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

新規発信研究集会

Kickoff Meeting■ Program ■Tuesday, June 17
10:00
|

10:00
Opening Remarks

10:30
10:40
|
11:30
11:40
|
12:30

Reinhard FARWIG(TU Darmstadt)

|
15:20

|

(Kyushu University)

Ulrich KOHLENBACH(TU Darmstadt)

11:00

Toshiaki HISHIDA

|

14:00
|

|
16:40

川島秀一
(Kyushu University)(Nagoya University)

Mitsuru SUGIMOTO
杉本充

14:50

(Nagoya University)

15:00

Masao YAMAZAKI

|
15:50

Shuichi KAWASHIMA

菱田俊明

Lunch BreakCoffee Break
15:50

儀我美一
(the University of Tokyo)

Yoshiyuki KAGEI
隠居良行

Yoshikazu GIGA

10:50

11:50

Lunch Break
14:30

Wednesday, June 18

山崎昌男(Waseda University)
Coffee Break

16:20|Yoshihiro SHIBATA
柴田良弘

17:10

(Waseda University)

17:20

Closing Remarks

18:00 ～Reception

Kickoﬀ Meeting

Reinhard FARWIG
Technical University of Darmstadt, Darmstadt
farwig@mathematik.tu-darmstadt.deTitle:
The fundamental solution of compressible and incompressible ﬂuid ﬂow
past a rotating obstacle
Abstract:
We consider the ﬂow of either an incompressible or a compressible ﬂuid aroundor past a rotating rigid body in the whole space R3 . Using a global coordinatetransform and a linearization the problem reduces to a linear PDE system in a timeindependent domain which may admit stationary solutions. For the new system
on R3 – with a virtual obstacle – we are looking for the fundamental solution andits estimates in Lq -space.
In the incompressible case the fundamental solution has been found by T. Hishida,D. Müller and myself via Fourier transformation and estimated in Lq -space for allq ∈ (1, ∞), see [1, 2]. Although in the compressible case the same strategy leads toan explicit solution in Fourier space, the corresponding multiplier function showsa very diﬀerent behaviour: it is not smooth outside of the origin so that usual
Lq -estimates are not at all obvious. Actually, we need a restriction on q, see [3].
References:[1] R. Farwig, T. Hishida, D. Müller: Lq -Theory of a Singular ”Winding” Integral
Operator Arising from Fluid Dynamics. Paciﬁc J. Math. 215, 297–312 (2004)[2] R. Farwig: An Lq –analysis of Viscous Fluid Flow past a Rotating Obstacle.
Tôhoku Math. J. 58, 129–147 (2005)[3] R. Farwig, M. Pokorný: A Linearized Model for Compressible Flow past a Rotating Obstacle: Analysis via Modified Bochner–Riesz Multipliers. FB Mathematik, TU Darmstadt, Preprint no. 2682 (2014)
Date:
Tuesday, June 17 10:40-11:30

2Kickoﬀ Meeting

Kickoﬀ Meeting

Yoshikazu GIGA
The University of Tokyo, Tokyo
Graduate school of Mathematical Sciences, The University of Tokyo,
3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
labgiga@ms.u-tokyo.ac.jp

Title:On the stokes semigroup in some non-Helmholtz domains
Abstract:
It is well-known that the Stokes operator associated with the Stokes equations
generates analytic semigroups in Lp type space in various domains. Recent study
shows that the Stokes operator not only generates an analytic semigroup but also
enjoy maximal regularity property provided that domain admits the Helmholtz
decomposition in Lp . Unfortunately, there are domains that the Helmholtz decomposition fails.
In this talk we would like to give an example that the Stokes operator generates
an analytic semigroup in Lp type for some two-dimensional domains where the LpHelmholtz decomposition is not allowed.
Our method appeals to L∞ theory, which is recently established in various domains including a bounded domain, an exterior domain and a perturbed half space.
It turns out that our method applies to some non-Helmholtz domains as well as
cylindrical domains. However, a recent study of Lorenz von Below shows that athree- or higher dimensional layer domain does not permit L∞ theory. This is a
joint work in progress with Ken Abe, Katharina Schade and Takuya Suzuki.
Date:
Wednesday, June 18 10:00-10:50

Kickoﬀ Meeting

3Kickoﬀ MeetingToshiaki HISHIDA
Nagoya University, Nagoya
Graduate School of Mathematics, Nagoya University
Nagoya 464-8602, Japanhishida@math.nagoya-u.ac.jp

Title:
Stability of time-dependent Navier-Stokes ﬂow and algebraic energy
decay
Abstract:
Let V = V (x, t) be a given time-dependent Navier-Stokes ﬂow of an incompressible viscous ﬂuid in the whole space R3 . As important examples of this basic
ﬂow V , we have the following in mind: forward self-similar solution, time-periodic
solution and global mild (eventually strong) solution of the Cauchy problem. Itis thus reasonable to assume that V ∈ L∞ (0, ∞; L3,∞ ) ∩ Cw ([0, ∞); L3,∞ ), where
L3,∞ denotes the weak-L3 space. The energy stability of small V in this class withrespect to any initial disturbance in L2σ has been recently investigated by Karch,Pilarczyk and Schonbek. It would be interesting to ﬁnd how fast the disturbancedecays in L2 as t → ∞ when the initial disturbance possesses better summability
at space inﬁnity. In this presentation it is proved that any weak solution u(x, t)with the strong energy inequality to the perturbed Navier-Stokes system, which)( 3the disturbance should obey, enjoys ∥u(t)∥2 = O t− 2 (1/q−1/2) as t → ∞ provided
the initial disturbance is taken from Lq ∩ L2σ for some q ∈ [1, 2). The proof relies
on a development of the Fourier splitting method combined with Lq -Lr estimate of
the evolution operator generated by the linearized operator around the basic ﬂow
V.
This talk is based on a joint work with Maria E. Schonbek (University of California, Santa Cruz).
Date:
Wednesday, June 18 11:00-11:50

4Kickoﬀ Meeting

Kickoﬀ Meeting

Yoshiyuki KAGEI
Kyushu University, Fukuoka
Kyushu University, Fukuoka 819-0395, Japan

Title:On asymptotic behavior of solutions to the compressible Navier-Stokes
equation in a periodic layer
Abstract:
We consider the following compressible Navier-Stokes equation∂t ρ + div (ρu) = 0, ρ(∂ v + v · ∇v) − µ△v − (µ + µ′ )∇div v + ∇(P (ρ)) = 0,tv|∂Ω = 0,(ρ, v)|t=0 = (ρ0 , v0 ).

(1)

Here Ω is a periodic layer in Rn :{}Ω := x = (x′ , xn ); x′ ∈ Rn−1 , ω1 (x′ ) ≤ xn ≤ ω2 (x′ ) ,where ω1 , ω2 are smooth functions satisfyingωj (x′ + a′i ) = ωj (x′ ) (j = 1, 2 ; i = 1, · · · , n − 1)for all x′ ∈ Rn−1 with ⃗a′i (i = 1, · · · , n − 1) given by constant vectors ⃗a′i = 2πei ,αi ⃗i⃗e′i = T (0, · · · , 1, · · · , 0) ∈ Rn−1 ; µ and µ′ are the viscosity coeﬃcients that satisfyµ > 0, n2 µ + µ′ ≥ 0; and P is the pressure that is a smooth function of ρ. We
assume thatP ′ (ρ∗ ) > 0
for a given positive constant ρ∗ .
We discuss large time behavior of solutions to (1) around the constant state(ρ∗ , 0). It will be shown that the L2 norm of the perturbation (ρ(t) − ρ∗ , v(t))
n−1decays in the order O(t− 4 ) as t → ∞ if the initial perturbation (ρ0 − ρ∗ , v0 ) is
suﬃciently small; in fact, (ρ(t) − ρ∗ , v(t)) behaves diﬀusively in large time.
Date:
Tuesday, June 17 11:40-12:30

Kickoﬀ Meeting5

Kickoﬀ Meeting

Shuichi KAWASHIMA
Kyushu University, Fukuoka
Faculty of Mathematics, Kyushu University

Title:
Dissipative structure for symmetric hyperbolic systems with relaxation
Abstract:
We discuss the dissipative structure and the corresponding decay property for aclass of symmetric hyperbolic systems with relaxation. The Shizuta - Kawashimastability condition gives the characterization of the standard dissipative structure
for systems with symmetric relaxation. Recently, we found several intereting systems with non-symmetric relaxation which have diﬀerent dissipative structure. Inthis talk, we ﬁrst review the standard theory based on the Shizuta-Kawashima
condition. Then we discuss those new examples whose decay structure is of theregularity-loss type. Finally, we report the recent progress on the stability theory
for a class of symmetric hyperbolic systems with non-symmetric relaxation.
A part of this talk is based on the joint works with Yoshihiro Ueda, Renjun Duan,and Naofumi Mori.
Date:
Tuesday, June 17 15:50-16:40

6

Kickoﬀ Meeting

Kickoﬀ Meeting

Ulrich KOHLENBACH
Technical University of Darmstadt, Darmstadt
Technische Universität Darmstadt
Schlossgartenstraße 7D-64289 Darmstadt, Germany
kohlenbach@mathematik.tu-darmstadt.de

Title:
Logical extraction of eﬀective bounds in nonlinear analysis: metastability and abstract Cauchy problems
Abstract:
During the last two decades a systematic program of ‘proof mining’ emerged
which uses tools from mathematical logic (so-called proof interpretations) to systematically extract explicit quantitative information (e.g. rates of convergence)from prima facie nonconstructive proofs (e.g. convergence proofs). This has been
applied particularly successful in the context of nonlinear analysis (ﬁxed point theory, ergodic theory, topological dynamics, see e.g. [2]).
Together with my PhD student A. Koutsoukou-Argyraki in the IRTG 1529, werecently adapted this approach to results on the asymptotic behavior of solutionsof abstract Cauchy problems ([1]) that are given by suitable forms of accretive
operators in Banach space ([3]).
References:[1] Garcı́a-Falset, J., The asymptotic behavior of the solutions of the Cauchy problem generated by ϕ-accretive operators. J. Math. Anal. Appl. 310, pp. 594-608,2005.
[2] Kohlenbach, U., Applied Proof Theory: Proof Interpretations and Their Usein Mathematics. Springer Monograph in Mathematics. xx+536pp., 2008.
[3] Kohlenbach, U., Koutsoukou-Argyraki, A., Rates of convergence and metastability for abstract Cauchy problems generated by accretive operators. Preprint
2014.
Date:
Tuesday, June 17 14:30-15:20

Kickoﬀ Meeting

7

Kickoﬀ Meeting

Yoshihiro SHIBATA
Waseda University, Tokyo
Title:
Compressible and Incompressible two phase problem including the phasetransition problem
Abstract:In this talk, I would like to talk about the local wellposedness of the incompressiblecompressible two phase problem with sharp interface. The problem is described
as a certain free boundary problem.
To talk about the main point of my talk, forgetting the Fourier part, I would liketo talk about the Navier-Stokes part. Without phase transition case, I transfer theproblem to the problem in the ﬁxed domain by the Lagrange transform. The main
point of my approach is to prove the maximal Lp - Lq regularity for the linearized
equations. Then, I have the local wellposedness without any restriction of the sizeof the initial data.
On the other hand, with phase transition case, we can not use the Lagrange
transform, so that I have to use so called Hanzawa transform to transform theproblem to the ﬁxed domain. Then, the mass density beeing represented by thevelocity ﬁeld in the compressible part, we transfer the problem to the nonlinear
Lame-system in the compressible parts and Navier-Stokes equations in the incompressible part with suitable jump condition and one more equations on theinterface. I also prove the maximal Lp - Lq regularity for the linearized equations,but the local well-posedness holds only for the small initial data because of theHanzawa transform.
The main part of my talk is devoted to explaining the analysis of the Lopatinski
determinant for the model problem. Because, nowadays once getting the maximal
regularity for the model problem, it is rather well-known to prove local wellposedness for the quasilinear equations with help of maximal regularity for the linearizedequations.
The essential diﬀerence in the model problem between the without phase transition and with phase transition is the following: Without phase transition, wehave to treat the diﬀerence of the normal component of the velocity ﬁelds throughthe interface, but with phase transition, this part is given as the phase transition,
but we have one more condition written with the ﬁrst order partial diﬀerential
equations.
Date:
Wednesday, June 18 16:20-17:10

8

Kickoﬀ Meeting

Kickoﬀ Meeting

Mitsuru SUGIMOTO
Nagoya University, Nagoya
Title:
On a systematic understanding of smoothing estimates for water wave
equations
Abstract:
Since a pioneering work of T. Kato which established a smoothing estimate
for generalized Korteweg-de Vries equations, smoothing eﬀect for dispersive equations, including many other kind of water wave equation, has been intensivelyinvestigated in the last three decades. Such analysis is particularly important inapplications to nonlinear evolution equations, especially to those with derivatives
in the nonlinearity.
This talk describes a new approach to global smoothing problems for dispersive
and non-dispersive evolution equations based on the global canonical transforms
and the underlying global microlocal analysis. For this purpose, the Egorov-type
theorem is established with canonical transformations in the form of a class ofFourier integral operators, and their mapping properties are derived. This allows
us to globally reduce general dispersive equations to normal forms in one or twodimensions. Then, a new comparison principle for evolution equations is introduced. In particular, they allow us to relate diﬀerent smoothing estimates bycomparing certain expressions involving their symbols. As a result, it is shownthat the majority of smoothing estimates for diﬀerent equations are equivalent toeach other. Moreover, new estimates as well as several reﬁnements of known results
are obtained. The proofs are considerably simpliﬁed. A comprehensive analysis
is presented of smoothing estimates for dispersive and also non-dispersive equations with constant coeﬃcients. Applications are given to the detailed descriptionof smoothing properties of Korteweg-de Vries, Benjamin-Ono, Davey-Stewartson,Shrira, and other equations.
This is joint work with Michael Ruzhansky (Imperial College London).
Date:
Wednesday, June 18 14:00-14:50

Kickoﬀ Meeting

9Kickoﬀ Meeting

Masao YAMAZAKI
Waseda University, Tokyo
Title:
Two-dimensional Navier-Stokes exterior problem with some symmetry
conditions
Abstract:
This talk is concerned with the statonary Navier-Stokes equations on the wholeplane or on the two-dimensional exterior domain with nontrivial external forces
and the boundary values.
The existence of solutions of this problem is proved under symmetry of two coordinate axes. However, the existence of stationary solutions with decay suﬃcient for
the uniqueness and the stability (which correspond with the physically reasonable
solutions in the three-dimensional problem) is proved only under more stringent
decay conditions.
In this talk we give a new condition on the symmetry suﬃcient for the existenceof stationary solutions with adequate decay conditions. This condition generalized
the aforementioned strong conditions, and neither necessary nor suﬃcient for thesymmetry of coordinate axes. We emphasize that the new condition is independentof the choice of coordinate axes, and can naturally include the solutions withrotational symmetry.
The uniqueness in the large and the stability under initial L2 -perturbation of thestationary solutions are also discussed.
Date:
Wednesday, June 18 15:00-15:50

10

Kickoﬀ Meeting