Last update June 11, 2013

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 ■Mon, June 17Tue, June 18

Wed, June 19

Thu, June 20

9:30-10:40

Hi Jun CHOE ①(Yonsei Univ.)

Hi Jun CHOE ②

Hi Jun CHOE ③

Hi Jun CHOE ④

11:00-12:10

Juergen SAAL ①(TU Darmstadt)

Juergen SAAL ②

Juergen SAAL ③

Juergen SAAL ④

12:30-13:00

Tatsuo IGUCHI(Keio Univ.)

Teppei KOBAYASHI(Meiji Univ.)

Matthias GEISSERT(TU Darmstadt)

Hiroya ITO(Univ. of ElectroCommunications)

Lunch break

13:00-15:00

15:00-16:10

Yasushi TANIUCHI ①(Shinshu Univ.)

Yasushi TANIUCHI ②

Yasushi TANIUCHI ③

Yasushi TANIUCHI ④

16:40-17:10

Tsukasa IWABUCHI(Chuo Univ.)

Tohru OZAWA(Waseda Univ.)

Reinhard FARWIG(TU Darmstadt)

Erika USHIKOSHI(Tamagawa Univ.)

17:30-17:45

Lorenz von BELOW(TU Darmstadt)

Hana MIZEROVA(Univ. of Mainz)

Jonas SAUER(TU Darmstadt)

17:50-18:05

Moritz REINHARD(TU Darmstadt)

Bangwei SHE(Univ. of Mainz)

Katharina SCHADE(TU Darmstadt)

18:10-18:25

Hirokazu SAITO(Waseda Univ.)＊18：00～Reception

Miho MURATA(Waseda Univ.)●→Main-course●→30min talk●→15min talkDiscussion room → open 9:30 - 17:00 63 Bldg. 1F
Meeting room for Mathematics and Applied Mathematics

Main-Course

Main-Course

Hi Jun CHOE
Yonsei University, Seoul
Title:
Regularity question of the incompressible Navier-Stokes equations
Abstract:
We are interested in the regularity questions of the Navier-Stokes equations. First
we study the existence of Leray-Hopf solutions and the suitable weak solutions.
We introduce Galerkin method and related compactness in solution spaces forLeray-Hopf solution, and approximation of nonlinearity and its convergence in
Lesgues-Sobolev spaces and their embeddings.
For the regularity method, we ﬁrst look over history including minimal surface,harmonic map, elliptic and parabolic systems and analyzed the main steps. Then,
we move to partial regularity of the Navier-Stokes equations and introduce iteration
scheme to get one dimensional Hausdorﬀ measure. To improve Hausdorﬀ measure
by logarithmic factor, we follow higher integrability technique of reverse Holder for
Ap weight.
Finally, we consider various known sresults including self similarity, scaling andvorticity direction. In particular, Serrin class, Constantin-Feﬀerman’s vorticity
direction and axis symmetric ﬂow will be discussed.
Date:⃝
1 Monday, June 17 9:30-10:40⃝
3 Wednesday, June 19 9:30-10:40⃝
2 Tuesday, June 18 9:30-10:40⃝
4 Thursday, June 20 9:30-10:40

Juergen SAAL
Technical University of Darmstadt, Darmstadt
Title:
Some Progress on Fluid Dynamics in Singular Domains
Abstract:
Objective of this mini course is to present recent progress on models arisingin ﬂuid dynamics in non-smooth domains. Fluid ﬂow in singular domains hasquite some signiﬁcance in applications. For instance, models of cyclones, suchas hurricanes or tornados, are considered in a cylinder which is, by the presenceof edges, a so-called weakly singular domain. Another important application are
wetting and de-wetting phenomena. Here in general three-phase dynamic contactlines (gas/ﬂuid/solid, ﬂuid/ﬂuid/solid, etc.) appear, which at the end lead to2Main-Course

Main-Course

systems of equations on domains having at most a Lipschitz boundary. In spiteof their signiﬁcance there exist still fundamental open problems concerning both,modeling and rigorous mathematical treatment. The aim of the course is not togive a full solution to all these interesting but also very diﬃcult problems, but tointroduce concepts which might lead to a better understanding of the diﬃculties
and to present some new results for somewhat less intricate related ﬂuid models.
Within the course we will - as far as possible on an elementary level - introduce
and develop methods and techniques suitable for this purpose and diﬀerent fromclassical methods merely working for smooth regions. In this context we will discuss, e.g., the (non-commuting) operator sum method and oﬀ-diagonal estimates.
Based on these concepts we derive Lp maximal regularity for the linerized Stokes
system subject to partial slip type boundary conditions. We will prove this basic
result in optimal regularity classes for the case of cylindrical domains, wedges, andgeneral (graph) Lipschitz domains. A basic ingredient here also is the fact thatfor perfect slip boundary conditions the Stokes equations reduce to a standard
parabolic system. The results then extend to a more general class of partial slip
boundary conditions by employing perturbation arguments to the corresponding
reduced Stokes system.
Then we will demonstrate how the maximal regularity applies in order to provewell-posedness and stability of related non-linear Navier-Stokes type systems. In
particular, we will obtain this type results for some of the models important for
the applications mentioned above.
Date:⃝
1 Monday, June 17 11:00-12:10⃝
3 Wednesday, June 19 11:00-12:10⃝
2 Tuesday, June 18 11:00-12:10⃝
4 Thursday, June 20 11:00-12:10

Yasushi TANIUCHI
Shinshu University, Matsumoto
Title:
Uniqueness of mild solutions to the Navier-Stokes equations in unbounded domains
Abstract:
We discuss uniqueness of mild solutions bounded on the whole time axis to
the Navier-Stokes equations in 3-dimensional unbounded domains. For example, we consider time-periodic and almost periodic-in-time solutions. Thus far,uniqueness of almost periodic-in-time solutions to the Navier-Stokes equations in
unbounded domains, roughly speaking, is known only for a small almost periodicin-time solution in BC(R; L3w ) within the class of solutions which have suﬃciently
Main-Course3

Main-Course
small L∞ (L3w )-norm. We show that a small almost periodic-in-time solution in
BC(R; L3w ∩ L6,2 ) is unique within the class of all almost periodic-in-time solutions in BC(R; L3w ∩ L6,2 ). Here Lp,q denotes the Lorentz space. We also consider
uniqueness of mild solutions u with precompact range {u(t); t ∈ R} in L3w .
The proof of our uniqueness theorems are based on the method of dual equations.
Date:⃝
1 Monday, June 17 15:00-16:10⃝
3 Wednesday, June 19 15:00-16:10

4⃝
2 Tuesday, June 18 15:00-16:10⃝
4 Thursday, June 20 15:00-16:10

Main-Course30 minutes talks

30 minutes talks

Reinhard FARWIG
Technical University of Darmstadt, Darmstadt
with Hermann Sohr (Univ. Paderborn)with Werner Varnhorn (Univ. Kassel)
Title:
Besov space regularity conditions for weak solutions of the NavierStokes equations
Abstract:
Consider a bounded domain Ω ⊆ R3 with smooth boundary, an initial valueu0 ∈ L2σ (Ω), and a weak solution u of the Navier-Stokes system in [0, T ) × Ω withvanishing external force. Our aim is to develop regularity conditions for u which
are based on the Besov space{}(∫)
Bq,s (Ω) :=v ∈ L2σ (Ω); ∥v∥Bq,s (Ω) =∞∥e−τ A v∥sq dτ

1/s

<∞ ,

0

2 < s < ∞, 3 < q < ∞, 2s + 3q = 1, where A means the Stokes operator. Bq,s (Ω) has
been introduced in [1,2] concerning regularity properties for u; it coincides with the
−2/s
−2/s
well known Besov space Bq,s (Ω) of solenoidal vector ﬁelds in Bq,s (Ω). Further,B q,s (Ω) coincides with the space Bδq,s (Ω) which is obtained replacing ∥v∥Bq,s (Ω) by
(∫
)1,s
δ
, δ > 0.
∥v∥Bq,s (Ω) = 0 ∥e−τ A v∥sq dτδ
Our results extend regularity properties proved in [1,2]: E.g., the property u ∈∞Lloc ([0, T ); Bq,s (Ω)) together with lim ∥u∥L∞ (0,S;B q,s (Ω)) = 0 for each 0 < S < Tδ→0δ

is suﬃcient for regularity. Further, the property u ∈ C([0, T ); Bq,s (Ω)) or theproperty u ∈ L∞ (0, T ; Bδq,s (Ω)) with suﬃciently small norm is suﬃcient.
q,sThe weak solution u is unique if u ∈ L∞
loc ([0, T ); B (Ω)) and satisﬁes the strong
energy inequality on [t0 , T ) for each (!) t0 ∈ [0, T ).
References:[1] R. Farwig, H. Sohr, W. Varnhorn: On optimal initial value conditions for local
strong solutions of the Navier-Stokes equations. Ann. Univ. Ferrara 55, 89–110(2009)[2] R. Farwig, H. Sohr, W. Varnhorn: Extensions of Serrin’s uniqueness and regularity conditions for the Navier-Stokes equations. J. Math. Fluid Mech. 14,
529–540 (2012)[3] R. Farwig, H. Sohr, W. Varnhorn: Besov space regularity conditions for weaksolutions of the Navier-Stokes equations. TU Darmstadt, Preprint (2013)
Date:
Wednesday, June 19 16:40-17:10
30 minutes talks

5

30 minutes talks

Matthias GEISSERTTechnical University of Darmstadt, Darmstadt
with M. Nesensohn (TU Darmstadt)
with Y. Shibata (Waseda Univ.)
Title:On global Lp -solutions for a ﬂuid model of Oldroyd kind
Abstract:Let Ω ⊂ Rn be a bounded domain with a suﬃciently smooth boundary. We
consider the following system describing the ﬂow of a ﬂuid of Oldroyd kind in abounded domain Ω ⊂ Rnρ(∂t u + u.∇u) − α∆u + ∇π = β Div τ
in (0, ∞) × Ω,
div u = 0in (0, ∞) × Ω,∂t τ + u.∇τ + γτ = δEu + g(∇u, τ )in (0, ∞) × Ω,
(1)

u = 0
on (0, ∞) × ∂Ω,





u(0) = u0
in Ω,



τ (0) = τ0in Ω.
Here, the function u : [0, ∞)×Ω → Rn is the Eulerian velocity ﬁeld, π : [0, ∞)×Ω →R is the pressure, and τ : [0, ∞) × Ω → Rn×n is the elastic part of the stress tensor.
The symmetric part of the gradient is denoted by Eu = 21 (∇u + ∇uT ). Given are
the constants ρ, α > 0, β, δ > 0, γ ≥ 0 and the functiong(∇u, τ ) = τ W u − W uτ − a(τ Eu + Euτ ),where a ∈ [−1, 1] and W u = 21 (∇u − ∇uT ) denotes the antisymmetric part of thegradient.
We show existence and uniqueness of global strong solutions in the Lp -setting for
2−2/p
1 (Ω), where 1 < p, q < ∞ satisfy
small initial values u0 ∈ Bq,p (Ω) and τ0 ∈ Hq,n< 1. In the case γ > 0 we show that the solution converges
q > n and p1 + 2q
exponentially fast to zero. This is not the case for γ = 0, since system (1) has
non-trivial solutions. But we can show that even in this case the solution converges
to some equilibrium of (1).
To prove our main result, we transform (1) to Lagrangian coordinates and discuss
the Oldroyd operator()αρ−1 Aq −βρ−1 Pq Div
Aγ =
−δE
γ
where Aq denotes the Stokes operator and Pq the Helmholtz projection. We showthat this operator matrix admits maximal Lp -Lq regularity and that the spectrumof Aγ is mainly determined by γ. More precisely, 0 is in the resolvent of Aγ if
γ > 0, and 0 is in the spectrum of A0 if γ = 0. To prove global existence in the case6

30 minutes talks

30 minutes talks

γ = 0, we improve the linear theory and show a maximal regularity result in spacesof non-decaying functions in time. A ﬁxed point argument will then conclude theproof.
Date:
Wednesday, June 19 12:30-13:00

Tatsuo IGUCHI
Keio University, Yokohama
Title:
Solvability of the initial value problem to a model system for water
waves
Abstract:
The water wave problem is mathematically formulated as a free boundary problem
for an irrotational ﬂow of an inviscid and incompressible ﬂuid under the gravitational ﬁeld. The basic equations for water waves are complicated due to the nonlinearity of the equations together with the presence of an unknown free surface.
Therefore, until now many approximate equations have been proposed and analyzed to understand natural phenomena for water waves. Famous examples of suchapproximate equations are the shallow water equations, the Green–Nagdhi equations, Boussinesq type equations, the Korteweg–de Vries equation, the Kadomtsev–
Petviashvili equation, the Benjamin–Bona–Mahony equation, the Camassa–Holmequation, the Benjamin–Ono equations, and so on. All of them are derived fromthe water wave problem under the shallowness assumption of the water waves,which means that the mean depth of the water is suﬃciently small compared tothe typical wavelength of the water surface.
On the other hand, it is well-known that the water wave problem has a variational
structure. In fact, J. C. Luke (1967) gave a Lagrangian in terms of the velocity
potential and the surface variation, and showed that the corresponding Euler–
Lagrange equations are the basic equations for water waves. M. Isobe (1994)
and T. Kakinuma (2000) derived model equations for water waves without any
shallowness assumption. The model equations are the Euler–Lagrange equations
to an approximated Lagrangian, which is obtained by approximating the velocity
potential in Luke’s Lagrangian.
In this talk, we consider one of the model equations and report the solvabilityof the initial value problem. The model are nonlinear dispersive equations andthe hypersurface t = 0 is characteristic for the model equations. Therefore, theinitial data have to be restricted in an inﬁnite dimensional manifold in order tothe existence of the solution, and we show that the manifold is invariant under thetime evolution.
30 minutes talks

7

30 minutes talks

This talk is based on the joint research with my former student Yuuta Murakami.
Date:
Monday, June 17 12:30-13:00

Hiroya ITO
The University of Electro-Communications, Tokyo
Title:
Some remarks on generalized Korn’s inequalities
Abstract:
Korn’s inequality, well-known for its fundamental roll in dealing with various
equations appearing in continuum mechanics, asserts that there exists a constantC > 0 such that∥ε(u)∥2L2 (Ω) + ∥u∥2L2 (Ω) ≥ C ∥∇u∥2L2 (Ω)
for all H 1 vector ﬁelds u on a bounded Lipschitz domain Ω in Rn , where ε(u)represents the symmetric part of the gradient ∇u of u. We generalize Korn’s inequality by replacing ε(u) with a ﬁrst-order homogeneous partial diﬀerential system
in general form with constant coeﬃcients for vector-valued functions; generalization yields not only inequalities applicable to continuum mechanics but also those
applicable to diﬀerential geometry and general relativity. From J. Nečas’s work
we know some necessary and suﬃcient conditions (on the coeﬃcients) for such an
inequality to hold. The main purpose of this talk is to present a new necessary andsuﬃcient condition, which is in general easier to check than the known conditions.
Although the establishment of the new condition is not so easy in the sense that it
relies on a certain tool in algebraic geometry, the condition gives us a lot of useful
information about what are vague in Nečas’s argument.
Date:
Thursday, June 20 12:30-13:00

Tsukasa IWABUCHI
Chuo University, Tokyo
Title:
Global solutions for the Navier-Stokes equations in the rotational framework
Abstract:
We consider the Cauchy problems for the Navier-Stokes equations with the Coriolis force in the homogeneous Sobolev spaces Ḣ s (R3 ) and show the existence of
8

30 minutes talks

30 minutes talks

unique global solutions. Without the Coriolis force, it is known that the uniqueglobal solutions are obtained if the initial data is suﬃciently small in the Sobolev
1
space Ḣ 2 (R3 ). The regularity index 1/2 is scaling critical case from the view pointof scaling invariant property for the Navier-Stokes equations without the Coriolis
force. In this talk, the unique global solutions are obtained for large initial data
if the speed of rotation is suﬃciently large. In particular, we show the suﬃcientcondition for the existence of global solutions by the use of the norm of initial datain Ḣ s (R3 ) and the speed of the rotation in the case s > 1/2. In the case s = 1/2,
the suﬃcient condition is not characterized by the norm of initial data but is given
1by the precompact sets of Ḣ 2 (R3 ).
Date:
Monday, June 17 16:40-17:10

Teppei KOBAYASHI
Meiji University, Tokyo
Title:
Jeﬀery-Hamel’s ﬂows in the plane
Abstract:
Jeﬀery-Hamel’s ﬂow is an exact solution of the Navier-Stokes equations in adomain between two inclined lines. L. Rosenhead [1] treated such a problem.
The solutions are given in the Jacobian elliptic functions and the behavior of thesolutions is investigated.
In this talk, it is shown that in the domain there exists a solution of the NavierStokes equations applying the relation between the ﬂux constant and the amplitudeof the two lines.
References:[1] L. Rosenhead, The steady two-dimensional radial flow of viscous fluid between
two inclined plane walls, Proc. R. Soc. Lond. A, 175 (1940), 436-467.
Date:
Tuesday, June 18 12:30-13:00

Tohru OZAWA
Waseda University, Tokyo
with Shuji Machihara (Saitama Univ.)
with Hidemitsu Wadade (Gifu Univ.)
Title:
Hardy type inequalities on balls
30 minutes talks9

30 minutes talks

Abstract:
This talk is based on my recent joint-work with Shuji Machihara and Hidemitsu
Wadade. We revisit the Hardy inequalities on balls with radius R at the origin
in Rn with n ≥ 2. We describe how the behavior of functions on the boundaryaﬀects the Hardy type inequalities. A special attention is paid on the case n = 2
with logarithmic correction.
References:[1] O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow,
Second English edition, revised and enlarged. Translated from the Russian byRichard A. Silverman and John Chu. Mathematics and its Applications, Vol.
2 , Gordon and Breach, Science Publishers, New York-London-Paris, (1969).
[2] J. Leray, Etude de diverses équations integrales non linéaires et de quelques
problèmes que pose l’hydrodynamique, J. Math. Pures Appl. 12 (1933), 1–82.
[3] S. Machihara, T. Ozawa and H. Wadade, Hardy type inequalities on balls,
Tohoku Math. J. (in press)
Date:
Tuesday, June 18 16:40-17:10

Erika USHIKOSHI
Tamagawa University, Tokyo
Title:
New approach to the Hadamard variational formula for the Green function of the Stokes equations
Abstract:
We consider the stationary Stokes equations in the bounded domain Ω ⊂ Rd . The
purpose of this talk is to give a new proof of the ﬁrst variation of the Hadamard
variational formula for the velocity and the pressure, and establish the second
variation for them. More precisely, for every ε > 0, we deﬁne a family {Ωε }ε≥0of perturbed domains deﬁned by Ωε = Φε (Ω) with vol.Ωε = vol.Ω, where Φε isthe volume preserving diﬀeomorphism from the domain Ω to Ωε . We consider theGreen function of the Stokes equations which satisfy∆x Gε,n (x, y) − ∇x Rε,n (x, y) = en δ(x − y), (x, y) ∈ Ωε × Ωε ,div Gε,n (x, y) = 0,(x, y) ∈ Ωε × Ωε ,Gε,n (x, y) = 0,x ∈ ∂Ωε , y ∈ Ωεfor n = 1, · · · , d, where Gε,n (x, y) = {Giε,n (x, y)}i,n=1,··· ,d is the Green matrix forthe velocity, Rε,n (x, y) = {Rε,n (x, y)}n=1,··· ,d is the Green function for the pressure
10

30 minutes talks

30 minutes talksand {en }n=1,··· ,d denotes a canonical basis in Rd . Under the volume preserving perturbation, we give a more reﬁned proof of the the ﬁrst variation formulas for thevelocity δG and the pressure δR as δG(x, y) := limε→0 Gε (x,y)−G(x,y), δR(x, y) :=εRε (x,y)−R(x,y)
limε→0, and we derive the second variation of the Green function forεthe Stokes equations δ 2 G and δ 2 R as δ 2 G(y, z) := limε→0 Gε (y,z)−G(y,z)−εδG(y,z),ε2Rε (y,z)−R(y,z)−εδR(y,z)
2δ R(y, z) := limε→0. Our method gives a new systematicε2
proof of the Hadamard variational formula, which enables us to deal with thehigher derivatives with respect to the perturbation of domains.
Date:
Friday, June 20 16:40-17:10

30 minutes talks

11

15 minutes talks

15 minutes talks

Lorenz von BELOWTechnical University of Darmstadt, Darmstadt
Title:The spin coating system in the singular limit of vanishing surface tension
Abstract:
In [1] Denk, Geißert, Hieber, Saal, and Sawada studied a model for the spin
coating process. The model they proposed is a one-phase free boundary value
problem for a Newtonian ﬂuid and it is essentially given by the following set of
equations:∂t u + u · ∇u − ∆u + ∇θ = f,
div u = 0 in Ω(t)






S(u, θ)ν − σκν = 0
on Γ+ (t)(1)V =u·ν
on Γ+ (t)




u=0
on Γ−




u(0) = u0in Ω(0)for t > 0, where Ω(t) is a moving domain close to an inﬁnite layer R2 × (0, h), withﬁxed lower boundary Γ− and moving upper boundary Γ+ (t) parametrised by an
unknown function η. Here u denotes the velocity ﬁeld, θ the pressure, σ > 0 thesurface tension parameter, κ the mean curvature, S(u, θ) the stress tensor and V
the velocity of the upper boundary in normal direction.
In this talk we present results concerning the inﬂuence of the surface tension
parameter σ on solutions of (1) and its linearisation, in particular in the singular
limit σ → 0 of vanishing surface tension. Our results rely to a large extent on acareful analysis of an explicit solution formula for the linearised problem.
References:[1] Robert Denk, Matthias Geissert, Matthias Hieber, Jürgen Saal, and Okihiro
Sawada. The spin-coating process: Analysis of the free boundary value problem.
Communications in Partial Diﬀerential Equations, 36(7):1145–1192, 2011.
Date:
Monday, June 17 17:30-17:45

Hana MIZEROVA
Johannes Gutenberg University Mainz, Mainz
with Maria Lukacova (Mainz Univ.)

12

15 minutes talks

15 minutes talks

Title:
Error estimates for the Peterlin viscoelastic model
Abstract:
We consider the equation of a motion of an incompressible viscoelastic ﬂuidρ∂v
+ ρ(v · ∇)v = η∆v + div T − ∇p∂tdiv v = 0

on a bounded smooth domain Ω ⊂ R3 . The elastic stress tensor T can be expressedusing the conformation tensor C in the following way
T = tr C.C,
where C is symmetric positive-deﬁnite tensor, which satisﬁes the equation of theform:∂C+ (v · ∇)C − (∇v)C − C(∇v)T = tr C.I − (tr C)2 C + ϵ∆C.
∂t
We impose homogenous Dirichlet condition for the velocity v = 0 and no-ﬂux
condition for the conformation tensor ∂C∂n = 0 on the boundary ∂Ω. We consider
v0 and C0 to be the enough smooth initial data.
The existence of the weak solutionv ∈ L2 (0, T ; H10,div (Ω)) ∩ L∞ (0, T ; L2div (Ω))
C ∈ L2 (0, T ; H1 (Ω)) ∩ L∞ (0, T ; L2 (Ω))
is proven using the Galerkin method and the classical energy estimates. Our aim
is to analyze the error of a suitable approximation. We use a conforming ﬁnite
element discretization. For suﬃciently smooth solution and its approximation by
P k elements we get the order of convergence hk , where h is a mesh size.
Date:
Tuesday, June 18 17:30-17:45

Miho MURATA
Waseda University, Tokyo
Title:On the sectorial R-boundedness of the Stokes operator for the compressible viscous fluid flow and its application

15 minutes talks

13

15 minutes talks

Abstract:
We consider the initial boundary value problem of the Stokes equations for thecompressible viscous ﬂuid ﬂow with slip boundary condition in a bounded domain.
In order to consider a global in time unique existence theorem for a nonlinear
problem with some initial data close to a constant state in a bounded domain, theexponential stability of solutions to the Stokes equations is required. In this talk,we prove the exponential stability employing the sectorial R-boundedness of the
Stokes operator and a homotopic argument.
Date:
Wednesday, June 19 18:10-18:25

Moritz REINHARD
Technical University of Darmstadt, Darmstadt
Title:
Water entry of a flat elastic plate at high horizontal speed
Abstract:
Interactions of elastic structures with the ﬂuid ﬂow are diﬃcult to compute since
the elastic deformations of the body are coupled with the hydrodynamic forces.
In problems where a body with small deadrise angle impacts onto an undisturbed
water surface of inﬁnite depth, hydroelastic eﬀects can be accounted for using an
asymptotic hydrodynamic model. In the asymptotic hydrodynamic model? where
the ﬂuid is supposed to be inviscid and incompressible ? the boundary conditions
on the free surface and on the wetted part of the body are linearised and imposedon the initial equilibrium position of the liquid surface. The positions of the ﬂuid
spray-root regions are determined by Wagner’s condition. In this talk we consider
the problem where an inclined two-dimensional elastic plate enters a water surface
at high horizontal speed. Newton’s second law and Euler’s beam equation subject
to free-free boundary conditions govern both the rigid an elastic motions of theplate. The analysis is based on the normal-mode method.
We show that hydrodynamic pressures below atmospheric value can be found in
the contact region for the rigid-plate impact. However, elastic vibrations of theplate lead to signiﬁcant lower pressures. We conclude that elasticity of the plate
may promote cavitation of the ﬂuid, and the detachment of the ﬂuid from the solid
surface in the low-pressure regions during the early stage of impact.
Date:
Monday, June 17 17:50-18:05

Hirokazu SAITO
Waseda University, Tokyo14

15 minutes talks

15 minutes talks

Title:
On some decay properties of solutions for the Stokes problem with surface tension
Abstract:
In this talk, we would like to consider decay properties of solutions for the Stokes
problem which comes from a free boundary value problem in the half space. We
divide the solution into three parts that is the low, middle and high frequency partin the Fourier space. The analysis of the low frequency part is the most importantbecause the Lopatinski diterminant has roots which converge into the origin. We,
therefore, concentrate the analysis of the low frequency part in this talk.
Date:
Monday, June 17 18:10-18:25

Jonas SAUER
Technical University of Darmstadt, Darmstadt
Title:
Navier-Stokes Flow in Infinite Cylinders with Non-Constant Cross Section
Abstract:
We investigate maximal regularity in Lq -spaces of the Stokes operator in inﬁnite
cylindrical domains of non-constant cross-section. Based on results by Farwig andRi [1, 2] for the straight cylinder with bounded cross-section of class C 1,1 we obtaincorresponding resolvent estimates also in the case of perturbed cylinders, given
usual smallness assumptions on the perturbation. The calculations are carried out
in cylindrical coordinates in order to describe the cross-sections in terms of theazimuthal angle and the height of the cylinder. Since there are resolvent estimatesavailable in Lq -spaces even for non-homogeneous divergence, these estimates carryover to the perturbed cylindrical domains by classical perturbation arguments.
Finally, we establish a bounded H ∞ -calculus in Lqσ by a general result for theshifted Stokes operator Aq + c, c > 0, due to Abels [3], implying in particularmaximal regularity.
References:[1] Farwig, R. and Ri, M.-H.: The resolvent problem and H ∞ -calculus of the
Stokes operator in unbounded cylinders with several exits to inﬁnity. J. Evol.
Equ., 7(3) (2007), 497–528.
[2] Farwig, R. and Ri, M.-H.: Stokes resolvent systems in an inﬁnite cylinder.
Math. Nachr., 280(9-10) (2008), 1061–1082.

15 minutes talks

1515 minutes talks[3] Abels, H.: Bounded imaginary powers and H∞ -calculus of the Stokes operator
in unbounded domains. In: Nonlinear elliptic and parabolic problems. Progressin Nonlinear Diﬀerential Equations and Their Applications, Birkhäuser, 64(2005), 1–15.
Date:
Wednesday, June 19 17:30-17:45

Katharina SCHADE
Technical University of Darmstadt, Darmstadt
Title:
Lp -theory for nematic liquid crystal systems
Abstract:
Liquid crystals are matter in a state between liquid and solid. They ﬂow like aliquid, but their molecular structure is similar to that of a solid crystal. In themost common nematic phase, molecules possess an orientation but no positional
order. As such, the Ericksen-Leslie model describes the ﬂow of nematic liquid
crystals with Navier-Stokes equations coupled with a diﬀusion equation governing
the behaviour of the direction ﬁeld, see [2].
We consider a simpliﬁed Ericksen-Leslie model going back to Lin and Liu in 1995
[3]. We are able to treat the problem in a quasilinear setting, which has not been
considered before. We show the local existence of strong solutions using a variantof Clément-Li together with well-known maximal Lp -regularity results. We also
investigate stability of equilibria with the generalized principle of linearized stability. We obtain global existence of strong solutions which are eventually boundedon their maximal interval of existence by exploiting the Lyapunov-properties ofthe system’s energy. Furthermore, we obtain real-analyticity of the solution using
Angenent’s trick.
We give an outlook to a temperature-dependent extension of this model which
satisﬁes the ﬁrst and second law of thermodynamics.
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:
Wednesday, June 19 17:50-18:05
16

15 minutes talks

15 minutes talks

Bangwei SHE
Johannes Gutenberg University Mainz, Mainz
Title:
Numerical tests on some viscoelastic flows
Abstract:
We presented a new combined ﬁnite volume and ﬁnite diﬀerence method for theviscoelastic ﬂow. The high Weissenberg number problem is overcome by using
the log-conformation representation. However, the convergence is still an open
problem. As a complementary approach we use the kinetic theory to approximate complex viscoelastic stress tensor. The corresponding micro-model based on
the Fokker-Plank equation is studied. We present a new multiscale method andillustrate ﬂow behaviour through a series of numerical results.
Date:
Tuesday, June 18 17:50-18:05

15 minutes talks

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