Proof of Multiscale Convergence

By the Lipschitz assumption(A2)we have

kU(·,0;q)u⁰−U(·,0; ˜q)u⁰kC(I_ε,X)≲kq−q˜kC(I_ε) ku⁰k+λ(T_ε) . Thus for the initial Lipschitz estimate:

kSε(q)−Sε(˜q)kC(I_ε)≲εTεkq−q˜kC(I_ε)(1 +λ(Tε)).

With T_ε sufficiently small, this implies that S_ε is a contraction and hence Banach’s fixed-point theorem yields existence and uniqueness ofq_ε.

For the breakdown result, note that by g’s boundedness, limt↑T_εqε(t) ex-ists. If qε(Tε) ∈ Q we could extend qε beyond Tε locally, contradicting the

maximality of Tε. Henceqε(Tε)∈∂Q. □

3.4 Proof of Multiscale Convergence

We use the following classical result for averaging of an ordinary differential equations with periodic right-hand side, cf. [SVM07, Theorem 2.8.1]:

Theorem 3.4.1. LetQ⊂Rⁿ be connected, bounded and open. Letf: R×Q→ Rⁿ be 1-periodic, continuous in its first and uniformly Lipschitz continuous in its second argument. Then for anyx⁰∈Qthere is aT >0such that a solution x0: [0, ε⁻¹T]→Qof

x^′₀(t) =εf(x₀(t)) :=ε Z 1

0

f(s, x₀(t))ds, x₀(0) =x⁰

exists. The solution x0 is a first order approximation of the solution xεof x^′_ε(t) =εf(t, x_ε(t)), x_ε(0) =x⁰

in the sense that there is an ε^∗>0 such that for any0 < ε < ε^∗ the solution xεexists for all t∈[0, ε⁻¹T] and there holds

max

t∈[0,ε⁻¹T]|x_ε(t)−x₀(t)|≲ε.

Proof. The existence of limit solutions on the O(ε⁻¹)-timescale follows by rescaling to slow time, τ := εt, and using that x 7→ R1

0 f(s, x)ds is Lips-chitz continuous, since f is. The main ingredient for the averaging result is a comparison ofx0 andxεby keepingxεfixed on small intervals, see [Art07] for

details. □

Corollary 3.4.2. There exists T > 0 independent of ε such that the limit equation

q^′₀(t) =ε Z 1

0

g(q0(t), uπ(s;q0(t)))ds, q0(0) =q⁰ (3.2) has a unique solutionq₀: [0, ε⁻¹T]→Q. Furthermore, there is anε^∗>0such that for all0< ε < ε^∗ the solutionq_ε,π of

q_ε,π^′ (t) =εg(qε,π(t), uπ(t;qε,π(t))), qε,π(0) =q⁰. (3.7) exists for t∈[0, ε⁻¹T] with

max

t∈[0,ε−1T]|qε,π(t)−q0(t)|≲ε.

Proof. Define G(t, q) := g(q, u_π(t;q)). Then q_ε,π^′ (t) = εG(t, q_ε,π(t)) and G is continuous and1-periodic intand Lipschitz continuous inq, the latter following from the Lipschitz continuity of g by assumption and ofu_π byLemma 3.3.2.

The time-average ofG is the right-hand side of the limit equation and hence

we can apply the averaging theorem. _□

With this result, it remains to prove that qε is close to qε,π. For this, we first make the additional assumption that g is bounded by M > 0. By careful analysis of the constants involved, we will lift this restriction later on.

To this end, let the constant in ≲ be independent ofM. We will first prove that uε(t) := U(t,0;qε)u⁰ reaches anε-neighborhood of uπ(t;qε(t), ε) after a boundary layer of sizeO(|lnε|)and stays there, i.e.uεis slaved touπ. Lemma 3.4.3. Let K ∈N and (ak)^K_k=0 be non-negative with ak ≤bak−1+c forb, c≥0andk= 1, . . . , K. Then ak≤b^ka0+cPk−1

j=0b^j fork= 1, . . . , K.

Proof. Follows by induction. □

Lemma 3.4.4. There existsC >0 such that there holds

kU(t,0;q_ε)u⁰−u_π(t;q_ε(t))k≲e^−αtku⁰−u⁰_π(q⁰)k+εM for all t∈Iε with constants independent of|Iε|.

Proof. First, using the assumption of exponential stabilityA1we get kU(t,0;qε)u⁰−U(t,0;qε)u⁰_π(q⁰)k≲e^−αtku⁰−u⁰_π(q⁰)k.

Hence it remains to prove that kU(t,0;qε)u⁰_π(q⁰)−uπ(t;qε(t))k ≲ εM. For this, letL∈Nand the constant in≲in the following be independent ofL. Let Ik:= [tk, tk+1)fork= 0, . . . , K with0 =t0< t1< . . . < tK+1=Tε such that

|Ik|=Lfork= 0, . . . , K−1 and|IK| ≤L.

Let t ∈ Ik for k = 0, . . . , K and define q_ε^k := qε(tk). Note that by the boundedness assumption ongand sincet−tk ≤L we have

|qε(t)−q_ε^k|=|qε(t)−qε(tk)| ≤ε Z t

tk

|g(qε(s), uε(s))|ds≤εLM. (3.8)

We estimate

kU(t,0;qε)u⁰_π(q⁰)−uπ(t;qε(t))k

≤ kU(t,0;qε)u⁰_π(q⁰)−uπ(t;q^k_ε)k+kuπ(t;qε(t))−uπ(t;q^k_ε)k. The Lipschitz property ofuπ byLemma 3.3.2andineq. (3.8) yields

ku_π(t;q_ε(t))−u_π(t;q^k_ε)k≲|q_ε(t)−q_ε^k|≲εLM and hence

kU(t,0;qε)u⁰_π(q⁰)−uπ(t;qε(t))k ≤ kU(t,0;qε)u⁰_π(q⁰)−uπ(t;q_ε^k)k+εLM.

To estimate the first term on the right we add and subtractU(t, t_k;q_ε)u⁰_π(q^k_ε):

kU(t,0;qε)u⁰_π(q⁰)−uπ(t;q^k_ε)k ≤ kU(t,0;qε)u⁰_π(q⁰)−U(t, tk;qε)u⁰_π(q^k_ε)k +kU(t, t_k;q_ε)u⁰_π(q^k_ε)−u_π(t;q_ε^k)k. (3.9) For the first term, we split the evolution on[0, t]into[0, t_k]and the remainder [t_k, t]. Using the assumption of exponential stability(A1)this results in

kU(t,0;qε)u⁰_π(q⁰)−U(t, tk;qε)u⁰_π(q_ε^k)k

≲e^{−α(t−tk)}kU(tk,0;qε)u⁰_π(q⁰)−u⁰_π(q_ε^k)k. (3.10) For the second term the periodicity andtk∈Nimply

u_π(t;q^k_ε) =U(t, t_k;q_ε^k)u_π(t_k;q_ε^k) =U(t, t_k;q^k_ε)u⁰_π(q^k_ε) and thus the Lipschitz assumption(A2)yields

kU(t, tk;qε)u⁰_π(q_ε^k)−uπ(t;q_ε^k)k≲kqε−q^k_εkC(t_k,t) ku⁰_π(q^k_ε)k+λ(L)

≲εL(1 +λ(L))Mku_π(·;q^k_ε)kC_π(X)

≲εLλ(L)M,

where we used Lemma 3.3.3 and assumed, without restriction of generality, that λ(L)≥1. Using the last estimate andineq. (3.10)inineq. (3.9) leads to

kU(t,0;q_ε)u⁰_π(q⁰)−u_π(t;q_ε(t))k

≲e^{−α(t−tk)}kU(tk,0;qε)u⁰_π(q⁰)−u⁰_π(q_ε^k)k+εLλ(L)M (3.11) for anyt∈Ik. By passing to the limitt↑tk+1 fork < K this yields

kU(tk+1,0;qε)u⁰_π(q⁰)−uπ(tk+1;q^k+1_ε )k

≲e^−αLkU(tk,0;qε)u⁰_π(q⁰)−u⁰_π(q^k_ε)k+εLλ(L)M.

This is an inequality of the form of Lemma 3.4.3. Together with kU(tk,0;qε)u⁰_π(q⁰)−u⁰_π(q_ε^k)k= 0

fork= 0, an application of this lemma yields fork= 0, . . . , K that kU(t_k,0;q_ε)u⁰_π(q⁰)−u_π(t_k;q_ε^k)k≲εLλ(L)M

k−1

X

j=0

(Ce^−αL)^j

with a constantCindependent of LandM. We may chooseLindependent of M such thatCe^−αL<1. This implies, with≲now depending onL, that

kU(tk,0;qε)u⁰_π(q⁰)−uπ(tk;q_ε^k)k≲εM

fork= 0, . . . , K. Using this estimate inineq. (3.11)yields the claimed bound

for allt∈I_k and concludes the proof. □

We now use the previous result to establish thatqε is close to qε,π, where qε,π was defined, repeated here for convenience, as solution of

q_ε,π^′ (t) =εg(qε,π(t), uπ(t;qε,π(t))), qε,π(0) =q⁰. (3.7) Lemma 3.4.5. LetT >0 be fixed and|Iε| ≤ε⁻¹T. Then we have

maxt∈I_ε|q_ε(t)−q_ε,π(t)|≲εM, with constant depending onT.

Proof. Using the Lipschitz continuity ofg andu_π, we have fort∈I_ε that

For the second term we use our estimate fromLemma 3.4.4to see that ε Lemma 3.3.3. By Lipschitz continuity ofuπ there holds for the third term:

ε and application of Gronwall’s inequality yields the claimed estimate

|qε(t)−qε,π(t)| ≤εMe^Cεt ≲εM,

Proof. By the previous result and Corollary 3.4.2 we see that the estimate is valid on a possibly ε-dependent interval Iε for ε small enough. It remains to prove that the interval of existence may be extended to [0, ε⁻¹T]. Since the trajectory of the averaged equation Γ0 := q0([0, ε⁻¹T]) is compact and contained inQ, we can findη >0such that aη-neighborhood ofΓ0is contained in Q. Choosing ε^∗ small enough such that εM ≲ η for all 0 < ε < ε^∗ the previous corollary implies thatq_ε cannot leave this neighborhood (and hence Q) for any such ε. ByLemma 3.3.4solutions only cease to exist if q_ε crosses the boundary ofΩ. Hence for all0< ε < ε^∗ the solutionq_εmust exist at least

untilε⁻¹T. □

A simple consequence of our results is an estimate on the fast component, albeit limited by a boundary layer att= 0:

Lemma 3.4.7. For any δ∈(0,T)there exists ε^∗ =ε^∗(δ, M)>0, such that for all 0< ε < ε^∗ there holds

max

t∈[ε⁻¹δ,ε⁻¹T]kU(t,0;q_ε)u⁰−u_π(t;q₀(t))k≲εM.

Proof. ByLemma 3.4.4, we have fort∈[0, ε⁻¹T]that

kU(t,0;qε)u⁰−uπ(t;qε(t))k≲e^−αtku⁰−u⁰_πk+εM.

Using the Lipschitz continuity of u_π and the estimate for q_ε−q₀ from the Corollary 3.4.6, we hence get

kU(t,0;qε)u⁰−uπ(t;q0(t))k

≤ kU(t,0;qε)u⁰−uπ(t;qε(t))k+kuπ(t;qε(t))−uπ(t;q0(t))k

≲e^−αtku⁰−u⁰_πk+εM +|q_ε(t)−q₀(t)|

≲e^−αtku⁰−u⁰_πk+εM.

Fort≥t^∗(ε) =−α⁻¹ln(ε(1+M))we have e^−αt≤ε(1+M). Sinceε⁻¹δ > t^∗(ε) for all 0 < ε < ε^∗ if ε^∗ =ε^∗(δ, M) is sufficiently small, the claimed estimate

follows. □

Proof for unbounded g

We will now prove that one can drop the assumption ofg being bounded. We reduce this to the previous case by replacing g with a bounded function g_R outside a ball of radius R in X. Careful analysis of the preceding estimates then shows that forRsufficiently large andεsufficiently small this change has no effect on the solutions.

Lemma 3.4.8. ForR >0 definePR:X →X byPR(u) :=uforkuk ≤Rand PR(u) :=R_∥^u_u∥ for kuk> R. Then PR is Lipschitz continuous with constant 1.

Proof. Letu₁, u₂∈Xand assume without restriction of generality thatku₁k ≤ ku2k. If ku2k ≤ R we have kPR(u1)−PR(u2)k = ku1−u2k. For the case ku2k> Randku1k< Rletv∈X denote the orthogonal projection ofu1onto Ru2. Then it is easy to see that kv−PR(u2)k ≤ kv−u2kand hence

kPR(u1)−PR(u2)k²=ku1−PR(u2)k²≤ ku1−vk²+kv−u2k²=ku1−u2k².

If ku1k > R first assume that ku1k =ku2k, i.e. there existsα > Rˆ such that u1= ˆα_∥^u_u¹

1∥ andu2= ˆα_∥^u_u²

2∥. Sinceα7→αk_∥^u_u¹_1∥−_∥^u_u²_2∥kis increasing the claim follows since α > R. In the last case,ˆ R <ku₁k <ku₂k, we use the already proven cases together withP_R◦P_∥u1∥=P_R to see that

ku₁−u₂k ≥ ku₁−P_∥u1∥(u₂)k ≥ kP_R(u₁)−[P_R◦P_∥u1∥](u₂)k

≥ kPR(u1)−PR(u2)k. □

Lemma 3.4.9. For any R > 0 there exists a bounded, Lipschitz continuous function gR:Q×X → Rⁿ such that gR(q, u) = g(q, u) for all q ∈ Q and kuk ≤Rwith the same Lipschitz constant as g.

Proof. DefinegR(q, u) :=g(q, PR(u)). ThengR(u) =g(u)forkuk ≤Rand the Lipschitz continuity follows from the previous lemma. Since

|gR(q, u)|=|g(q−q⁰+q⁰, PR(u−0))|≲|q−q⁰|+kPR(u)k+|g(q⁰,0)|≲1 +R, where we used thatQis bounded,gR is bounded. □

Lemma 3.4.10. ForR >0large enough the limit solution q0 solves q^′₀(t) =ε

Z 1 0

gR(q0(t), uπ(s;q0(t)))ds, q0(0) =q⁰ for all t∈[0, ε⁻¹T].

Proof. The set Γ :={uπ(s;q0(t)) | s∈ [0,1], t∈ [0, ε⁻¹T]} ⊂X is bounded.

Hence we can choose R >0 large enough such thatkuk ≤Rfor allu∈Γ and henceg(q₀(t), u_π(s;q₀(t))) =g_R(q₀(t), u_π(s;q₀(t))). □

Now letq_ε,R denote the solution with right-hand sideg_RwhereR >0is at least as large as required by the previous lemma. For the bound M >0 ofg we have M ≲1 +R ≲R for R sufficiently large. Then by the results of the previous section there is anε^∗(R)>0such that for all0< ε < ε^∗ the solution qε,R exists for[0, ε⁻¹T]and there holds

kqε,R(t)−q0(t)k≲εR,

kU(t,0;q_ε,R)u⁰−u_π(t;q_ε,R(t))k≲e^−αtku⁰−u⁰_πk+εR

for allt∈[0, ε⁻¹T], where we implicitly used thatq₀is also the solution of the limit equation forg_R. WithR-independent constantsC >0 we get

kU(t,0;q_ε,R)u⁰k ≤C ku⁰−u⁰_πk+ku_π(t;q_ε,R(t))k+εR

≤C ku⁰−u⁰_πk+kuπ(t;q0(t))k+|qε,R(t)−q0(t)|+εR

≤C ku⁰−u⁰_πk+kuπ(t;q0(t))k+εR . Fixing R > 0 such that C ku⁰−u⁰_πk+kuπ(t;q0(t))k

≤ ^R₂ and making the corresponding ε^∗(R) > 0 small enough such that ¹₂ +Cε ≤ 1 holds for all 0< ε < ε^∗ we getkU(t,0;qε,R)u⁰k ≤Rfor allt∈[0, ε⁻¹T]and hence

qε,R(t)^′ =εgR(qε,R(t), U(t,0;qε,R)u⁰) =εg(qε,R(t), U(t,0;qε,R)u⁰).

But this implies thatqε,Rfor suchRand0< ε < ε^∗solves the originaleq.(3.1), henceqε,R =qεand of course the estimates from the previous section still hold.

Im Dokument Temporal Multiscale Methods for a Model of Atherosclerosis (Seite 66-72)