JSPS-DFG Japanese-German Graduate Externship
第 5 回 日独流体数学国際研究集会
The 5th Japanese-German International Workshop on
Mathematical Fluid Dynamics
June 11 - 15, 2012 at Waseda University, Nishi-Waseda Campus
63 Bldg. 04 Conference Room

■ Program ■Mon, June 11
10：30-11：30↑Maremonti ①Tue, June 12
Geissert ①

11：30

12：00

Course-13：00Maekawa①

Maremonti ②

13：00

↓

-15：00

Kozono ①

Maekawa②

15：00

Maremonti ④

Kozono ②

Maekawa

Maekawa③

Maremonti ③

Kozono ③

15：30

Hirofumi Notsu

Takahiro Okabe

Ryo Takada

Bin Xie-16：00(Waseda Univ.)(Hirosaki Univ.)(Kyoto Univ.)(Shinshu Univ.)

Ken AbeErika Ushikoshi

Kohei Soga

(Univ. of Tokyo)(Tohoku Univ.)(Waseda Univ.)-16：30↑

Geissert ③④

Geissert ④

Coffee Break-15：30

16：00

Fri, June 15

Lunch Break (1H)

-14：00
14：00

Geissert ②

Thu, June 14

Coffee Break-12：00Mini

Wed, June 13

Tomoyuki
Nakatsuka(Nagoya Univ.)

Kozono ④

16：30

Coffee Break-17：00Scientific

17：00

Hirokazu Saito

Miho Murata

Masashi Ohnawa

Pen-Yuan Hsu*17:00-Talks-17：30(Waseda Univ.)(Waseda Univ.)(Waseda Univ.)(Tokyo Univ.)

Reseption

17：30

Karen Yeressian

Bangwei She

Loenz v. Below

Dario Goetz-17：50(TU Darmstadt)(JGU Mainz)(TU Darmstadt)(TU Darmstadt)

17：50

Jan-Erik Stecher

Georg

Rebekka

Manuel

-18：10(TU Darmstadt)

Schoechtel

Burkholz

Nesensohn(TU Darmstadt)(TU Darmstadt)(TU Darmstadt)

↓

18：10

Martin Sauer-18：30(TU Darmstadt)

Mini-Cource

Matthias GEISSERTTechnische Universität Darmstadt
Title:
Analytical Aspects of Complex Fluids
Abstract:
In this mini-course we discuss existence and uniqueness of solutions of a class ofviscoelastic ﬂuids. We start by presenting several models describing the motionof viscoelastic ﬂuids. Roughly speaking, the motion of a viscoelastic ﬂuid can be
described by the Navier-Stokes equation coupled with a transport equation for
the elastic part of the stress tensor. Hence, we have to investigate a (nonlinear)
hyperbolic-parabolic coupled system. We then present diﬀerent approaches toinvestigate the existence and uniqueness of solutions of the latter system.
Date:⃝
1 Tuesday, June 12 10:30-11:30⃝
3 Thursday, June 14 10:30-11:30⃝
2 Wednesday, June 13 10:30-11:30⃝
4 Friday, June 15 14:00-15:00

Hideo KOZONO
Waseda University
Title:
New function space in general unbounded domains and its applications
to the Navier-Stokes equations
Abstract:
We consider problems on the mathematical ﬂuid mechanics in general unboundeddomains Ω with non-compact boundaries. In such domains Ω, it is known thatthe usual Helmholtz decompositions in Lr (Ω), 1 < r < ∞ does not hold exceptr = 2, and hence we need to introduce another function space L̃r (Ω) deﬁned byL̃r (Ω) = Lr (Ω) + L2 (Ω) for 1 < r ≤ 2 and L̃r (Ω) = Lr (Ω) ∩ L2 (Ω) for 2 ≤ r <∞, respectively. This new function space L̃r (Ω) plays a substitutionary role for
Lr (Ω) so that the Helmholtz decomposition holds. Deﬁning the Stokes operator
A in L̃r (Ω), we can develop well-known techniques like analyticity of the semigroup {e−tA }t>0 and the maximal regularity theorem on ∂t + A as well as thecharacterization of the domains of the fractional powers Aα , 0 < α < 1. Asapplications of the theory on L̃r (Ω), we are able to treat fundamental problems on
uniqueness, regularity and decay properties of weak solutions of the Navier-Stokes
equations in Ω.
Date:⃝
1 Monday, June 11 14:00-15:00⃝
3 Thursday, June 14 14:00-15:00
2⃝
2 Wednesday, June 13 12:00-13:00⃝
4 Friday, June 15 15:30-16:30

Mini-Cource

Mini-Cource

Yasunori MAEKAWA
Kobe University
Title:
Mathematical analysis on some coherent structures of vorticity fields
for viscous incompressible flows
Abstract:
There are typical structures observed in two or three dimensional viscous incompressible ﬂows described by the Navier-Stokes equations. These structures areoften expressed in terms of vorticity ﬁelds, i.e., the curl of velocity ﬁelds of the
ﬂuid. In this lecture we discuss some of these coherent structures and present
recent mathematical results on the related topics. In particular, we consider thefollowing problems.
(i) Stability of the Lamb-Oseen vortex in two-dimensional exterior domains
It is known that the Lamb-Oseen vortex describes the large time behavior ofviscous incompressible ﬂows in the whole plane when the initial vorticity is
suﬃciently localized. We extend this result to the case of the two-dimensional
exterior domains when the circulation at inﬁnity is suﬃciently small.
(ii) Stability of the Burgers vortex in three-dimensional whole space
The Burgers vortex is known as a simple model of vortex tubes observedin the three-dimensional turbulent ﬂows. We present some results on the
spectrum of the linearized operator around the Burgers vortex. The Burgers
vortex is considered as a stationary ﬂow in the presence of a speciﬁc linear
strain, and the linear strain plays an important role in the stability property.
Motivated by this fact, we also consider the ﬂows with general linear strains
and discuss the existence and the nonexistence of the associated stationary
ﬂows.
(iii) Vorticity creation and formation of the vortex line along the boundary at theinviscid limit in the half plane
The inviscid limit problem of the Navier-Stokes equations is a classical issue
in the ﬂuid dynamics. However, there is a serious diﬃculty when the no-slip
boundary condition is imposed on the velocity ﬁeld, due to the appearanceof the boundary layer. In the boundary layer region the vorticity is highly
created and forms a vortex line, whose mechanism is formally explained by
the Prandtl theory. We present a rigorous description of this behavior at
the inviscid limit when the support of the initial vorticity is away from theboundary.
Date:⃝
1 Monday, June 11 12:00-13:00⃝
3 Thursday, June 14 12:00-13:00⃝
2 Tuesday, June 12 14:00-15:00⃝
4 Friday, June 15 12:00-13:00

Mini-Cource

3Mini-Cource

Paolo MAREMONTI
Seconda Università degli Studi di Napoli
Title:
Stokes and Navier-Stokes IBVP with nondecaying data
Abstract:
We develop four lectures concerning a special study of the Stokes and Navier-Stokes
initial boundary value problem. We assume Ω ⊆ Rn , n ≥ 3, unbounded domain
and nondecaying initial data. The aim is essentially to show the well posedeness(in the Hadamar sense) of the problem. We structure the topics as follows:(i) We recall some proprieties of singular transformation in Hölder spaces; wedevelop some results concerning the Stokes semigroup properties with an initial data in L1 (Ω). Both these results are employed to prove some pointwise
estimates for the pressure ﬁeld and the velocity ﬁeld respectively.
(ii) A maximum modulus theorem in exterior domains for the Stokes problem isthe ﬁrst problem that we discuss for nondecaying data. It is approached bymeans of methods of the functional analysis, and it is a natural developmentof some results for bounded domains obtained by Abe and Giga in their recent
paper.
(iii) We discuss the well posedeness of the initial boundary value problem with
nondecaying data in exterior domains. We prove that some restrictions areneeded to obtain the uniqueness, even in the case of the Stokes problem.
(iv) We discuss the existence and the uniqueness of the solutions to the NavierStokes Cauchy problem and we give the idea for the proof of the case of an
exterior domain.
The lectures are given by using a projector of pdf ﬁles and blackboard.
Date:
⃝
⃝
1 Monday, June 11 10:30-11:30
2 Tuesday, June 12 12:00-13:00⃝⃝
3 Wednesday, June 13 14:00-15:00
4 Friday, June 15 10:30-11:30

4

Mini-Cource

Scientific Talks

Ken ABEThe University of Tokyo
Title:The Stokes ﬂow in exterior domains with non-decaying initial velocity
Abstract:
Analyticity of semigroups is one of fundamental properties for semigroups associated to parabolic equations. In this talk we study analyticity of the Stokes semigroup in spaces of bounded functions both in bounded and unbounded domains.
In particular, we discuss the problem in an exterior domain as a typical example
of an unbounded domain. Even existence of solution is non-trivial in that case.
We derive analyticity of the Stokes semigroup by solving the unique existence for
the Stokes equations with non-decayng initial data. This talk is based on a joint
work with Professor Yoshikazu Giga.
Date:
Tuesday, June 12 16:00-16:30

Lorenz von BELOWTechnische Universität Darmstadt
Title:The spin-coating process with heat transfer
Abstract:
In [1] Denk, Geissert, Hieber, Saal and Sawada proved wellposedness of a modelfor the spin coating process. The model they studied is a one-phase free boundary
value problem for a Newtonian ﬂuid which takes rotational eﬀects into account.
We present an extension of their model which includes the eﬀect of heat transfer
and prove existence of a uniquely determined solution on an arbitrary time interval(0; T ) provided the initial data are suﬃciently small.
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:
Wednesday, June 13 17:30-17:50

Rebekka BURKHOLZTechnische Universität Darmstadt
Scientific Talks

5

Scientific Talks

Title:
A Stochastic FitzHugh-Nagumo Model
Abstract:
We study a stochastic version of the FitzHugh-Nagumo model, a stochastic semilinear evolution equation with dissipative nonlinearity and additive (Gaussian) noise
which describes the signal propagation in the axon of a neural cell.
Thus it is of the form√dX(t) = (AX(t) + F (X(t)))dt + QdW (t),X(0) = x ∈ D(A) ∩ D(F ) ⊂ H = L2 (0, 1) × L2 (0, 1).
We take advantage of the fact that A generates an analytic semigroup (of contractions) to prove the existence and uniqueness of a mild solution with deterministic
initial value of the system above. Although the nonlinearity F does not fulﬁl thestandard linear growth condition, this mild solution exists for all x ∈ H.
With additional assumptions on the coeﬃcients our analysis results in a proof ofexistence and uniqueness of an invariant measure µ for the random dynamicalsystem. Then in the last step we identify the associated Kolmogorov operator on
the set of cylindrical functions.
Date:
Wednesday, June 13 17:50-18:10

Dario GOETZ
Technische Universität Darmstadt
Title:On the compressible Navier-Stokes ﬂow with Neumann boundary conditions
Abstract:
The mathematical analysis of a compressible viscous ﬂuid ﬂow with a free boundary
relies on a thorough investigation of the corresponding linear compressible NavierStokes ﬂow with Neumann boundary conditions. In this talk, we are concernedwith showing R-sectoriality of the appearing Stokes operator in the half-space. Dueto the presence of a non-constant density-term, one has to deal with λ-dependent
coeﬃcients in this problem.
Date:
Thursday, June 14 17:30-17:50

Pen-Yuan HSU
The University of Tokyo
6

Scientific Talks

Scientific Talks

Title:On nonexistence for stationary solutions to the Navier-Stokes equations
with a linear strain
Abstract:
In this work we consider stationary solutions to the three-dimensional NavierStokes equations for viscous incompressible ﬂows in the presence of a linear strain.
For certain class of strains we prove a Liouville type theorem under suitable decay
conditions on vorticity ﬁelds.
Date:
Thursday, June 14 17:00-17:30

Miho MURATA
Waseda University
Title:On the sectorial R -boundedness of the Stokes operator for the compressible viscous ﬂuid ﬂow in the half-space with slip boundary condition
Abstract:
We consider the maximal Lp -Lq regularity for the compressible viscous ﬂuid ﬂow
in the half-space with slip boundary condition. The key of the proof of maximal
Lp -Lq regularity is to consider the sectorial R-boundedness of the Stokes operator
for the corresponding resolvent problem. Therefore in this talk, we mainly would
like to report the sectorial R-boundedness of the Stokes operator.
Date:
Tuesday, June 12 17:00-17:30

Tomoyuki NAKATSUKA
Nagoya University
Title:
Uniqueness of steady Navier-Stokes ﬂows in exterior domains
Abstract:
In this talk, we present a uniqueness theorem for stationary solutions to the NavierStokes equations in three-dimensional exterior domains. We establish the regularity theory for the perturbed Stokes equations in the Lorentz space, and then we
give the proof of the theorem employing the bootstrap argument.
Date:
Monday, June 11 16:00-16:30

Scientific Talks

7

Scientific Talks

Manuel NESENSOHNTechnische Universität Darmstadt
Title:
Lp -theory for a class of viscoelastic ﬂuids with a free surface
Abstract:
The investigated model is a generalization of the Oldroyd-B model. An approachto the corresponding free boundary problem without surface tension in Lagrangian
coordinates is presented. Local in time existence of a unique strong Lp -solution
for arbitrary large initial data is shown.
Date:
Thursday, June 14 17:50-18:10

Hirofumi NOTSU
Waseda University
Title:
Analysis of characteristics ﬁnite diﬀerence schemes for convection-diﬀusion
problems
Abstract:
Numerical schemes based on the method of characteristics have a special advantage, i.e., symmetry of the coeﬃcient matrix of each system of linear equations,which reduces computational costs. We will talk on an L2 -analysis of newly developed characteristics ﬁnite diﬀerence schemes of second order in time for convectiondiﬀusion problems.
Date:
Monday, June 11 15:30-16:00

Masashi OHNAWA
Waseda University
Title:
Decay rates towards traveling waves for a model system of radiating gas
Abstract:
The present research is concerned with the asymptotic behavior of a discontinuous
solution over R to a model system of radiating gas:
ut + uux + qx = 0,
−qxx + q + ux = 0.

8

Scientific Talks

Scientific Talks
We assume the initial data satisfy u0 (x) → u± as x → ±∞ with u− > u+ and|u− − u+ | ≤ 1/2. Under these asymptotic conditions, this system supports asmooth traveling wave solution uniquely up to shifts. We focus on the case when
the initial data have discontinuity at one point and show that the solution in thesense of Kruzkov converges to a traveling wave at exactly the same rate as the
decay rate of the initial perturbation from the traveling wave.
Date:
Wednesday, June 13 17:00-17:30

Takahiro OKABE
Hirosaki University
Title:
Initial proﬁle for the slow decay of the Navier-Stokes ﬂow in the halfspace
Abstract:
We consider the asymptotic behavior of the incompressible viscous ﬂuid in thehalf-space governed by the usual Navier-Stokes equations. Especially, we consider
the energy-decay problem of the weak solution to the Navier-Stokes equations. For
this direction, there are many results on the upper bound for the energy-decay. For
the whole space case, by the Fourier splitting method for the linear Stokes ﬂow,
we also have the lower bound of the energy decay. However, for the half-spacecase, it seems that there are few result on the lower bound of the decay problem.
Our previous result [1] derived the lower bound of the slow decay of the linear
Stokes ﬂow and [1] characterized the initial proﬁle which causes the lower bound.
However, since we consider speciﬁc initial data, tangential ﬂow, we do not include
two dimensional case. Our aim is to consider more general initial data and to
derive the lower bound for the two-dimensional ﬂow.
References:[1] T. Okabe, Lower bound of L2 decay of the Navier-Stokes flow in the half-space
Rn+ and its asymptotic behavior in the frequency space, submitted.
Date:
Tuesday, June 12 15:30-16:00

Hirokazu SAITO
Waseda University
Title:On the Lp -Lq maximal regularity of the Neumann-Dirichlet problem for
the Stokes equations in an inﬁnite layer
Scientific Talks

9

Scientific Talks

Abstract:
In this talk, I would like to consider the Lp -Lq maximal regularity of the Stokes
problem with Neumann-Dirichlet-type boundary condition in an inﬁnite layer.
This Stokes problem is linearized of a free surface problem which is called theocean problem. The free boundary problem in L2 -L2 framework is considered by
Beale and Nishida. They proved a unique existence of its global solution and its
decay properties in [1]. We need the Lp -Lq maximal regularity of the Stokes problem in order to deal with the free boundary problem in Lp -Lq -framework. In viewof this, I will report the Lp -Lq maximal regularity of the Stokes problem.
References:[1] J.T.Beale and T.Nishida, Large-time behavior of viscous surface waves, Recent
topics in nonlinear PDE, II (Sendai, 1984), 1-14, North-Holland Math. Stud.,
128, North-Holland, Amsterdam, 1985.
Date:
Monday, June 11 17:00-17:30

Martin SAUER
Technische Universität Darmstadt
Title:
Ergodicity for Generalized Newtonian Fluids
Abstract:
We study so called generalized Newtonian ﬂuids arising as a more general model ofthe Navier-Stokes equations in mathematical ﬂuid dynamics. A prominent example are the Power Law or Ladyzhenskaya ﬂuids having a power law type structurewith exponent p for the stress tensor. The main features of such equations besides
the better description of non-Newtonian behaviour are global uniqueness results
depending on the exponent p. Under stochastic forcing these equations have been
discovered only recently and in this talk we will consider them perturbed by additive noise. We study existence and uniqueness of stationary distributions andtheir ergodic properties.
Date:
Tuesday, June 12 18:10-18:30

Georg Schoechtel
Technische Universität Darmstadt
Title:
Simulation of stationary fractional Ornstein-Uhlenbeck process and application to turbulence
10

Scientific Talks

Scientific Talks

Abstract:
We investigate several methods to simulate the stationary fractional OrnsteinUhlenbeck (sfOU) process, i.e. the stationary solution of the Langevin equation
driven by a fractional Brownian motion (fBM) with arbitrary Hurst parameter in(0,1). In particular, we discuss the spectral approximate simulation method andthe applicability of the circulant embedding method, the fastest known exact simulation. As an application we introduce a two-dimensional random ﬂuid velocity
ﬁeld deﬁned as a series of sfOU processes, which satisﬁes a linear stochastic partial
diﬀerential equation driven by a fBM. The usefulness of such random velocity ﬁelds
in simulations is that we can create random velocity ﬁelds with desired statisticalproperties, thus generating artiﬁcial images of realistic turbulent ﬂows.
Date:
Tuesday, June 12 17:50-18:10

Bangwei SHE
Johannes Gutenberg- Universität Mainz
Title:
Kinetic ﬂux vector splitting(KFVS) method for compressible two-phaseﬂow containing non-conservative products
Abstract:
As the Baer-Nunziato model well prescribed the interaction between two phases,
it had become research hot spot. The non-conservative terms in the model can’tbe treated as ﬂux or common source terms, and none of existed discrete methods
for the non-conservative terms are mature. We propose a KFVS scheme in microscopic view. Then numerical ﬂuxes can be got by integrating particle velocity
distribution function. Besides, the non-conservative terms are explicitly introduced
into the evolution and construction of numerical ﬂuxes, thus well handled the nonconservative terms. We also prospect the same accuracy with schemes constructedby Riemann solver, but much less calculation loads.
Date:
Tuesday, June 12 17:30-17:50

Kohei SOGA
Waseda University
Title:
Time global stability of the Lax-Friedrichs scheme
Abstract:
I show time global stability of the Lax-Friedrichs scheme applied to initial value
Scientific Talks

11

Scientific Talks

problems of scalar conservation laws and Hamilton-Jacobi equations under theperiodic setting. We observe a stochastic and variational approach to the LaxFriedrichs scheme, formulating the value function for the discretized equations withrandom walks. This approach yields locally uniform convergence of approximate
entropy solutions to the exact ones, except any ”small” neighborhoods of shocks.
It is shown by techniques of viscosity solutions that exact entropy solutions at
t = 1 are bounded by a constant independent of initial data. Combining these
two results, we obtain several compactness of approximate solutions, which leads
to time global stability. It is very hard to obtain the results here by the usualfunctional analytic approach to the Lax-Freidrichs scheme.
Date:
Thursday, June 14 16:00-16:30

Jan-Erik STECHER
Technische Universität Darmstadt
Title:
Cirac Concentrations in Lotka-Volterra parabolic PDE
Abstract:
In this presentation a diploma thesis based on a paper written by Benoit Perthame
and Guy Barles will be discussed. It deals with a parabolic nonlinear PDE which
describes the evolution of species, including small mutations. We will illustrateexistence and uniqueness results and present convergence results by letting mutations vanish. This leads to a speciﬁc structure of the solution. It can be written
as a sum of Dirac measures, which can be interpreted as a mathematical way tosupport Darwin’s Law of Evolution.
Date:
Monday, June 11 17:50-18:10

Ryo Takada
Kyoto University
Title:
Time periodic solutions to the Navier-Stokes equations in the rotationalframework
Abstract:
In this talk, we consider the Navier-Stokes equations in the rotational frameworkwith the time periodic external force. Using a dispersive estimate for the Coriolis
term, we give a suﬃcient condition on the size of the external force for the existenceof time periodic solutions in terms of the Coriolis parameter. It follows from our
12

Scientific Talks

Scientific Talks

condition that the unique existence of time periodic solutions is guaranteed forlarge external forces provided the speed of rotation is suﬃciently fast.
Date:
Wednesday, June 13 15:30-16:00

Erika USHIKOSHI
Tohoku University
Title:
Hadamard variational formula for the Green function for the velocity
and pressure of the Stokes equations of the perturbations of domains
Abstract:
We consider the Stokes equations describing the motion of the incompressible ﬂuid
moving slowly on the bounded domain with the smooth boundary. In this talk,
we investigate the Hadamard variational formula of the Stokes equations for thevelocity and pressure by a new approach. Here the Hadamard variational formula
is ﬁrst introduced by J. Hadamard and gives the variation of the Green function
under a certain perturbation of its domain.
Date:
Wednesday, June 13 16:00-16:30

Karen YERESSIAN
Technische Universität Darmstadt
Title:
Spatial decay estimates for elliptic integro-diﬀerential equations
Abstract:
For elliptic integro-diﬀerential equations, which arise from Levy stochastic processes, we prove spatial decay estimates in the form of weighted norms. Our
approach is general, allowing diﬀerent decay rates. Applications of such estimates
include bounding the domain for numerical calculations and proving the existenceof solutions in unbounded domains with growing force term at inﬁnity.
Date:
Monday, June 11 17:30-17:50

Bin XIE
Shinshu University
Scientific Talks

13

Scientific Talks

Title:
Stochastic Fractal Burgers type equations with conservation laws
Abstract:
I will consider a fractal Burgers type equation driven by some noise. The operator
appeared in such equation is the generator of a symmetric α-stable process withα ∈ (0, 2). Such equation is a classical stochastic Burgers equation when α = 2
and it represents a scalar conservation law with random forcing as α converges to
0. Without the noise term, to the best of my knowledge such equation is initiallystudied by P. Biler, T. Funaki and W. A. Woyczynski in 1998 and the well-posed
problem is considered according to the index α. In this talk, I will add somenoise to their equation and consider the existence and uniqueness of its solutions
in diﬀerent frameworks, which depends on α.
Date:Thursday, June 14 15:30-16:00

14

Scientific Talks

Campus Map

63 Bldg. 2nd floor 04 conference room

Organizers :Yoshihiro Shibata (Waseda University)
Matthias Hieber (TU Darmstadt)
Contact Information
Shibata Lab. (Waseda Univ.)
Department of Mathematics, Waseda University
3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japann.ikezaki■kurenai.waseda.jp ■→@http://www.japan-germany.sci.waseda.ac.jp/event/201206workshop.html
Supported by: Research Institute of Nonlinear PDEs,
Organization for University Research Initiatives, Waseda University