Before we continue to the proof of our main theorem, we give some examples of evolution processes which satisfy the assumptions above.
LetX be a Hilbert space with scalar product(·,·). LetV be a separable, reflexive Banach space, densely embedded in X. Hence V ,→ X ,→ V∗ by the usual identification of X with X∗. The norms on V and V∗ are denoted as k · kV and k · kV∗, the duality product between V and V∗ by h·,·i. We define W(s, t) :={u∈ L2(s, t;V);∂tu∈ L2(s, t;V∗)}, being a Banach space with normkukW(s,t):=kukL2(s,t;V)+k∂tukL2(s,t;V∗)and note thatW(s, t),→ C([s, t], X).
We will examine families of monotone operators, see [Sho96] for an introduc-tion to monotone operators. LetA:Q→ {V →V∗} be a family of operators, such that
B1. For eachq∈Q,A(q)is radially continuous.
B2. The family is uniformly strongly monotone: There existsγ >0such that hA(q)u−A(q)v, u−vi ≥γku−vk2V
for allq∈Qand u, v∈V.
B3. The family has uniform linear growth: There existsM >0such that kA(q)ukV∗≤M(kukV + 1)
for allq∈Qand u∈V.
B4. The mapA is Lipschitz continuous in the sense that there isL >0 with kA(q)u−A(˜q)ukV∗≤L|q−q˜|(kukV + 1)
for allq,q˜∈Qandu∈V.
Note that (B4)implies(B3)if for oneq∈Qthe operatorA(q)grows linearly.
The evolution processU is generated by the solutions of the periodically-forced parabolic equation with operators A: For f ∈L2(0,1;V∗)and u0 ∈X we defineU(t, s;q)u0:=u(t), whereuis the solution to
∂ru(r) +A(q(r))u(r) =f(r) forr∈(s, t)a.e., u(s) =u0, (3.3)
with f extended periodically toR. We first give some more specific examples of operator familiesAwith the above properties, then prove well-posedness of the associated process and its fulfillment of our assumptions (A1)–(A3).
Remark 3.2.1. We could also study for1< p <∞the setting corresponding to
W(s, t) :={u∈Lp(I, V)|∂tu∈[Lp(I, V)]∗∼=Lq(I, V∗)}
for 1p+1q = 1, with appropriate changes to the assumptions.
Example 3.2.2. In case of linear operators A: Q → L(V, V∗), (B1)– (B4) reduce to uniform coercivity and boundedness, i.e. there exists γ > 0 and M >0 such that
hA(q)u, ui ≥γkuk2V, kA(q)ukV∗≤MkukV
for all q ∈ Q and u ∈ V, together with Lipschitz continuity of A: There is L >0 such that for allq,q˜∈Qwe have
kA(q)−A(˜q)kL(V,V∗)≤L|q−q˜|. Example 3.2.3. LetA0:V →V∗ be radially continuous and strongly mono-tone, which reduces to coercivity ifA0∈L(V, V∗). Letα:Q→Rbe Lipschitz continuous such that α≤α(·)≤αfor0< α < α <∞. Then A(q) :=α(q)A0
is an operator family satisfying (B1)– (B4).
Example 3.2.4. LetΩ⊂Rd withd∈Nbe a bounded Lipschitz domain and V :=H01(Ω), X :=L2(Ω). Let K:Q→ L∞(Ω,L(Rd,Rd)) be Lipschitz con-tinuous and K(q)be (q, x)-uniformly bounded and positive definite matrices, i.e. there existsγ >0 andM >0such that
[K(q)](x)ξ·ξ≥γ|ξ|2, kK(q)kL∞(Ω,L(Rd,Rd))≤M
for all q ∈ Q, ξ ∈ Rd and x ∈ Ω almost everywhere. Then the family of operatorsA(q) :=−div(K(q)∇·)satisfies(B1)– (B4), since byExample 3.2.2 we have to check:
• The family is uniformly bounded byM: Since forq∈Qandu, v∈V we have(K(q)∇u,∇v)≤ kK(q)kL∞(Ω,L(Rd,Rd))k∇ukk∇vk, there holds
kA(q)kL(V,V∗)≤ kK(q)kL∞(Ω,L(Rd,Rd))≤M.
• The operators are uniformly coercive: Since Kis uniformly positive def-inite, we have for allq∈Qandu∈V:
hA(q)u, ui= (K(q)∇u,∇u)≥γk∇uk2.
• The mapq7→A(q)is Lipschitz continuous, since forq,q˜∈Q
kA(q)−A(˜q)kL(V,V∗)≤ kK(q)−K(˜q)kL∞(Ω,L(Rd,Rd))≲|q−q˜|.
Example 3.2.5. Let Ω ⊂ Rd with d ∈ N be a bounded Lipschitz domain, V := H0,σ1 (Ω) and X := L2σ(Ω) be the solenoidal function spaces, i.e. those spaces with vanishing (weak) divergence. Let P:L2(Ω) →L2σ(Ω) denote the Helmholtz projection. Then the Stokes operator, A0 := −P∆ ∈L(V, V∗), is coercive and hence Stokes’ problem with slowly changing viscosities falls into
the described framework byExample 3.2.3.
Lemma 3.2.6. The evolution process U is well-defined in the sense that for t≥s,q∈C([s, t], Q)andu0∈X a unique solutionu∈W(s, t),→C([s, t], X) to problem(3.3)exists. We have the a-priori estimate
kukL∞(s,t;X)+kukL2(s,t;V)≲ku0k+kfkL2(s,t;V∗).
Proof. Note that for fixed q ∈ Q, we have by strong monotonicity (B2) for u∈V:
hA(q)u, ui=hA(q)u−A(q)0, u−0i+hA(q)0, ui ≥(γkukV −M)kukV
where we used thatkA(q)0kV∗≤M due to(B3). HencehA(q)u, uikuk−V1→ ∞ forkukV → ∞andA(q)is coercive with constants independent ofq. Existence of solutions for this particular non-autonomous Cauchy problem then follows from standard theory, see e.g. [Sho96, Proposition III.4.1]. Uniqueness follows from the proof of exponential stability, see below. For the a-priori estimate, we test as usual with the solutionuitself, to arrive at
1 2
d
drku(r)k2+hA(q(r))u(r), u(r)i=hf(r), u(r)i and apply the coercivity estimate from above, to get
1 2
d
drku(r)k2+γku(r)k2V ≤(kf(r)kV∗+M)ku(r)kV.
The estimate then follows by application of Young’s inequality with suitable
constants and integration in time. □
Lemma 3.2.7. The process is exponential stable as assumed by(A1).
Proof. Lets < t,q∈C([s, t], Q)andu0,u˜0∈X. Letu(t) := U(t, s;q)u0 and
˜
u(t) :=U(t, s;q)˜u0, such that both satisfy the correspondingeq. (3.3). Then 1
2 d
dtku−˜uk2+hA(q)u−A˜u, u−u˜i= 0, u(s)−u(s) =˜ u0−u˜0 and strong monotonicity(B2) andV ,→X yield dtdku−u˜k2+ku−u˜k2≲0.
By Gronwall’s inequality, this gives as claimed
ku(t)−u(t)˜ k2≤e−C(t−s)ku0−u˜0k. □ Lemma 3.2.8. The assumption(A2) of Lipschitz continuity is satisfied.
Proof. Let s < t, q,q˜∈C([s, t], Q)and u0 ∈X. Letu(t) :=U(t, s;q)u0 and
˜
u(t) :=U(t, s; ˜q)u0, such that both satisfy the correspondingeq. (3.3). Then
∂r(u−u)(r) +˜ A(q(r))u(r)−A(q(r))˜u(r) =A(˜q(r))˜u(r)−A(q(r))˜u(r)
withu(s)−u(s) = 0. By arguments as in the preceding results, this yields˜ ku(t)−u(t)˜ k≲kA(q)˜u−A(˜q)˜ukL2(s,t;V∗).
Applying the Lipschitz assumption (B4)pointwise, we have kA(q)˜u−A(˜q)˜ukL2(s,t;V∗)≲kq−q˜kC([s,t])(ku˜kL2(s,t;V)+√
t−s).
Together with the a-priori estimate for u˜ and using the periodicity of f, this
yields(A2). □
Lemma 3.2.9. The process is1-periodic as assumed by(A3).
Proof. This is evident due to the1-periodicity off. □
We conclude this overview with a simple application ofTheorem 3.1.1.
Example 3.2.10. WithΩ := (−1,1)2andQ:= (1,2)consider the problem
∂tuε(t, x)−(1 +qε(t))∆uε(t, x) =sin(2πt)sin(πx1)sin(πx2) inΩ, (3.4a) qε′(t) =εkuε(t)kL2(Ω)qε(t)(2−qε(t)) (3.4b) with initial values uε(0) = 0, qε(0) = q0 = 1 and uε = 0 on ∂Ω. The fast equation is of the type investigated inExample 3.2.4and hence a process with X =L2(Ω). The right-handg of the slow equation is Lipschitz continuous in X andQbut unbounded. ApplyingTheorem 3.1.12the limit equation is
q′0(t) =ε Z 1
0
kuπ(s;q0(t))kL2(Ω)ds
q0(t)(2−q0(t)) (3.5) withuπ(·;q)forq∈Qbeing the1-periodic solution to
∂suπ(s, x;q)−(1 +q)∆uπ(s, x;q) =sin(2πs)sin(πx1)sin(πx2).
The periodic problem has an autonomous differential operator, in contrast to the original non-autonomous problem. We can in particular apply a simple eigenfunction decomposition of the periodic problem to see that it has the explicit solution
uπ(s, x;q) = 1 2π
π(1 +q)sin(2πs)−cos(2πs)
π2(1 +q)2+ 1 sin(πx1)sin(πx2) and hence the time-average of the norm is given by
Z 1 0
kuπ(s;q0(t))kL2(Ω)ds= 1 π2p
π2(1 +q)2+ 1. Therefore, the limit eq. (3.5)can be written explicitly as
q0′(t) =ε q0(t)(2−q0(t)) π2p
π2(1 +q0(t))2+ 1, q0(0) = 1. (3.6) Thus only the solution of a simple ordinary differential equation, instead of the coupled, oscillatory system of partial and ordinary differential equations,
is necessary.
2Whileq0∈∂Qis not included in the theory, this is of course only a formal problem since the right-hand side ofeq. (3.4b)is non-negative and zero only ifuε= 0such thatqε(t)∈Q fort >0.
1e-05 1e-04 1e-03 1e-02 1e-01
1e-03 1e-02 1e-01 1e+00
linear convergence
‖ qε - q0 ‖∞ ε
Figure 3.1: Error between the numerical solutions toqε andq0 from Exam-ple 3.2.10 in terms ofε together with a reference linear conver-gence (dashed line).
We also investigated the convergence inExample 3.2.10 numerically, but keep this section brief and refer to Chapter 5 for a discussion of the issues when solving the limit equation for the more interesting plaque problem on moving domains. For the fast-slow system we used the second-order backward difference scheme (BDF2) for the temporal discretization and quadratic finite elements on a quadrilateral mesh for the spatial discretization. Temporal and spatial resolution were chosen such that the numerical discretization errors could be neglected. The direct numerical simulation was carried out using the finite element library deal.II [Alz+18], the limit eq. (3.6) was solved in MATLAB. As can be seen in Figure 3.1, the results are in good agreement with the theory.