独立行政法人日本学術振興会 日独共同大学院プログラム

第回 日独流体数学国際研究集会■ Program ■Tue, Nov. 5
9:30-10:40

10:55-12:05

12:20-13:00

John G. HEYWOOD①(Univ. of British
Columbia)
Konstantinas
PILECKAS ①(Vilnius Uni.)
Cameron TROPEA(TU Darmstadt)

13:00-15:00

15:00-16:10

Wed, Nov. 6

Thu, Nov. 7

Fri, Nov. 8

PILECKAS ②

PILECKAS ③HEYWOOD ③HEYWOOD ②TABATA ③

HEYWOOD ④

Carlo R. GRISANTI(Univ. of Pisa)

Francesca CRISPO(2nd Univ. of Naples)

Maria LUKACOVA(Univ. of Mainz)

Lunch break
Masahisa
TABATA ①(Waseda Univ.)

TABATA ②

16:40(-16:55)

PILECKAS ④TABATA ④

Martin RAPP(TU Darmstadt)

Bangwei SHE(Univ. of Mainz)

Pen-yuan HSU(Univ. of Tokyo)

Hana MIZEROVA(Univ. of Mainz)

Shinya NISHIBATA(Tokyo Tech)

Takayuki KUBO
(Univ. of Tsukuba)

17:30-17:45

Katharina SCHADE(TU Darmstadt)

Yuto IMAI(Waseda Univ.)

Naofumi MORI
(Kyusyu Univ.)

17:50-18:05

Jonas SAUER(TU Darmstadt)

Tomoyuki
NAKATSUKA(Nagoya Univ.)

Takahito
KASHIWABARA(Univ. of Tokyo)●→Main-course●→40min talk●→15min talk

(17:05-)-17:20＊18：00～Reception

Discussion room → open on Wed and Fri / 9:30 - 17:00 / 63 Bldg 1F
Meeting room for Mathematics and Applied Mathematics
Closed on Tue and Thu!!

Main-Course

Main-CourseJohn G. HEYWOOD
University of British Columbia, Vancouver
Title:
On a Conjectured Pointwise Bound for Solutions of the Stokes Equationsin Nonsmooth Domains
Abstract:Let Ω be an arbitrary open subset of R3 . Let D (Ω) = {φ ∈ C∞
0 (Ω) : ∇ · φ = 0} .
And let J (Ω) and J0 (Ω) be the completions of D (Ω) in the L2 -norm ∥·∥ and in
the Dirichlet-norm ∥∇·∥ , respectively. Given u ∈ J0 (Ω) , there is at most one
f ∈ J (Ω) such that (∇u, ∇φ) = − (f , φ) for all φ ∈ D (Ω) . If such a function fe and Wenzheng Xie has conjectured thatexists, it is denoted by △u,
sup |u|2 ≤Ω1e∥∇u∥ △u.
3π(1)

Xie proved (1) for Ω = R3 , and proved that its general validity would follow fromits validity for smoothly bounded domains. He also proved that the constant is
optimal, for any domain.
For general domains, Xie proved an analogue of (1) for the Laplacian, considering
it as a model problem. However, at one point of the proof he used the maximum
principle, in showing that for every point y in a smoothly bounded domain Ω,∥Gµ (·, y)∥L2 (Ω) ≤ ∥gµ (·, y)∥L2 (R3 ) ,(2)where Gµ is the Green’s function for the Helmholtz operator −△+µ in Ω, and gµ is
the corresponding fundamental singularity. Although he conjectured an analogueof (2) for the Stokes operator, it has yet to be proven. However, I have proven, evenfor the Stokes operator, that the ratio of the two sides of (2) tends to 1 as µ → ∞.
Consequently, to complete the proof of (1), it is now enough to consider smoothly
bounded domains and to show a tendency toward singularity, in the sense thate nµn ≡ △u2/ ∥∇un ∥2 → ∞,(3)

if {un } is a sequence of functions such that the ratio of the left to right sidesof (1) tends to its supremum. My eﬀorts to prove (3) have led to many further
conjectures.
It is well understood that if (1) is valid for arbitrary domains, then so is most of
the current existence and regularity theory for the Navier-Stokes equations.
Date:⃝
1 Tuesday, Nov. 5 9:30-10:40⃝
3 Friday, Nov. 8 9:30-10:40

2⃝
2 Wednesday, Nov. 6 10:55-12:05⃝
4 Friday, Nov. 8 10:55-12:05

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Konstantin PILECKAS
Vilnius University, Vilnius
Faculty of Mathematics and Informatics, Vilnius University

Title:
Stationary Navier–Stokes equations in bounded domains with multiply
connected boundaries. Leray’s problem
Abstract:

Lecture 1

During this lecture we study in a bounded domain Ω ⊂ Rn the stationary NavierStokes system with homogeneous boundary conditions −ν∆v + (v · ∇)v + ∇p = f in Ω,(1)
div v = 0 in Ω,v = 0 on ∂Ω,where ∇ =

n
)
∑
∂
,
a·b
=
ai bi , div v = ∇·v, ∆x = ∇·∇ is the Laplacian,
,
.
.
.
∂x1
∂xn

( ∂

i=1

v and p stand for the velocity vector and the pressure, f is the density of external
forces:v = (v1 , . . . , vn ),f = (f1 , . . . , fn ),ν > 0 means the constant viscosity of the liquid.
Let H(Ω) be a subspace of solenoidal vector ﬁelds belonging to W̊21 (Ω). By aweak solution of problem (1) we understand a vector function v ∈ H(Ω) satisfying
the integral identity∫∫∫ν ∇v · ∇η dx + (v · ∇)v · η dx = f · η dx ∀ η ∈ H(Ω),
(2)
Ω

Ω

Ω

where
∇v · ∇η =n ∑n∑∂vi ∂ηii=1 j=1∂xj ∂xj.

It will be shown that the integral identity (2) is equivalent to the operator equationin the space H(Ω)
v = Av(3)
with the compact operator A. In order to apply the Leray–Schauder Fixed Point
Theorem, we will prove that all possible solutions vλ of the equationvλ = λAvλ ,λ ∈ [0, 1],(4)

are uniformly (with respect to λ) bounded. Then from the Leray–Schauder it
follows that equation (3) has at least one solution.
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The solution of the integral identity (2) (or, equivalently, of the operator equation(3)) in general could be non-unique. However we prove the uniqueness of thissolution for small data.
∫
We also show that there exists a pressure function p ∈ L2 (Ω) such that p(x) dx =
∫

0 and

∫
∇v · ∇η dx +νΩ(v · ∇)v · η dx =
Ω

∫

Ω

∫
p div η dxΩf · η dx ∀ η ∈ W̊21 (Ω).

+

(5)

Ω

Moreover,2v ∈ W2,loc(Ω),

1p ∈ W2,loc(Ω)

and the pair (v, p) satisﬁes the Navier–Stokes equations (1) almost everywhere inΩ.

Lecture 2
The stationary Navier-Stokes system with nonhomogeneous boundary conditions −ν∆v + (v · ∇)v + ∇p = 0 in Ω,(6)
div v = 0 in Ω,v = a on ∂Ω,will be studied in the multi-connected domain Ω = Ω0 \ ∪N
j=1 Ωj , where Ω̄j ⊂ Ω,Ωj ∩ Ωi = ∅, j ̸= i.
The continuity equation div v = 0 implies the necessary compatibility condition
for the solvability of problem (6):∫a · n dS =N ∫∑a · n dS =

j=1 Γ

∂Ω

j

N
∑

Fj = 0,(7)

j=1

where n is a unit vector of the outward (with respect to Ω) normal to ∂Ω, Γj =∂Ωj . The compatibility condition (7) means that the net ﬂux of the ﬂuid over theboundary ∂Ω is zero.
In this lecturer we shall prove the existence of the solution under the stronger
than (7) condition which requires the all ﬂuxes Fj of the boundary value a to be
zero separately across each component Γj of the boundary ∂Ω:∫Fj = a · n dS = 0,
j = 1, 2, . . . , N,(8)ΓjNotice that the condition (8) does not allow the presence of sinks and sources.

4

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In this lecture we will study the method based on of the Leray–Hopf extensionof the boundary data. If the condition (8) is valid, then there exists a function bsuch thatrot b(x) ∂Ω = a(x).
We construct the Leray–Hopf cut–oﬀ function ζ(x, ε) which has following properties:(i) ζ(x, ε) = 1 f or x ∈ ∂Ω,ζ(x, ε) = 0 f or dist(x, ∂Ω) ≥ δ = δ(ε),(ii) 0 ≤ ζ(x, ε) ≤ 1,(iii) |∇ζ(x, ε)| ≤cεwith the constant c independent of ε.
dist(x, ∂Ω)

The Leray–Hopf’s extension function has the form()
B(x, ε) = rot ζ(x, ε)b(x) .

(9)

Then
div B(x, ε) = 0,

B(x, ε) ∂Ω = a(x).

We look for a week solution of the problem (6) in the form v = u + B, whereu ∈ H(Ω). Then for u we get the integral identity∫∫∫∫ν ∇u · ∇η dx + (u · ∇)u · η dx + (u · ∇)B · η dx + (B · ∇)u · η dx
Ω

Ω

∫
Ω

∫
(B · ∇)B · η dx − ν

f · η dx −

=

Ω

Ω

∫
Ω

∇B · ∇η dx ∀ η ∈ H(Ω).

(10)

Ω

The integral identity (10) is equivalent to the operator equationu = Buwith the compact operator B in the space H(Ω). In order to show that all possible
solutions of the operator equation with the parameter λ are uniformly bounded,
we prove and apply the following Leray–Hopf’s inequality∫∫(u · ∇)B · u dx ≤ cε |∇u|2 dx ∀u ∈ H(Ω),
(11)
Ω

Ω

where the constant c is independent of ε and u. We get the desired estimate bychoosing in (11) the parameter ε > 0 suﬃciently small.

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Lecture 3
We shall show that the Leray–Hopf’s extension of the boundary data is notpossible, if the condition (8) is violated (the counterexample of A. Takeshita will
be presented), i.e., we prove that if the ﬂuxes Fi of the boundary value a throw theconnected components Γi of the boundary ∂Ω are nonzero, then the Leray–Hopf’s
inequality (11), in general, is not valid. Thus in this case it is not possible to apply
the same as in the previous lecture method.
In this lecture we consider the method of getting an a priory estimate by acontradiction. In order to simplify the proofs, we still assume for a while that thecondition (8) is fulﬁlled. The main idea of the last method consist in the following.
Consider the integral identity corresponding to the operator equation with theparameter λ:
∫
∫
∫
( λ
)
( λ
)
λ
λ
ν ∇u · ∇η dx − λ(u + B) · ∇ η · u dx − λu · ∇ η · B dx
Ω

=λ

(∫ (

Ω

)

∫

∫
∇B · ∇η dx + ν

B · ∇ η · B dx − ν)
f · η dx

∀ η ∈ H(Ω). (12)

Ω

Ω

Ω

Ω

Here B is an arbitrary divergence free extension of the boundary value a.
Assume that the solutions of (12) are not uniformly bounded in H(Ω) with respectto λ ∈ [0, 1]. Then there exist sequences {λk }k∈N ⊂ [0, 1] and {uλk = uk }k∈N ∈H(Ω) such that
∫
∫
∫
(
)
(
)
uk · ∇ η · B dx
(uk + B) · ∇ η · uk dx − λk
ν ∇uk · ∇η dx − λk
Ω

Ω

(∫

= λk

(

)
B · ∇ η · B dx − ν∫∫

f · η dx

∇B · ∇η dx + ν
Ω

Ω

Ω

)

∀ η ∈ H(Ω), (13)Ωandlim λk = λ0 ∈ [0, 1],k→∞lim Jk = lim ∥∇uk ∥L2 (Ω) = ∞.

k→∞

k→∞

First, taking in (13) η = Jk−2 uk and passing to a limit we obtain the equality∫)(
b · B dx,
b·∇ u
(14)
u
ν = λ0Ωb is a weak limit in H(Ω) of the sequence {Jk−1 uk }.
where ub andSecond, taking in (13) η = Jk−2 ξ with arbitrary ξ ∈ H(Ω) we obtain that u
the corresponding pressure function pb satisfy the Euler equations:
 (
)
b·∇ u
b + ∇b
λ0 u
p = 0,

(15)
b = 0,div ub |∂Ω = 0.
u
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It follows from (15) thatpb |Γi = pbi , i = 1, . . . , N,
where pbi are constants.
Multiplying (15) by B and integrating by parts yields
∫
λ0

(

)

b·∇ u
b · B dx =uN∑

pbi Fi .

(16)

i=1

Ω

If the condition (8) is valid, i.e. Fi = 0, i = 1, . . . , N , then the right-hand side of(16) is equal to zero and (16) contradicts to (14). Thus, all possible solutions of(12) are bounded and by the Leray-Schauder Fixed Point Theorem there exists atleast one weak solution of problem (6).
Since by incompressibility of the ﬂuid
N∑

Fi = 0,i=1

the right-hand side of (16) will be zero also if
pb1 = pb2 = . . . = pbN .
It was shown by Ch. Amick constructing a counterexample that, in general, it
is not the case. However, if the domain Ω ⊂ R2 and the boundary data satisfy
certain symmetry conditions thenpb1 = pb2 = . . . = pbN
holds and we get the needed a priory estimate by the contradiction. Thus, For thesymmetric case the existence of the solution to (6) is proved (by Ch. Amick) only
under the necessary condition (7).
Finally, we prove the existence of the solution to problem (6) in the case of”suﬃciently small” ﬂuxes Fj .

Lecture 4
We will study the nonhomogeneous boundary value problem for Navier–Stokes
equations in a two–dimensional bounded multiply connected domain Ω ⊂ R2 . We
will prove that this problem admits at least one solution only under the necessary
condition (7) without imposing any smallness assumptions on the value of |F |.
The proof of the main result uses the Bernoulli law for a weak solution to theEuler equations. The detailed proof of the weak version of the Bernoulli law will
be presented. A priory estimate for possible solutions of (12) is proved by a contradiction method. We get the contradiction by using the co-area formula. This
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proof is essentially based on results concerning ”ﬁne properties” of Sobolev functions and level sets of W 2,1 -functions (a weak version of the Morse-Sard theorem)
which were obtained recently by J. Bourgain, M.V. Korobkov and J. Kristensen.
A short introduction to these topics will be presented in the lecture.
Date:⃝
1 Tuesday, Nov. 5 10:55-12:05⃝
3 Thursday, Nov. 7 9:30-10:40

8⃝
2 Wednesday, Nov. 6 9:30-10:40⃝
4 Thursday, Nov. 7 15:00-16:10

Main-Course

Main-Course

Masahisa TABATA
Waseda University, Tokyo
Department of Mathematics, Waseda University
3-4-1, Ohkubo, Shinjuku, Tokyo, 169-8555 Japantabata@waseda.jp

Title:
Galerkin-Characteristics Finite Element Methods for Flow Problems

Abstract:
A remarkable feature of fow problems is that the governing equations include thematerial derivative term
Dϕ∂ϕ≡
+ u · ∇ϕ,Dt∂twhere u is a function expressing the ﬂow ﬁeld and ϕ is an unknown physicalquantity such as the density, the velocity, or the energy. It makes the problems
asymmetric and nonlinear when the velocity ﬁeld is unknown, e.g., in the NavierStokes equations ϕ stands for each component ui of unknown velocity u, which leads
to the nonlinear term u · ∇ui . The combination of this term with the diﬀusion
term −ν∆u describes many important phenomena in sciences and engineering.
It produces various fruitful and interesting results, especially when the diﬀusion
constant ν is small, e.g., high Reynolds number problems in the Navier-Stokes
equations. In devising numerical schemes for the solution of those phenomena, itis well-known that the discretization of this term is crucial because the conventional
Galerkin ﬁnite element method and the centered diﬀerence method easily produce
unphysical oscillating solutions. Among remedies for the instability the methodof characteristics seems to be natural from the physical point of view since itapproximates the particle movement along the trajectory.
In this course we consider ﬁnite element methods based on characteristics, i.e.,
Galerkin-characteristics FEMs. Those methods lead to symmetric schemes which
are robust even for convection-dominated problems. Beginning with the basic idea
and fundamental properties of Galerkin-characteristics method, we discuss recent
results on the stability and convergence of the schemes for ﬂow problems.
Date:⃝
1 Tuesday, Nov. 5 15:00-16:10⃝
3 Thursday, Nov. 7 10:55-12:05⃝
2 Wednesday, Nov. 6 15:00-16:10⃝
4 Friday, Nov. 8 15:00-16:10

Main-Course940 minutes talks

40 minutes talks

Francesca CRISPO
Second University of Naples, Caserta

francesca.crispo@unina2.it

with Paolo Maremonti (Seconda Università degli Studi di Napoli)
Title:High regularity results of solutions to modified p-Navier-Stokes equations
Abstract:
We consider the following modified p-Navier-Stokes system−∇ · (|∇u|p−2 ∇u) + (u · ∇)u + ∇π = f ,∇ · u = 0 in R3 ,(1)

in the sub-quadratic case p ∈ (1, 2). This system was considered for the ﬁrst timein the sixties by Lions. Subsequently the system obtained by replacing ∇u with
its symmetric part has been much more studied, due to its connection with the
motion of shear-thinning ﬂuids. As this last system is usually called p-NavierStokes system, we refer to system (1) as modified p-Navier-Stokes.
We ﬁnd suﬃcient conditions for the existence of high regular solutions, in the
3psense of second derivatives in Lq (R3 ), for q ∈ ( 3−p, +∞). By embedding, we get
1,αC -regularity of solutions.
As for the Navier-Stokes system in R3 , the uniqueness in the existence class is
not achieved.
These results are part of a joint work with Paolo Maremonti (Second Univesrityof Naples).
Date:Thursday, Nov. 7 12:20-13:00

Carlo Romano GRISANTI
University of Pisa, Pisa

grisanti@dma.unipi.it

with Giovanni P. Galdi (University of Pittsburgh)
Title:
An inverse problem with time periodicity for a non-newtonian liquid in
an infinite pipe.
Abstract:
We study the motion of a non-newtonian liquid in a straight pipe of inﬁnite
length. The time-periodic case under consideration has an analogue in the newtonian framework since 1955 with the work of Womersley concerning the blood
motion in large arteries. In an inﬁnite pipe of constant cross section Σ, a velocity
ﬁeld parallel to the axis of the pipe and not depending on the axial coordinate
10

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40 minutes talks

has to be found. The only constraints are the no-slip boundary condition and theﬂow-rate, which is a prescribed, time periodic function α(t). The inverse problem
for the velocity v and the pressure gradient Γ is the following∂v= µ0 ∆v + ∇ · S(∇v) + Γ(t) in Σ × (0, T )


∂t




 ∫
v(x, t) dx = α(t)in [0, T ]


Σ







v|∂Σ = 0
in [0, T ]Under suitably assumptions of coercivity, growth and monotonicity on the extrastress tensor S(∇v), we give a positive answer to the well posedness of the problem.
Our result takes into account both the shear-thinning and shear-thickening behavior of the liquid.
Date:
Wednesday, Nov. 6 12:20-13:00

Takayuki KUBO
University of Tsukuba, Tsukuba
Title:On two phase problem: compressible-compressible model problem
Abstract:
We consider the model problem for the two phase problem in cases of compressible- compressible ﬂuid ﬂows without surface tension. In order to prove the local intime existence of this problem, the generation of analytic semigroup for linearlized
problem and its maximal Lp − Lq regularity are needed in our method. The key
step of our method is to prove the existence of R-bounded solution operator to
the generalized resolvent problem corresponding the linearized problem:1
λρ± + γ±div⃗u± = f±

in RN
±,

λ⃗u± − Div S± (⃗u± , ρ± ) = g±
⃗u+ |x =0+ − ⃗u− |x =0− = ⃗k

in RN
±,
on RN
0 ,

S+ (⃗u+ , ρ+ )n+ |xN =0+ − S− (⃗u− , ρ− )n− |xN =0− = −⃗h

on RN
0 .

N

N

Here, ρ± , ⃗u± = (u±,1 , . . . , u±,N )(N ≥ 2) are unknown mass density and unknown
2 ρ )I is stretching tensor,
velocity ﬁelds. S± (⃗u± , ρ± ) = 2µ1± D(⃗u± ) + (µ2± div⃗u± − γ±±D(⃗u) = (∇⃗u +T ∇⃗u)/2 is N × N matrix called the Cauchy deformation tensor andI denotes the N × N identity matrix. Moreover for N × N matrix function M =40 minutes talks11

40 minutes talks∑(Mij ), the i th component of DivM is deﬁned by N
n+ = (0, . . . , 0, −1)
j=1 ∂j Mij . ⃗niiis the unit outward normal to R+ and µ± , γ± (i = 1, 2) are constants such that12µ1± > 0, µ1± + µ2± > 0, γ±
, γ±
≥ 0.

The resolvent parameter λ varies in Λε,λ0 = Σε,λ0 ∩ Kε ,where Σε,λ0 = {λ ∈ C || arg λ| ≤ π − ε, |λ| ≥ λ0 } and{}Kε = λ ∈ C | (Reλ + γm + ε)2 + (Imλ)2 ≥ (γm + ε)2
( + +
)
γ−γ−
γ γ
with γm = max µ11+µ2 2 , µ11+µ2 2 .
++−−In this talk, we will report the existence of R-bounded solution operator tothis generalized resolvent problem. This is joint work with Yoshihiro Shibata andKouhei Soga(Waseda Univ.).
Date:
Wednesday, Nov. 6 16:40-17:20

Maria LUKACOVA
Johannes Gutenberg University Mainz, Mainz
Title:
Asymptotic preserving schemes for weakly compressible flows
Abstract:
Many geophysical ﬂows exhibit complex multiscale phenomena in particular when
low Mach or Froude number limits are considered. Consequently, construction ofeﬃcient problem-suited schemes for weakly compressible ﬂows is a challenging
problem.
In this talk we present new asymptotic preserving ﬁnite volume and discontinuous
Galerkin schemes within the framework of bicharacteristic schemes. The methods
couple a ﬁnite volume or discontinuous Galerkin formulation with approximate
evolution operators. The latter are constructed using the bicharacteristics of multidimensional hyperbolic systems, such that all of the inﬁnitely many directions ofwave propagation are considered explicitly. In order to take into account multiscale
phenomena nonlinear ﬂuxes are split into a linear part governing the acoustic andgravitational waves and to the rest nonlinear part that models advection. Time
integration is realized by the IMEX type approximation using the semi-implicit
second-order backward diﬀerentiation formulas (BDF2) scheme. We prove thatthe proposed large time step schemes are asymptotic preserving for small Machor Froude number ﬂows. Numerical experiments demonstrate stability, accuracy
and robustness of these new large time step with respect to small Mach or Froudenumbers. Comparisons with the standard one-dimensional approximate Riemann12

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40 minutes talks

solvers used for the ﬂux integration demonstrate better stability, accuracy as well
as reliability of our new multidimensional methods.
This work has been done in cooperation with S. Noelle (Aachen), K.R. Arun
(Trivandrum), L. Yelash (Mainz), G. Bispen (Mainz), F. Giraldo (Monterey) andA. Mueller (Monterey).
Date:
Friday, Nov. 8 12:20-13:00

Shinya NISHIBATATokyo Institute of Technology, Tokyo
Department of Mathematical and Computing Sciences,
Tokyo Institute of Technology
shinya@is.titech.ac.jp

Title:
Stationary waves to symmetric hyperbolic-parabolic systems in half
space
Abstract:
In this talk, we consider the large-time behavior of solutions to hyperbolicparabolic coupled systems in the half line. Assuming that the systems admit theentropy function, we may rewrite them to symmetric forms. For these symmetrizable hyperbolic-parabolic systems, we ﬁrst prove the existence of the stationary
solution. In the case where one eigenvalue of Jacobian matrix appeared in a stationary problem is zero, we assume that the characteristics ﬁeld corresponding to thezero eigenvalue is genuine non-linear in order to show the existence of a degeneratestationary solution. We also prove that the stationary solution is time asymptotically stable under a smallness assumption on the initial perturbation. The key tothe proof is to derive the uniform a priori estimates by using the energy method in
half space developed by Matsumura and Nishida as well as the stability conditionof Shizuta–Kawashima type. These theorems for the general hyperbolic-parabolic
system cover the compressible Navier–Stokes equation for heat conductive gas.
These results are obtained through the joint work with Dr. Tohru Nakamura
at Kyushu University.
Date:
Tuesday, Nov. 5 16:40-17:20

Cameron TROPEATechnical University of Darmstadt, Darmstadt
Institute of Fluid Mechanics and Aerodynamics

Title:New Horizons for Optimization in Fluid Mechanics
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40 minutes talks

Abstract:
Improved capabilities in both experimental and numerical ﬂuid dynamics suggestthat closer interaction between ‘mathematical’ and ‘engineering’ interests in ﬂuid
dynamics, pursuits which have often been followed quite remote from one another,is becoming indispensable. This applies to a wide range of problems, starting fromthe elusive description of a dynamic contact angle to problems of ﬂow geometry
optimization or closed-loop feedback control of ﬂow control devices. The focus of
this lecture will be on several experimental developments over the very recent past
which, while not being game-changing in character, have signiﬁcantly broadened
our ability to elucidate ﬂow physics in the laboratory and open up new avenues ofanalysis, especially for time-dependent ﬂow ﬁelds. The examples presented havebeen chosen to be representative of ﬁelds in which considerable scope for collaboration between the engineer and mathematician can immediately be recognized.
One main topic will be addressed, relating to our ability to capture three velocity components of a ﬂow ﬁeld over three dimensions, and the consequences thiscapability brings to our understanding of ﬂow ﬁelds. Two examples in this topicalarea will be given; one involving time resolved measurements, the other time averaged. The very complete ﬂow ﬁeld descriptions at heretofore unchartered time andlength scales allow models to be formulated which are much more universal in theirvalidity than models based only on ‘engineering’ correlations. It is at this model
formulation stage and/or for purposes of optimization where an accompanying
rigorous mathematical description of the ﬂuid dynamics can be invaluable.
Date:
Tuesday, Nov. 5 12:20-13:00

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15 minutes talks

Pen-Yuan Hsu
The University of Tokyo, Tokyo
Title:On Liouville problems for the planer Navier-Stokes equations with theno-slip boundary condition
Abstract:
In this work we prove a Liouville type result for the Navier-Stokes equations in
the half plane with the no-slip boundary condition. As an application, we extend
a geometric regularity criterion [?] of solutions to the Navier-Stokes equations in
R3 to the case when the domain is a half space R3+ with the Dirichlet condition by
applying our Liouville type theorem.
This talk is based on the joint work with Professor Yoshikazu Giga and Professor
Yasunori Maekawa.
References:[1] Y. Giga and H. Miura, On vorticity directions near singularities for the NavierStokes ﬂows with inﬁnite energy, Comm. Math. Phys. 303 (2011) 289-300.
Date:
Thursday, Nov. 7 17:05-17:20

Yuto IMAI
Waseda University, Tokyo
Title:
Quaternifiations and extensions of current algebras on S 3
Abstract:[]
Let ( g ,,) be a complex Lie algebra and U (g) be the enveloping algeg
bra of g. Let H be the quaternions and S 3 H be the space of H-valued mappings on S 3 . We introduce a Lie algebra structure on S 3 gH = S 3 H ⊗ U (g).
Then we introduce a 2-cocycle on S 3 gH and the corresponding central extension S 3 gH ⊕ (Ca). As a Lie subalgebra of S 3 H we have the Lie algebra ofLaurent polynomial spinors C[ϕ±(m,l,k) ] spanned by a complete orthogonal system of eigenspinors {ϕ±(m,l,k) }m,l,k of the tangential Dirac operator on S 3 . Then
C[ϕ±(m,l,k) ] ⊗ U (g) is a Lie subalgebra of S 3 gH . Its central extension bg(a) is obtained as a Lie subalgebra of S 3 gH ⊕ (Ca). Finally we have a Lie algebra b
g whichis obtained by adding to bg(a) a derivation d. When g is a simple Lie algebra, bgis an inﬁite dimensional simple Lie algebra. We shall investigate the root spacedecomposition and the Chevalley generators of bg.

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Date:
Wednesday, Nov. 6 17:30-17:45

Takahito KASHIWABARAThe University of Tokyo, Tokyo
Title:On the Stokes equation with a generalized Robin boundary condition
arising from fluid-structure interaction
Abstract:
We consider some “genelarized” Robin boundary condition for the incompressible
Stokes problem which reads:−ν∆u + ∇p = f , div u = 0in Ω,σ(u, p)n + αu − βdivΓ ΠΓ (u) + βκΠΓ (u)n = h on Γ = ∂Ω,
where σ and ΠΓ are ﬂuid and membrane stress tensors given byσ(u, p) = −pI + ν(∇u + (∇u)T ),ΠΓ (u) = λdivΓ u I + µ(∇Γ u + (∇Γ u)T ).
Here, α, β, λ, µ, ν are positive constants; n is the outer unit normal on Γ; ∇Γ anddivΓ are the surface gradient and divergence operators; κ = divΓ n is the mean
curvature of Γ. Such a boundary condition is motivated from some ﬂuid-structureinteraction problems in hemodynamics.
We establish the well-posedness of the PDE problem above and its ﬁnite element
approximation. The key feature is the introduction of Sobolev spaces which admit
equal-order regularity both in Ω and on Γ, i.e., H m (Ω; Γ) = {v ∈ H m (Ω) ; v|Γ ∈H m (Γ)}, m = 1, 2, . . . . This is a joint work with C. M. Colciago (EPFL) andProf. A. Quarteroni (EPFL and MOX).
Date:
Thursday, Nov. 7 17:50-18:05

Hana Mizerová
Johannes Gutenberg University Mainz, Mainz
with Mária Lukáčová (Mainz Univ.)
Title:
Numerical analysis of a viscoelastic fluid flow
Abstract:
We consider the viscoelastic model describing the behavior of some polymeric
ﬂuids. The polymer molecules are treated as two beads connected by a nonlinear
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spring. The Peterlin approximation of the spring force is used to derive the equation for the conformation tensor. The ﬁrst aim of this short talk is to present theexistence results for this model. The second part of the talk is dedicated to theerror estimates for the suitable approximation of some viscoelastic models. We
apply the method of characteristics on the convective term to get the numericalresults.
Date:
Friday, Nov. 8 17:05-17:20

Naofumi MORI
Kyusyu University, Fukuoka

Title:
Dissipative structure for the Timoshenko system with Cattaneo’s typeheat conduction
Abstract:
In this talk we consider the Timoshenko system by introducing the heat conduction satisfying the Cattaneo law in the one-dimensional whole space. The systemis written in the form

φtt − (φx − ψ)x = 0,





 ψtt − a2 ψxx − (φx − ψ) + bθx = 0,θt + qx + bψtx = 0, τ q + q + κθ = 0,0 txwhere t ≥ 0 is the time variable, x ∈ R is the spacial variable which denotes thepoint on the center line of the beam, φ is the transversal displacement, ψ is therotation angle of the beam, θ denotes the temperature, q denotes the heat ﬂow,
and a, b, κ and τ0 are positive constants.
We observe that the dissipative structure of the system is of the regularity-loss
type which is a little diﬀerent from that of the dissipative Timoshenko system
studied in [?], though they are both regarded as symmetric hyperbolic systems.
Besides, we also see that the dissipative structure of the system is the same as thatof the system with Fourier’s type heat conduction but the condition in which the
decay property is the regularity-loss type is diﬀerent.
Moreover, we show the optimal L2 decay estimates of the solution in a generalsituation. The key of the proof is to show the detailed pointwise estimates of thesolution in the Fourier space.
References:[1] K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss typefor dissipative Timoshenko system, Math. Models Meth. Appl. Sci., 18 (2008),
647-667.
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Date:
Thursday, Nov. 7 17:30-17:45

Tomoyuki NAKATSUKA
Nagoya University, Nagoya
Title:On uniqueness of symmetric Navier-Stokes flows around a body in theplane
Abstract:
In this talk, we consider the uniqueness of symmetric weak solutions to the stationary Navier-Stokes equation in two-dimensional exterior domains. It is known
that, under suitable symmetry conditions on the domain and the data, the problem
admits at least one symmetric weak solution tending to zero at inﬁnity. Thus far,the existence of a weak solution tending to zero at inﬁnity is not obtained without
symmetry.
Given two symmetric weak solutions u and v, we show that if u satisﬁes theenergy inequality ∥∇u∥22 ≤ (f, u) and sup(|x| + 1)|v(x)| is suﬃciently small, thenu = v. As an application, our uniqueness theorem, together with the recent
result of Yamazaki(2011), gives some information on the asymptotic behavior of asymmetric weak solution corresponding to small data. The proof of the uniqueness∞ andtheorem relies upon the density property for the solenoidal vector ﬁeld C0,σ
the Hardy inequality for symmetric functions.
Date:
Wednesday, Nov. 6 17:50-18:05

Martin RAPP
Technical University of Darmstadt, Darmstadt
Title:
Variable Exponent Spaces and Their Applications to Fluid Dynamics
Abstract:
We are interested in solving equations modelling the behaviour of electrorheological ﬂuids in a bounded domain Ω. These ﬂuids have a viscosity which depends
on the strength of an electrical ﬁeld which we assume to be non-constant in spacesuch that the diﬀusive term is modelled by the p(·)-Laplacian− div (δ + ∇u)p(·)−2 ∇u,where p : Ω → [1, ∞) is a variable exponent and δ ≥ 0.
First the theory of Lebesgue and Sobolev Spaces with variable exponents will be

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introduced. Then we will solve the Navier-Stokes equations for electrorheologicalﬂuids and the following convection-diﬀusion equation− div (δ + ∇u)p(·)−2 ∇u + a · ∇u = f
u = 0

in Ω
on ∂Ω.

r(·)

for some given a ∈ Lσ (Ω) by the developed theory.
Date:
Thursday, Nov. 7 16:40-16:55

Jonas SAUER
Technical University of Darmstadt, Darmstadt
Fachbereich Mathematik, Technische Universität Darmstadt, 64283 Darmstadt,
jsauer@mathematik.tu-darmstadt.de

Title:
Navier-Stokes Flow in Spatially Periodical Domains
Abstract:
We investigate maximal regularity in Lq (G)-spaces of the abstract Stokes operator
AG,q in the locally compact abelian group G := Rn−1 × T , where T is the onedimensional torus and the abstract Stokes operator is deﬁned in an obvious way on
Lqσ (G). This is achieved by using abstract harmonic analysis in order to establish aconcept of the class of Muckenhoupt weights Ap (G) similar to (and including) theclassical concept in the setup of Rn . In fact, this concept carries over to a muchlarger class of locally abelian groups. With the Muckenhoupt weights at hand, we
proof a group version of the extrapolation theorem, corresponding to the classicalextrapolation theorem due to Garcı́a-Cuerva and Rubio de Francia [1]. Using this,we show the R-boundedness of the operator family λ(λ − AG,q )−1 in the weighted
space Lqω,σ (G) of solenoidal vector functions.
The group ansatz also furnishes us with unique solvability of the linear Stokes
equations in G even for non-homogeneous divergence data. This result may ben−1transferred in the fashion of Farwig and Sohr [2,3] to the periodic half space R+
×T and due to the non-homogeneous divergence data also to periodical cylindrical
domains with non-constant cross-section.
References:[1] Garcı́a-Cuerva, J. and Rubio de Francia, J. L.: Weighted norm inequalities
and related topics. North-Holland Publishing Co., Amsterdam, Volume 116,
1985
[2] Farwig, R. and Sohr, H.: Weighted Lq -theory for the Stokes resolvent in exterior domains. J. Math. Soc. Japan, 49(2) (1997), 251–288

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15 minutes talks[3] Farwig, R. and Sohr, H.: Generalized resolvent estimates for the Stokes system
in bounded and unbouded domains. J. Math. Soc. Japan, 46(4) (1994), 607–643
Date:
Tuesday, Nov. 5 17:50-18:05

Katharina SCHADE
Technical University of Darmstadt, Darmstadt
with M. Hieber and J. Prüss
Title:A thermodynamically consistent extension of the simplified EricksenLeslie model
Abstract:
The simpliﬁed Ericksen-Leslie model [?, ?] describes the ﬂow of nematic liquid
crystals. By imposing the ﬁrst and second law of thermodynamics we obtaina temperature-dependent extension which is hence a priori thermodynamicallyconsistent.
By treating the problem in a quasilinear setting similarly to [?], we subsequently
show the local existence of strong solutions and global existence of strong solutions
near equilibria as well as global existence of solutions which are eventually bounded.
Furthermore, we obtain real-analyticity of the solution using Angenent’s trick.
References:[1] M. Hieber, M. Nesensohn, J. Prüss and K. Schade, Dynamics of Nematic Liquid
Crystals: The Quasilinear Approach, submitted.
[2] F.M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational
Mech. Anal. 28(1968), 265–283.
[3] F.H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow ofliquid crystals, Comm. Pure Appl. Math. 48(1995), 501–537.
Date:
Tuesday, Nov. 5 17:30-17:45

Bangwei SHE
Johannes Gutenberg University Mainz, Mainz
Title:
Numerical simulation on some viscoelastic flows
Abstract:
The aims of this short presentation are twofolds:
ﬁrst we give an overview of some models for viscoelastic ﬂuids and discuss existence results of weak solutions. We will also highlight a connection between open
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analytical questions and numerical simulations indicating blow up. Moreover we
present some stabilization techniques used in the numerical methods. In the second part, we will introduce the characteristic method and present its application
for these models.
Date:
Friday, Nov. 8 16:40-16:55

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