, we obtain the estimate
Z T 0
Z
Ω
div(ϕn⊗ vn) · ζ d(x, t) ≤ C Z T
0
k∇ϕnkL3kvnkL2+ kϕnkL2 kζkH1 dt
≤ C Z T
0
kϕnk3/4H1kϕnk1/4H3kvnkL2+ kϕnkL2 kζkH1 dt
≤ C
kϕnk3/4L∞(0,T ;H1)kϕnk1/4L2(0,T ;H3)kvnkL2(0,T ;L2)+ kϕnkL∞(0,T ;L2)
kζkL8/3(0,T ;H1)
≤ C kζkL8/3(0,T ;H1).
Taking the supremum over all ζ ∈ L8/3(0, T ; H1) with kζkL8/3(0,T ;H1) ≤ 1, we thus conclude the uniform estimate
kdiv(ϕn⊗ vn)kL8/5(0,T ;(H1)0)≤ C. (5.5) Now, by a comparison argument, we infer from (3.34b) (written for the functions with index n) that
k∂tϕnkL8/5(0,T ;(H1)0)≤ C. (5.6)
Since L1 is continuously embedded in (W1,4)0, we infer from (5.2) that
kdiv(σn⊗ vn)kL1(0,T ;(W1,4)0)≤ C. (5.7) By means of a comparison argument, we eventually conclude from (3.34d) (written for the functions with index n) that
k∂tσnkL1(0,T ;(W1,4)0)≤ C. (5.8)
Combining (5.2) with (5.5)–(5.8), we eventually obtain the uniform estimate
kϕnkW1,8/5(0,T ;(H1)0)∩L∞(0,T ;H1)∩L2(0,T ;H3)+ kσnkW1,1(0,T ;(W1,4)0)∩L∞(0,T ;L2)∩L2(0,T ;H1)
+ kσnkL4(0,T ;L2Γ)+ kµnkL2(0,T ;H1)+ kvnkL2(0,T ;L2)+
pηn(ϕn) Dvn
2
L2(0,T ;L2)
+ kpnkL4/3(0,T ;L2)+ kΨϕ(ϕn)kL4(0,T ;L2)∩L2(0,T ;L6)+ k∇Nσ(ϕn, σn)kL2(0,T ;L2)
+ kdiv(ϕn⊗ vn)kL4/3(0,T ;L3/2)∩L8/5(0,T ;(H1)0)+ kdiv(σn⊗ vn)kL1(0,T ;L1)≤ C. (5.9)
Step 2: Passing to the limit. The next step is to pass to the limit as n → ∞. From the uniform estimate (5.9), we infer the existence of a quintuplet (ϕ, σ, µ, v, p) as well as limits τ and ϑ such that for all s ∈ [0, 1),
ϕn → ϕ weakly-∗in L∞(0, T ; H1),
weakly in W1,85(0, T ; (H1)0) ∩ L2(0, T ; H3), a.e. in Q,
and strongly in C([0, T ]; Hs) ∩ L2(0, T ; H2+s), (5.10a) ϕn|Γ → ϕ|Γ strongly in C([0, T ]; L2Γ), and a.e. on Σ, (5.10b)
σn → σ weakly-∗in L∞(0, T ; L2),
weakly in W1,1(0, T ; (W1,4)0) ∩ L2(0, T ; H1), a.e. in Q,
and strongly in C([0, T ]; (W1,4)0) ∩ L2(0, T ; Hs), (5.10c) σn|Γ → σ|Γ weakly in L4(0, T ; L2Γ), strongly in L2(0, T ; L2Γ), and a.e. on Σ, (5.10d)
µn → µ weakly in L2(0, T ; H1), (5.10e)
vn → v weakly in L2(0, T ; L2div), (5.10f)
pn → p weakly in L43(0, T ; L2), (5.10g)
div(ϕn⊗ vn) → τ weakly in L43(0, T ; L32) ∩ L85(0, T ; (H1)0), (5.10h)
div(σn⊗ vn) → ϑ weakly in L1(0, T ; L1), (5.10i)
ηn(ϕn) → 0 strongly in L∞(Q), and a.e. in Q, (5.10j)
λn(ϕn) → 0 strongly in L∞(Q), and a.e. in Q, (5.10k)
as n → ∞, along a non-relabeled subsequence. The strong convergences in (5.10a) and (5.10c) are a direct consequence of the Aubin–Lions–Simon lemma (see [10, Theorem II.5.16]), and the strong convergences in (5.10b) and (5.10d) are obtained through the trace theorem. We further point out that the convergence vn→ v in L2(0, T ; L2div) (see (5.10f)) already entails that
div(vn) → div(v) weakly in L2(0, T ; L2) (5.11) as n → ∞. Recalling the assumptionsA2–A8, we use the above convergences to infer that
C(ϕn, σn) → C(ϕ, σ), D(ϕn, σn) → D(ϕ, σ) a.e. in Q, (5.12a) Nϕ(ϕn, σn) → Nϕ(ϕ, σ), Nσ(ϕn, σn) → Nσ(ϕ, σ) a.e. in Q, (5.12b)
∇Nσ(ϕn, σn) → ∇Nσ(ϕ, σ), Sv(ϕn, σn) → Sv(ϕ, σ) a.e. in Q, (5.12c)
Ψϕ(ϕn) → Ψϕ(ϕ) a.e. in Q, (5.12d)
SΓ(ϕn, σn) → SΓ(ϕ, σ) a.e. on Σ, (5.12e)
Λϕ(ϕn, σn) → Λϕ(ϕ, σ), θϕ(ϕn, σn) → θϕ(ϕ, σ) a.e. in Q, (5.12f) Λσ(ϕn, σn) → Λσ(ϕ, σ), θσ(ϕn, σn) → θσ(ϕ, σ) a.e. in Q, (5.12g) as n → ∞, after extracting a subsequence. Now, using (5.12d), (5.12c), the uniform estimate (5.9), and the uniqueness of the limit, we further conclude that
Ψϕ(ϕn) → Ψϕ(ϕ) weakly in L4(0, T ; L2) ∩ L2(0, T ; L6), (5.13)
∇Nσ(ϕn, σn) → ∇Nσ(ϕ, σ) weakly in L2(0, T ; L2) (5.14) as n → ∞, after another subsequence extraction.
Let now δ ∈ Cc∞(0, T ), η ∈ H1(Ω; Rd), ζ, θ ∈ H1(Ω; RL), ξ ∈ W1,4(Ω; RM) ,→ L∞(Ω; RM), and q ∈ H1(Ω) be arbitrary test functions. We now test the weak formulation (3.34) (written for (ϕn, µn, σn, vn, pn) and the viscosities ηn and λn) with δη, δζ, δθ and δξ, and we integrate the resulting equations with respect to time from 0 to T . We further multiply the identity (3.34e) (written for vn, ϕn and σn) by δq and integrate
the resulting equation over Q. In summary, we obtain
Our next goal is then to pass to the limit n → ∞ in this variational formulation. Invoking the convergence properties (5.10) and (5.12), and using Lebesgue’s dominated convergence theorem,A5–A7, we deduce that
C(ϕn, σn)δ∇ζ → C(ϕ, σ)δ∇ζ, D(ϕn, σn)δ∇ξ → D(ϕ, σ)δ∇ξ in L2(Q), (5.16a) θϕ(ϕn, σn)δ∇ζ → θϕ(ϕ, σ)δ∇ζ, θσ(ϕn, σn)δ∇ξ → θσ(ϕ, σ)δ∇ξ in L2(Q), (5.16b)
θΓ(ϕn, σn)δ∇ξ → θΓ(ϕ, σ)δ∇ξ, in L2(Σ) (5.16c)
as n → ∞. Furthermore, proceeding as in Step 4 of the proof of Theorem3.5, we use Lebesgue’s generalized convergence theorem [4, Sec. 3.25] to conclude that
Nϕ(ϕn, σn) · δη → Nϕ(ϕ, σ) · δη in L2(Q), (5.17a) Nϕ(ϕn, σn) · δθ → Nϕ(ϕ, σ) · δθ in L2(Q), (5.17b) Λϕ(ϕn, σn) · δ∇ζ → Λϕ(ϕ, σ) · δ∇ζ in L2(Q), (5.17c) Λσ(ϕn, σn) · δ∇ξ → Λσ(ϕ, σ) · δ∇ξ in L2(Q), (5.17d) ΛΓ(ϕn, σn) · δ∇ξ → ΛΓ(ϕ, σ) · δ∇ξ in L2(Σ). (5.17e) For most of the terms in (5.15), we can simply use the convergences (5.10), (5.13), (5.14), (5.16) and (5.17) to pass to the limit n → ∞. However, some of the terms require a closer investigation.
In the terms depending on the viscosity functions ηn and λn, we can pass to the limit n → ∞ as follows:
Z
Z
Q
δλn(ϕn)div(vn)I : ∇η d(x, t)
≤ C kλn(ϕn)kL∞(Q)kSv(ϕn, σn)kL2(0,T ;L2)kδkL∞([0,T ])kηkH1 → 0. (5.19) Furthermore, we still need to recover the identities τ = div(ϕ ⊗ v) and ϑ = div(σ ⊗ v) almost everywhere in Q. To prove the latter identity, we first deduce from (5.10i) that
Z
Let us now consider an arbitrary test function ξ0 ∈ Cc∞(Ω; RM). Performing an integration by parts, we obtain
Due to (5.10c) and (5.10f), we may pass to the limit on the right-hand side by the weak-strong convergence principle. After another integration by parts, we get
Z in Q. In particular, this proves that
Z
Proceeding similarly with the convection term associated with the phase-field variable, we conclude that τ = div(ϕ ⊗ v) almost everywhere in Q, and
We can now use the convergences (5.10)–(5.14), (5.16)–(5.19), (5.21), and (5.22), along with the identities τ = div(ϕ⊗v) and ϑ = div(σ ⊗v) a.e. in Q, to pass to the limit n → ∞ in the variational formulation (5.15).
Since δ ∈ C∞([0, T ]) was arbitrary, this proves that the quintuplet (ϕ, µ, σ, v, p) satisfies the equations 0 =
+ Z
Ω
Sσ(ϕ, σ, µ) · ξ dx − Z
Γ
SΓ(ϕ, σ) · ξ dS (5.23d)
almost everywhere in (0, T ), for all test functions η ∈ H1(Ω; Rd), ζ, θ ∈ H1(Ω; RL), ξ ∈ W1,4(Ω; RM), as well as the identity
div(v) = Sv(ϕ, σ) a.e. in Q. (5.23e)
Testing (5.23a) with any function η0∈ Cc∞(Ω; Rd), we deduce that Z
Ω
p div(η0) dx = Z
Ω
νv − (∇ϕ)>µ − (∇σ)>Nσ(ϕ, σ) · η0dx.
Since
νv − (∇ϕ)>µ − (∇σ)>Nσ(ϕ, σ)
L1(0,T ;L3/2)
≤ C kvkL2(0,T ;L2)+ k∇ϕkL2(0,T ;L2)kµkL2(0,T ;L6)
+ C k∇σkL2(0,T ;L2) kϕkL2(0,T ;L6)+ kσkL2(0,T ;L6)+ 1
≤ C,
we conclude that ∇p exists in the weak sense with
∇p = νv − (∇ϕ)>µ − (∇σ)>Nσ(ϕ, σ) ∈ L1(0, T ; L32) a.e. in Q.
Plugging this identity into (5.23a) and integrating the resulting expression by parts, we infer that 0 = −
Z
Ω
p(t) div(η) + ∇p(t) · η dx = − Z
Γ
p(t)η · n dS (5.24)
for almost all t ∈ (0, T ) and all η ∈ H1. For any q ∈ C1(Γ), we have −qn ∈ H1 and we may thus choose η = −qn. We thus obtain
0 = Z
Γ
p(t) q n · n dS = Z
Γ
p(t) q dS (5.25)
for all q ∈ Cb1(Γ) and almost all t ∈ (0, T ), which directly proves that p|Σ = 0 a.e. on Σ. In summary, we have
p ∈ L43(0, T ; L2) ∩ L1 0, T ; W01,32, (5.26) and thus, all regularities in (3.37) are established. In particular, via integration by parts, (5.23a) can be replaced by the equivalent formulation
0 = Z
Ω
∇p · η + νv − (∇ϕ)>µ − (∇σ)>Nσ(ϕ, σ) · η dx. (5.27) We thus conclude that the quintuplet (ϕ, µ, σ, v, p) satisfies the weak formulation (3.38).
As a further consequence of the convergences ϕn → ϕ in C([0, T ]; L2) from (5.10a) and σn → σ in C([0, T ]; (W1,4)0) from (5.10c) we have, as n → ∞,
ϕ0= ϕn|t=0→ ϕ|t=0 in L2(Ω; RL),
hσ0, ΦiH1= hσn|t=0, ΦiW1,4 → hσ|t=0, ΦiW1,4 for all Φ ∈ W1,4(Ω; RM), meaning that ϕ and σ satisfy the initial conditions (3.38f) and (3.38g).
This proves that the limit (ϕ, µ, σ, v, p) is a weak solution of the multiphase Cahn–Hilliard–Darcy system in the sense of Definition 3.6. We further point out that the additional regularity property (3.41) can be verified by arguing exactly as in the proof of Theorem3.5. Thus, the proof of Theorem3.7is complete.
Acknowledgment
The authors want to thank Harald Garcke for helpful discussions. Patrik Knopf was partially supported by the RTG 2339 “Interfaces, Complex Structures, and Singular Limits” of the German Science Foundation (DFG). Their support is gratefully acknowledged.
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