Bull. London Math. Soc. 2014 Matioc 1145 55

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Bull. London Math. Soc. 46 (2014) 1145–1155 C 2014 London Mathematical Societydoi:10.1112/blms/bdu068

Non-uniform continuity of the semiflow map associated to theporous medium equation

Bogdan-Vasile Matioc

Abstract

We prove that the semiflow map associated to the evolution problem for the porous mediumequation (PME) is real-analytic as a function of the initial data in H s(S), s > 7

2, at any fixed

positive time, but it is not uniformly continuous. More precisely, we construct two sequences of exact positive solutions of the PME which at initial time converge to zero in H s(S), but suchthat the limit inferior of the difference of the two sequences is bounded away from zero in H s(S)at any later time.

1. Introduction and the main result

We consider herein the evolution problem associated to the one-dimensional porous mediumequation (PME)

ut = (uux)x, x ∈ S and t > 0, (1.1)

in the periodic case, that is, with S denoting the unit circle S := R/(2πZ). The initial conditionfor (1.1) is given by the relation

u(0) = u0. (1.2)

Equation (1.1) and its generalization

ut = ∆um, t > 0, (1.3)

with m > 1, have received in the last decades lots of attention from people working inmathematics and other fields. A systematic presentation of the mathematical theory for thePME is presented in the book by Vazquez [28]. There are nevertheless still many interestingfeatures related to (1.3) which have not been studied yet.

For the choice m = 2 made here, the PME is a model describing groundwater flows [2] anddates back to Boussinesq’s derivation in 1903. Its two-phase version has been only recentlyderived in [10] as the lubrication approximation of the Muskat problem. The question we areinterested in is whether the semiflow map [(t, u0) → u(t; u0)] associated to (1.1), cf. Theorem2.1, is uniformly continuous in H s(S) as a function of the initial data when keeping the (positive)

time fixed. The uniform continuity of the flow map has been investigated recently in thecontext of several hyperbolic models for water waves: the Camassa–Holm equation [14, 15],the equation for the wave surface corresponding to the Camassa–Holm equation [7, 8], the Eulerequations [16], the b-equation [12], the µ-b equation [23], the hyperelastic rod equation [19], theNovikov equation [13], the modified Camassa–Holm equation [11], the modified Camassa–Holmsystem [24], the answer being always negative. We should emphasize that all these hyperbolicmodels can be written as first-order non-linear equations, the solutions breaking some times infinite time, cf., for example, [6, 25]. On the other hand, equation (1.1) is parabolic (degeneratewhen u becomes zero), of second order, and it possesses globally defined strong solutions which

Received 6 June 2014; revised 25 June 2014; published online 28 August 2014.

2010 Mathematics Subject Classification 35B30, 35G25, 35K65 (primary), 76S05 (secondary).



1146 BOGDAN-VASILE MATIOC

converge towards flat states.† Moreover, keeping the positive time fixed, the semiflow map isreal-analytic with respect to the initial data, cf. Theorem 2.1. Let us also recall that in thesetting of non-negative L1(R)-solutions with finite mass [18], one can associate a semiflow mapto (1.3), and this map is in fact a contraction at each fixed t > 0, that is,

u(t; u0) − u(t; v0)L1(R) u0 − v0L1(R).

Additionally, it is a contraction also with respect to all Wasserstein distances W p, with p ∈[1, ∞],

W p(u(t; u0), u(t; v0)) W p(u0, v0),

cf., for example, [3, 5], the W 2-contractivity in arbitrary space dimensions being establishedin [4]. In higher space dimensions, W p-contractivity with p large does not hold [27]. For thesereasons, the non-uniform continuity property established in Theorem 1.1 for the semiflow mapassociated to (1.1) and (1.2) is surprising, the more because for the linear correspondent of (1.1), that is, the heat equation, the semiflow map is a contraction at any fixed positive time,cf. Remark 1.3.

To establish our result, we first construct two sequences of positive approximate solutionsof (1.1) which are sufficiently close to the exact solutions of (1.1) defined by the value of theapproximate ones at t = 0. The sequences of approximate and exact solutions are approachingwhen n → ∞ the regime where the equation becomes degenerate. Nevertheless, using parabolicmaximum principles for the solutions of (1.1) and for their first spatial derivatives (this is thereason why m = 2 is so important), commutator estimates, and interpolation properties of Sobolev spaces, we show that the difference between the exact solutions that we have found isbounded away from zero, in the limit n → ∞, at any positive time, although it converges tozero at t = 0. As far as we know, this is the first result that proves, in high order Sobolev spaces,the non-uniform continuity with respect to the initial data for the semiflow map correspondingto a parabolic evolution equation.

In order to present our main result, let u(·; u0) denote the unique strong solution of (1.1)and (1.2) associated to an initial data

u0 ∈ V s := {u ∈ H s(S) : minS

u > 0},

whereby for technical reasons we are restricted to considering s > 72 (see Lemma 3.2). Then,

the mapping

[[0, ∞) × V s (t, u0) → u(t; u0) ∈ V s]

defines a global continuous semiflow, cf. Theorem 2.1. Moreover, the semiflow is real-analyticin (0, ∞) × V s, that is, u ∈ C ∞((0, ∞) × V s, V s) and for all (t0, u0) ∈ (0, ∞) × V s there existsr > 0 such that

u(t0 + s; u0 + v0) = n∈N

∂ n

u(t0; u0)[(s, v0)]n

n! ,

for all (s, v0) ∈ R × H s(S) with (s, v0)R×H s < r.The main result of this paper is the following theorem.

†Poincare’s inequality and parabolic maximum principles ensure that any of the solutions u of (1.1) found inTheorem 2.1 satisfies

d

dtΛ(u−[u])2L2 = −2

S

u(u2x + u2xx) dx −2minu(0)Λ(u−[u])2L2 in (0,∞),

with [u] denoting the mean integral value of u over one period and Λ := Λ1. This implies exponential convergencein H 1(S). The principle of linearized stability, cf., for example, [22, Theorem 9.1.2], can be additionally used

to prove exponential convergence in stronger Sobolev norms, provided that the initial data are close to theirmean value in these norms.



FLOW PROPERTIES OF THE PME 1147

Theorem 1.1 (Non-uniform continuity of the semiflow map). Let s > 72 be fixed. Then,

at any positive time t, the mapping

[V s u0 → u(t; u0) ∈ V s]

associated to the semiflow defined by the evolution equation (1.1) and (1.2) is real-analytic , butit is not uniformly continuous. More precisely , there exist two sequences of positive solutions

(un)n, (vn)n ⊂ C ([0, ∞), V s) ∩ C ∞([0, ∞) × S),

and for each T > 0 a positive constant C > 0 with the following properties:

supn∈N

maxt∈[0,T ]

(un(t)H s + vn(t)H s) C,

limn→∞

un(0) − vn(0)H s = 0,(1.4)

but

lim inf n→∞

inf t∈[δ,T ]

un(t) − vn(t)H s √

π for each δ ∈ (0, T ]. (1.5)

Before starting our analysis, we make some remarks.

Remark 1.2. Theorem 1.1 shows that the semiflow map associated to the PME on the p-torus

T p := S× . . . × S

p-times

,

p ∈ N, p 2, is not uniformly continuous in H s(T p) for any s > max{72 , ( p + 2)/2}. This follows

from the fact that the sequences (un)n and (vn)n defined by

un(t, x1, . . . , x p) := un(t, x1), vn(t, x1, . . . , x p) := vn(t, x1),

for t 0 and (x1, . . . , x p) ∈ T p, with (un)n and (vn)n determined as in Theorem 1.1, satisfysimilar properties as (un)n and (vn)n. We note that the condition s > ( p + 2)/2 is sufficient toguarantee that the semiflow map

[[0, ∞) × V s (t, u0) → u(t; u0) ∈ V s],

whereby V s is now given by V s := {u ∈ H s(T p) : minTp u > 0}, is well defined, cf. [1].The case when the exponent m in (1.3) satisfies m = 2 appears to be more difficult, as no

maximum principle can be used for the derivatives ∂ xun and ∂ xvn. Additionally, the methodsused in the proofs of Lemmas 2.3–3.2 do not seem to be applicable when m is not an integer.

Remark 1.3. It is well known that in the case of the linear heat equation

ut = uxx, x ∈ S and t > 0. (1.6)

the estimate u(t; u0)H s u0H s holds true for all t 0, cf., for example, [17]. Hence, thesemiflow map is a contraction at any fixed positive time. This suggests that the non-uniformcontinuity property derived in Theorem 1.1 is due to the non-linear character of the PME.

Remark 1.4. Let us point out that the construction presented in Theorem 1.1 hasno correspondent within the setting of ordinary differential equations (ODEs). Indeed, wenote first that the right-hand side of (1.1) corresponds to an operator [u

→ f (u)] with

∂f (u)L(H s+δ,H s+δ−2) K uH s for some positive constant K and some δ ∈ (0, 12) with




s + δ − 2 0†. Now consider an autonomous ODE

u = φ(u),

whereby E is a Banach space, φ

∈ C 1(D, E ), and D

⊂ E is open. Assuming

∂φ(u)

K

u

for all u ∈ D, and if un, vn : [0, T ] → D, n ∈ N, are solutions of u = φ(u) satisfying (1.4) (withH s(S) replaced by E ), then it is easy to see that

un(t) − vn(t) C (K, T )un(0) − vn(0) for all t ∈ [0, T ], n ∈ N,

so that (1.5) can never be achieved.

Notation. Throughout this paper, we shall denote by C positive constants which may dependonly upon s and T . Furthermore, H r(S), with r ∈ R, is the L2-based Sobolev space on theunit circle. Given r ∈ R, we let Λr := (1 − ∂ 2x)r/2 denote the Fourier multiplier with symbol((1 + |k|2)r/2)k∈Z, which is an isometric isomorphism Λr : H q(S) → H q−r(S) for all q, r ∈ R.Finally, we let QT := [0, T ]

×S.

Before proceeding with the analysis, we recall the following Kato–Ponce commutatorestimate:

[Λr, f ]gL2 C r(f xL∞Λr−1gL2 + Λrf L2gL∞), (1.7)

which is valid for r > 32 and all f, g ∈ C ∞(S), cf. [20, 26]. Hereby, [·, ·] is the commutator

defined by [S, T ] := ST − T S.Outline. In Section 2, we establish first the well-posedness of the evolution problem (1.1) and

(1.2), and then we introduce two sequences of approximate solutions of the latter problem. InSection 3, we estimate the error between the approximate solutions and the exact solutions of (1.1) and (1.2) associated to the approximate solutions in several Sobolev norms. The proof of the main result Theorem 1.1 is presented in Section 4.

2. Well-posedness and approximate solutions for (1.1)

The following well-posedness result is based on the theory of quasilinear parabolic problems aspresented in [1], cf. Theorem 12.1 and the discussion following Theorem 12.6. That the strongsolutions of (1.1) are globally defined can be seen by arguing along the lines of [9, Theorem 2.1],for example.

Theorem 2.1 (Global well-posedness). Let s > 32 be given. Then, the problem (1.1) and

(1.2) possesses for each u0 ∈ V s a unique global solution u(·; u0) within the class

C ([0, ∞), V s) ∩ C ∞

((0, ∞) × S).Moreover , the mapping

[[0, ∞) × V s (t, u0) → u(t; u0) ∈ V s]

defines a global semiflow which is continuous on [0, ∞) × V s and real-analytic on (0, ∞) × V s.

In the remaining of this paper, s > 72 and T > 0 are kept fixed. In a first step, we construct

two sequences of approximate positive solutions of equation (1.1). The first sequence (U n)n of

†To establish the well-posedness of the evolution problem (1.1) and (1.2), it is natural to exploit the quasilinear

structure of (1.1) and write f (u) = −A(u)u with the operator A(v)u := −(vux)x, v ∈ V s and u ∈ H s+δ

(S),satisfying A ∈ C ω(V s,H(H s+δ(S),H s+δ−2(S))), cf., for example, [1].




approximate solutions is defined by

U n(t, x) := n−3 + n−s cos(nx), (2.1)

for (t, x) ∈ QT and n ∈ N, n 2. We note that U n is in fact independent of the time variable

and that for n large the term involving the cosine has a high spatial frequency. Moreover,these approximate solutions approach the boundary of V s when n → ∞, where equation (1.1)becomes degenerate. The second sequence (V n)n of approximate solutions is given by

V n(t, x) := n−1 + e−ntn−s cos(nx), (2.2)

for (t, x) ∈ QT and n ∈ N, n 2. The term involving the cosine has again high spatial frequencywhen n is large, but now it decays very fast with respect to tn. These approximate solutionsalso approach the boundary of V s when letting n → ∞.

Before we estimate the error associated to the approximate solutions, we note

cos(nx)H r =√

π(1 + n2)r/2, (2.3)

for all r ∈ R and n

∈ N, n 2. This property can be easily deduced from the definition† of the

norm in H r(S), cf., for example, [8]. The same arguments show also that 1H r = √ 2π. It isnow immediate to see that

U n(0) − V n(0)H s =√

2π(n−1 − n−3) −→n→∞ 0, (2.4)

and that for all t 0 we have

U n − V n(t)H s (1 − e−nt)n−s cos(nx)H s −√

2π(n−1 − n−3)

=√

π(1 − e−nt)

1 + n2

n2

s/2

−√

2π(n−1 − n−3). (2.5)

These properties are both related to (1.4) and (1.5). We derive now several properties forthe approximate solution sequences (U n)n and (V n)n. It turns out that though the supremumnorm of V n is considerably larger than that of U n, as n2U nL

∞/V nL

∞→n→∞ 1, these

approximate solutions satisfy surprisingly similar Sobolev estimates.

Lemma 2.2 (Estimates for the sequence (U n)n). Let s > 72 be given and n ∈ N, n 2 be

arbitrary. Then, we have

n−3

4 U n 2n−3 in S, (2.6)

|∂ xU n| n−s+1 in S. (2.7)

Additionally , there is a constant C > 0 independent of n such that

U nH r Cnr−s for all r s, (2.8)

and such that the error term

E U n := ∂ tU n − ∂ x(U n∂ xU n) (2.9)

satisfies

E U nH 1 Cn−s. (2.10)

†The functions ek := eik·/√

2π, k ∈ Z, form an orthonormal basis of L2(S) and

uH r = ΛruL2 =

k∈Z

(1 + |k|2)r|u|ekL2(S)|21/2

,

for all r ∈ R and u ∈ H r(S).




Proof. The estimates (2.6)–(2.8) are obvious consequences of the definition (2.1) and of (2.3). In order to establish (2.10), we compute that

E U n = −U n∂ 2xU n − (∂ xU n)2

= n−s−1

cos(nx) + n−2s+2

cos(2nx),and therefore the desired conclusion (2.10) follows from (2.3), since s 3.

Correspondingly, we have the following estimates for the sequence (V n)n.

Lemma 2.3 (Estimates for the sequence (V n)n). Let s > 72 be given and n ∈ N, n 2 be

arbitrary. Then, we have

n−1

4 V n 2n−1 in QT , (2.11)

|∂ xV n| n−s+1 in QT . (2.12)

Additionally , there is a constant C > 0 independent of n such that

maxt∈[0,T ]

V n(t)H r Cnr−s for all r s, (2.13)

and such that the error term

E V n := ∂ tV n − ∂ x(V n∂ xV n) (2.14)

satisfies

maxt∈[0,T ]

E V n(t)H 1 Cn−s. (2.15)

Proof. The estimates (2.11)–(2.13) are obtained directly from (2.2) and (2.3). Concerning

(2.15), we note that the terms with the highest norms cancel each other

E V n = ∂ tV n − V n∂ 2xV n − (∂ xV n)2

= −ne−ntn−s cos(nx) + n2e−ntn−s cos(nx)(n−1 + e−ntn−s cos(nx))

− n2e−2ntn−2s sin2(nx)

= n−2s+2e−2nt cos(2nx),

and the desired result follows from (2.3) and the fact that s 3.

3. Exact solutions and error estimates

The sequences (un)n and (vn)n from Theorem 1.1 are defined by letting un and vn denote theglobal solutions of (1.1) which satisfy initially

un(0) = U n and vn(0) = V n(0), (3.1)

respectively. Because the initial data are smooth, the exact solutions un and vn share this pro-perty, that is, un, vn ∈ C ∞([0, ∞) × S). Moreover, as U n and V n(0) are even, the uniquenessstatement in the latter theorem guarantees that un(t) and vn(t) are even for all t 0. Parti-cularly, we have that

∂ xun(t, ±π) = ∂ xvn(t, ±π) = 0, (3.2)

for all t 0 and n ∈ N, n 2. The parabolic maximum principles yield now that un and vnsatisfy estimates similar to some of those satisfied by U

n and V

n. Indeed, the strong maximum

principle and Hopf’s theorem applied in the cylinder [−π, π] × [0, T ], cf. [21], and the relations




(3.2) yield that

n−3

4 un 2n−3 in QT , (3.3)

n

−1

4 vn 2n−1 in QT . (3.4)

On the other hand, differentiating (1.1) with respect to x we see that ∂ xun is the solutionof the problem

zt = unzxx + 3∂ xunzx for t 0 and x ∈ [−π, π],z(t, ±π) = 0 for t 0,z(0, x) = ∂ xU n(x) for x ∈ [−π, π].

Hence, the strong maximum principle implies

|∂ xun| n−s+1 in QT . (3.5)

Using a similar argument, we obtain that

|∂ xvn| n−s+1 in QT . (3.6)

The estimates (3.3)–(3.6) together with the Kato–Ponce commutator estimate (1.7) are thekey ingredient in the proof of the following lemma.

Lemma 3.1 (The errors U n − unH r and V n − vnH r). Let r s > 72 and let n ∈ N,

n 2 be arbitrary. Then, there is a constant C > 0 independent of n such that

maxt∈[0,T ]

U n − un(t)H r Cnr−s, (3.7)

maxt∈[0,T ] V n(t) − vn(t)H r Cnr−s. (3.8)

Proof. In order to prove (3.7), it suffices to show, cf. (2.8), that

maxt∈[0,T ]

un(t)H r Cnr−s.

Therefore, we compute that

1

2

d

dtun2H r =

S

ΛrunΛr∂ x(un∂ xun) dx = − S

Λr(∂ xun)Λr(un∂ xun) dx

= − S un|Λ

r

(∂ xun)|2

dx − S Λ

r

(∂ xun)[Λ

r

, un]∂ xun dx,

in [0, T ]. Using Young’s inequality together with (1.7), (3.3), and (3.5), we then get

1

2

d

dtun2H r −n−3

4 Λr(∂ xun)2L2

+ Λr(∂ xun)L2[Λr, un]∂ xunL2

−n−3

4 Λr(∂ xun)2L2

+ C Λr(∂ xun)L2unH r∂ xunL∞

−n−3

4 Λr(∂ xun)2L2

+ Cn−s+1Λr(∂ xun)L2unH r

−n−3

4 Λr(∂ xun)2L2

+ n−3

4 Λr(∂ xun)2L2 + Cn−2s+5un2H r

Cn−2s+5un2H r ,




in [0, T ]. Since s 52 , we conclude that

maxt∈[0,T ]

un(t)H r C un(0)H r ,

and the desired claim (3.7) follows from (2.8). We still have to show that (3.8) holds true. Thisproperty follows by using the same arguments as in the proof of (3.7), because vn and V n satisfysimilar estimates to those verified by un and U n, respectively (cf. (2.13), (3.4), and (3.6)).

We remark that the estimate (2.15) can be improved in the sense that we may add amultiplicative term e−2nt on the right-hand side of (2.15). However, it is not clear from theproof above how to carry this property over to the sequence (vn)n. Based on Lemma 3.1, weobtain below estimates for the errors in the H 1-norm.

Lemma 3.2 (The errors U n − unH 1 and V n − vnH 1). Let s > 72 be given and n ∈ N

with n 2 arbitrary. Then, there exists a constant C > 0 independent of n such that

maxt∈[0,T ]

U n − un(t)H 1 Cn−s, (3.9)

maxt∈[0,T ]

V n(t) − vn(t)H 1 Cn−s. (3.10)

Proof. Denoting by w the difference between the approximate solution U n and the associ-ated exact solution un, that is, w := U n − un, we see that w is a solution of the initial valueproblem

wt = −(wwx)x + (U nwx)x + (w∂ xU n)x + E U n for t > 0,

w(0) = 0,(3.11)

whereby E U n is the error term defined by (2.9). With Λ := Λ1 and using (3.11), we find that

1

2

d

dtw2H 1 =

S

ΛwΛwt dx

= − S

ΛwΛ(wwx)x dx +

S

ΛwΛ(U nwx)x dx +

S

ΛwΛ(w∂ xU n)x dx

+ S ΛwΛE U n dx

=: I 1 + I 2 + I 3 + I 4,

in [0, T ]. We estimate the terms I i, 1 i 4, separately. Using integration by parts, theboundedness of (U n − un)n in H s(S), cf. (3.7), and the fact that H s(S) → C 3(S), we get that

I 1 =

S

wxΛ2(wwx) dx =

S

ww2x dx −

S

wx(wwx)xx dx

=

S

ww2x dx − 3

S

wxxw2x dx −

S

wxxxwwx dx

C w2H 1 . (3.12)




The same arguments, the fact that U n is positive and that (U n)n is bounded in H s(S), cf. (2.6)and (2.8), yield that

I 2 = − S

wxΛ2(U nwx) dx = − S

U nw2x dx +

S

wx(U nwx)xx dx

= − S

U nw2x dx −

S

wxx(U nwxx + ∂ xU nwx) dx

− S

U nw2xx dx −

S

∂ xU nwxwxx dx

1

2

S

∂ 2xU nw2x dx

C w2H 1 (3.13)

and

I 3 =

− S

wxΛ2(w∂ xU n) dx =

− S

∂ xU nwwx dx + S

wx(w∂ xU n)xx dx

= − S

∂ xU nwwx dx − S

wxx(w∂ 2xU n + wx∂ xU n) dx

= − S

∂ xU nwwx dx + 1

2

S

∂ 2xU nw2x +

S

∂ 3xU nwwx dx +

S

∂ 2xU nw2x dx

C w2H 1 . (3.14)

Finally, recalling (2.10), we have

I 4 =

S

ΛwΛE U n dx ΛwL2E U nH 1 Cn−swH 1 . (3.15)

Gathering (3.12)–(3.15), we see thatd

dtw2H 1 C (w2H 1 + n−swH 1) in [0, T ],

and therefored

dtw2H 1 C (w2H 1 + n−2s) in [0, T ].

Since w(0) = 0, the desired estimate (3.9) follows from Gronwall’s inequality. For the proof of (3.10), we argue analogously, as the difference V n − vn solves the same problem (3.11), butwith E U n replaced by E V n (see (2.14)). Moreover, all the properties of E U n , U n, and U n − unthat where used in the proof of (3.9) are inherited by E V n , V n, and V n − vn, respectively, cf.Lemmas 2.2, 2.3, and 3.1.

4. Proof of the main result

In the remaining part, we prove that the functions un and vn defined in the previous section,cf. (3.1), satisfy the properties required in Theorem 1.1 (for n 1 we simply set un = vn = 1).Indeed, the assertions (1.4) follow directly from (2.4) and Lemmas 2.2, 2.3, and 3.1. In orderto prove (1.5), we use the triangle inequality to get that

un(t) − vn(t)H s U n − V n(t)H s − U n − un(t)H s − V n(t) − vn(t)H s , (4.1)

for all t ∈ [0, T ] and n 2. The first term is estimated from below as in (2.5). For the last twoterms, we use the well-known interpolation inequality

uH s u(r−s)/(r−1)H 1 u(s−1)/(r−1)

H r ,




with r > s chosen arbitrarily, but fixed. Combined with (3.7) and (3.9), the latter inequalityyields

maxt∈[0,T ]

U n − un(t)H s U n − un(t)(r−s)/(r−1)H 1 U n − un(t)(s−1)/(r−1)

H r

Cn−s(r−s)/(r−1)n(r−s)(s−1)/(r−1)

Cn(s−r)/(r−1), (4.2)

and similarly we obtain from (3.8) and (3.10) that

maxt∈[0,T ]

V n(t) − vn(t)H s Cn(s−r)/(r−1). (4.3)

Gathering (2.5), (4.1)–(4.3), we have established that

un(t) − vn(t)H s √

π(1 − e−nt)

1 + n2

n2

s/2

−√

2π(n−1 − n−3) − Cn(s−r)/(r−1),

for all t ∈

(0, T ] and n 2. The desired final claim (1.5) follows now when letting n → ∞

.

Acknowledgements. The author thanks the anonymous referees for the suggestions andcomments which have improved the quality of the article.

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Bogdan-Vasile Matioc Institut fur Angewandte Mathematik Leibniz Universitat Hannover Welfengarten 130167 Hannover Germany

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