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, Jan. 9

Wed, Jan. 10

Thu, Jan. 11

Fri, Jan. 12

10:50~ Opening

9:30-10:40
Yoshihiro SHIBATA①(Waseda Univ.)

9:30-10:40
Fanghua LIN③

9:30-10:40
Yoshihiro SHIBATA③

11:00-12:10
Fanghua LIN①(New York Univ.)

10:55-12:05
Fanghua LIN②

10:55-12:05
Yoshihiro SHIBATA②

12:20-12:50
Theodore D. DRIVAS(Princeton Univ.)

12:15-12:45
Dieter BOTHE(TU Darmstadt)

12:15-12:45
Reinhard FARWIG(TU Darmstadt)

10:55-11:25
Amru HUSSEIN(TU Darmstadt)
11:35-12:05
Takahito
KASHIWABARA(Univ. of Tokyo)
12:15-12:45
Okihiro SAWADA(Gifu Univ.)

Lunch Break
14:30-15:00
Ryo TAKADA(Kyushu Univ.)

14:30-15:00
Masahiro KUNIMOTO(Waseda Univ.)

15:10-15:40
Kohei SOGA
(Keio Univ.)

15:10-15:40
Shinya UCHIUMI(Waseda Unv.)

14:30-15:00
Thieu Huy NGUYEN(Hanoi Univ. of Sci.
and Tech.)
15:10-15:40
Takahiro OKABE(Hirosaki Uni.)

14:30-15:00
Patrick TOLKSDORF(TU Darmstadt)
15:10-15:40
Tsukasa IWABUCHI(Tohoku Uni.)

Coffee Break
16:15-16:45
Hajime KOBA
(Osaka Univ.)

16:15-16:45
Hirokazu SAITO(Waseda Univ.)

16:15-16:45
Ken ABE(Osaka City Univ.)

16:55-17:25
Kazuyuki TSUDA(Osaka Univ.)

16:55-17:25
Miho MURATA(Kanagawa Univ.)

16:55-17:25
Erika USHIKOSHI(Yokohama National
Univ.)

17:35-18:15
Keiichi WATANABE(Waseda Univ.)
Naoto KAJIWARA(Univ. of Tokyo)
Klaus KRESS(TU Darmstadt)
Mathis GRIES(TU Darmstadt)

17:35-18:15
Suma'inna(Waseda Univ.)
Ken FURUKAWA(Univ. of Tokyo)
Anton SEYFERT(TU Darmstadt)
Yuka TERAMOTO(Kyushu Univ.)

17:35-18:15
Kohei NAKAO(Shinshu Univ.)
Hiroyuki TSURUMI(Waseda Univ.)
David WEGMANN(TU Darmstadt)
Andreas SCHMIDT(TU Darmstadt)●→Main-course ●→30min talk

16:15-16:55
Marc WRONA(TU Darmstadt)
Mitsuo HIGAKI(Kyoto Univ.)
Hana MIZEROVA(Czech Acad. Sci.)
Sebastian ZAIGLER(TU Darmstadt)

17:00 Closing
17:30~
Closing Reception●→10min talk

Main-Course

Main-Course

Fanghua LIN
New York University, New York
Title:
Existence and Regularity Theory of Suitable Weak Solutions of the Incompressible Navier-Stokes equations.
Abstract:
After a brief discussion on the classical Leray-Hopf weak solutions, the energy equality and inequality, we shall concentrate mainly on the existence and regularity(partial
regularity) of suitable weak solutions. The notion of suitable weak solutions were introduced in the works of Sheaﬀer, Caﬀarelli-Kohn-Nirenberg. The latter also established
its existence. Here I shall present some other constructions of suitable weak solutions.
We then discuss the regularity of these suitable weak solutions and to see how they
connected to the earlier contributions of Serrin, Prodi, and others and recent works bySeregin, Sverak and others.
Date:⃝
1 Tuesday, Jan. 09 11:00-12:10⃝
3 Thursday, Jan. 11 09:30-10:40⃝
2 Wednesday, Jan. 10 10:55-12:05

Yoshihiro SHIBATA
Waseda University, Tokyo
Title:
Maximal regularity and free boundary problem for the Navier-Stokes equations
Abstract:
I will give the following talks.
1st lecture: Lp -Lq maximal regularity for the Stokes equations with free boundary
conditions and the local well-posedness for some free boundary problem of the NavierStokes equations.
2nd lecture: The global well-posedness for free boundary problem of the Navier-Stokes
equations with surface tension in the case that the reference domain is closed to a balland initial data are small enough.
3rd lecture: The global well-posedness for free boundary problem of the Navier-Stokes
equations without surface tension in the case that the reference domain is an exterior
domain and initial data are small enough.
Date:⃝
1 Wednesday, Jan. 10 09:30-10:40⃝
3 Friday, Jan. 12 09:30-10:40

2⃝
2 Thursday, Jan. 11 10:55-12:05

Main-Course30 minutes talks

30 minutes talks

Ken ABE
Osaka City University
Title:
Axisymmetric flows in an exterior domain
Abstract:
We consider the three-dimensional Navier-Stokes equations for axisymmetricinitial datau0 (x) = ur0 (r, z)er (θ) + uθ0 (r, z)eθ (θ) + uz0 (r, z)ez .
Here (r, θ, z) denotes the cylindrical coordinate and er , eθ , ez are the basis in R3 .
The azimuthal component of velocity uθ is referred as the swirl component. It is
known that the Cauchy problem of the Navier-Stokes equations is globally wellposed for large axisymmetric initial data in L3 with finite energy, if the swirl
component of initial velocity is identically zero (with no swirl). For initial datawith swirl, unique solvability of the Cauchy problem in L3 ∩ L2 is unknown.
In this talk, we study axisymmetric flows with swirl in an exterior domainΠ = {(x1 , x2 , x3 ) ∈ R3 | xh = (x1 , x2 ), |xh | > 1},

subject to the slip boundary condition (D(u)n)tan = 0, u · n = 0 on ∂Π, whereD(u) = (∇u + ∇T u)/2 is the deformation tensor, n is the unit normal and ftan =f − n(f · n) denotes the tangential component of a vector field f on ∂Π. We
report unique existence of global solutions for large axisymmetric data in L3 ∩ L2
satisfying the decay condition of the swirl component |xh |uθ0 ∈ L∞ . This talk isbased on a joint work with Prof. Seregin (St. Petersburg/ Oxford).
Date:
Thursday, Jan. 11 16:15-16:45

Dieter BOTHE
Technical University of Darmstadt
Title:
Thermodynamically consistent modeling of multicomponent diﬀusion
Abstract:
We revisit the modelling of multicomponent diﬀusion within the framework of
30 minutes talks3

30 minutes talks

thermodynamics of irreversible processes with main emphasis on class-I models
which comprise partial mass balances for all chemical constituents, but a common
momentum and energy balance. Consistency of the mass fluxes with the continuity
equation requires the fluxes to sum up to zero. Within the standard approach one
flux is eliminated, leading to the generalized Fickean multicomponent diﬀusion
fluxes. This procedure breaks the symmetry between the diﬀerent species andtherefore requires a non-diagonal, fully coupled closure. The (cross-)mobilities
necessarily depend on the composition in a complicated, singular manner. In
particular, constant mobilities are inconsistent with the physically required forward
invariance of the cone of non-negative partial densities.
Alternatively, the Maxwell-Stefan theory also fits within the same framework, butuses a so-called resistance form in which the closure is performed for the diﬀusion
velocities. In contrast to the Fickean closure, the resulting fluxes are consistent to
non-negativity of the partial densities even if constant Maxwell-Stefan diﬀusivities
are employed. But, similar as for the Fickean approach, the closure requires a fully
occupied matrix of MS-diﬀusivities. From experiments as well as from molecular
dynamics simulations it is known that MS-diﬀusivities are not constant but depend
especially on the mixture composition, although in a non-singular manner. A
theoretical model for this dependence was recently proposed as a multicomponent
extension of the Darken equation known from diﬀusion in solids.
After reviewing these two known diﬀusion theories, we propose and discuss a noveland more direct closure which avoids the tedious inversion of the Maxwell-Stefan
equations. More importantly, it also gives rise to a diagonal closure in which only
those cross-eﬀects are present that are necessary to guarantee consistency withtotal mass conservation plus a compositional dependence of the diﬀusivity. We
show that the latter case corresponds precisely to the multicomponent Darken
equation.
Date:
Wednesday, Jan. 10 12:15-12:45

Theodore D. DRIVAS
Princeton University

tdrivas@math.princeton.edu

Title:An Onsager Singularity Theorem for the Compressible Euler Equations
Abstract:
We prove that bounded weak solutions of the compressible Euler equations willconserve thermodynamic entropy, unless the solution fields have suﬃciently low
space-time Besov regularity. A quantity measuring kinetic energy cascade will
also vanish for such Euler solutions, unless the same singularity conditions are
satisfied. It is shown furthermore that strong limits of solutions of compressible
4

30 minutes talks

30 minutes talks

Navier-Stokes equations that are bounded and exhibit anomalous dissipation are
weak Euler solutions. These inviscid limit solutions have non-negative anomalous
entropy production and kinetic energy dissipation, with both vanishing when solutions are above the critical degree of Besov regularity. Stationary, planar shocks
in Euclidean space with an ideal-gas equation of state provide simple examples
that satisfy the conditions of our theorems and which demonstrate sharpness of
our L3 -based conditions. This talk is based on joint work with G. Eyink.
Date:
Tuesday, Jan. 09 12:20-12:50

Reinhard FARWIG
Technical University of Darmstadt
farwig@mathematik.tu-darmstadt.de

Chenyin Qian
Department of Mathematics, Zhejiang Normal University, China

Title:The global attractor for autonomous quasi-geostrophic equations
with fractional dissipation in R2
Abstract:
Consider the autonomous quasi-geostrophic equation with fractional dissipationin R2
θt + u · ∇θ + (−∆)α θ = f (x, θ)in the subcritical case 1/2 < α ≤ 1, with initial condition θ(x, 0) = θ0 and given
external force f (x, θ). Here the real scalar function θ is the so-called potential
temperature, and the incompressible velocity field u = (u1 , u2 ) = (−R2 θ, R1 θ) is
determined from θ via Riesz operators. Our aim is to prove the existence of thecompact global attractor A in the Bessel potential space H s (R2 ) when s > 2(1−α).
The construction of the attractor is based on the existence of an absorbing set
in L2 (R2 ) and H s (R2 ) where s > 2(1 − α). A second major step is usually basedon compact Sobolev embeddings which unfortunately do not hold for unboundeddomains. To circumvent this problem we exploit compact Sobolev embeddings onballs BR ⊂ R2 and uniform smallness estimates of solutions on R2 \ BR . In theliterature the latter estimates are obtained by a damping term λθ, λ < 0, as partof the right hand side f to guarantee exponential decay estimates. In our approach
we exploit a much weaker nonlocal damping term of convolution type ρ ∗ θ withnegative Fourier transform ρb.
Date:Thursday, Jan. 11 12:15-12:45

30 minutes talks

5

30 minutes talks

Amru HUSSEIN
Technical University of Darmstadt
Title:
Nematic Liquid Crystals in Lipschitz domains
Abstract:
Liquid crystals are materials which behave with some respects like a liquid andwith others like a solid crystal, and they have become present in everyday life by
their practical application in liquid crystal displays (LCD).
The Ericksen-Leslie theory is a continuum theory for liquid crystals coupling a
fluid equation to an equation for the director field which describes the orientationof the rod-like crystals.
We consider the simplified Ericksen-Leslie model in three dimensional bounded
Lipschitz domains. Applying a semilinear approach, we prove local and global wellposedness (assuming a smallness condition on the initial data) in critical spaces
for initial data in L3σ for the fluid and W 1,3 for the director field. The analysis ofsuch models, so far, has been restricted to domains with smooth boundaries, andwell-posedness in critical spaces has not been proven for bounded domains.
This talk is based on a joint work with Anupam Pal Choudhury and Patrick
Tolksdorf.
Date:
Friday, Jan. 12 10:55-11:25

Tsukasa IWABUCHI
Tohoku University
Title:
Ill-posedness for the compressible Navier-Stokes equations
under the barotropic condition
Abstract:
We consider the compressilbe Navier-Stokes system in the critical Besov spaces.
It is known that the system is (semi-)well-posed in the scaling semi-invariant spaces
n

n

−1

p
p
of the homogeneous Besov spaces Ḃp,1
× Ḃp,1
for all 1 ≤ p < 2n. On the other
hand if the data is in a larger scaling invariant class such as p > 2n, then thesystem is not well-posed. In this talk, we show that for the critical case p = 2n thesystem is ill-posed by showing that there exists an explicit example of initial datasuch that the corresponding strong solution is strongly discontinuous as t → 0.

Date:
Friday, Jan. 12 15:10-15:40

630 minutes talks

30 minutes talks

Takahito KASHIWABARAThe University of Tokyo
Title:
Global strong well-posedness of the primitive equations
pin L∞
xy Lz -type spaces
Abstract:
Primitive equations, which describe large scale motion of ocean or atmosphere,
are obtained by assuming the hydrostatic balance in the vertical direction of the
3D Navier-Stokes equations. In our previous results, existence and uniqueness of a
global-in-time strong solution to the primitive equations within the Lp -framework
have been established either by analytic semigroup approach or by maximal regularity. In this talk, we will discuss cases corresponding to the end-point casep = ∞. More precisely, the Banach space which our analysis is based on is L∞
in the horizontal variables x, y and Lp in the vertical one z. We prove that one
can construct via the Fujita-Kato method a unique strong solution for the initialpdata in Cxy Lpz , where arbitrary perturbation in L∞
xy Lz is allowed provided it issmall. The proofs are quite diﬀerent depending on boundary conditions (Dirichlet
or Neumann).
Date:
Friday, Jan. 12 11:35-12:05

Hajime KOBA
Osaka University

iti@sigmath.es.osaka-u.ac.jp

Title:On the Helmholtz-Weyl decomposition on surfaces and its application
to multiphase flow
Abstract:
We study the Helmholtz-Weyl decomposition of vector-valued functions on asurface. We prove that we can decompose an L2 -vector field on the surface into
the three parts such as a surface divergence-free part, surface gradient part, and amean curvature part. We also apply the Helmholtz-Weyl decomposition to consider
a mathematical modeling of multiphase flow with surface tension.
A part of this talk is based on joint researches with Yoshikazu Giga (Universityof Tokyo), Chun Liu (Illinois Institute of Technology, Chicago), and Kazuki Sato
(Osaka University).
References:[1] H. Koba, C. Liu, and Y.Giga, Errata to Energetic variational approaches forincompressible fluid systems on an evolving surface. to appear in Quart. Appl.
Math.
30 minutes talks

7

30 minutes talks[2] H. Koba and K. Sato, Energetic variational approaches for non-Newtonian
fluid systems. preprint, arXiv:1705.06956.
Date:
Tuesday, Jan. 09 16:15-16:45

Masahiro KUNIMOTO
Waseda University
Research Organization for Nano & Life Innovation, Waseda Univesity, Japan,
kmasahiro@aoni.waseda.jp

Title:
Modeling and Spectroscopic Analysis of the Liquid-liquid Extractionof Boric Acid in Micro Channel Devices
Abstract:
For the mechanistic understanding of chemical reactions at the liquid-liquidinterface of aqueous and organic phases, theoretical modeling and analysis are
conducted for the interfacial reaction taking place in a Y-shaped micro flow channel. As a target reaction, a condensation of boric acid (aqueous side) with 2,2,4trimethyl-1, 3-pentanediol (organic side), as “TMPD”, is chosen, which serves as
one of the extraction processes to eliminate boron impurities from the aqueous
solution. TMPD is one of the hydrophobic diol compounds. It is worth notingthat boric acid and TMPD are expected to react at the liquid-liquid interfacial
area to produce hydrophobic species to be transferred to the organic phase, which
results in the removal of boron from the aqueous phase. The modeling strategy is
as follows. Regarding the flow inside, the typical parabolic profile describing laminar flow that is common in the case of micro channel systems is employed, basedon the following assumptions: (i) the flow is incompressible, (ii) the gravity eﬀect
is suﬃciently small to be neglected, (iii) there is no surface tension at the liquidliquid interface, and (iv) the flow regime inside the channel is a well-developed.
Regarding the reaction rate of the condensation, the rate constant of the secondorder reaction is estimated from the activation free energy obtained from quantum
chemical calculations. The flow profile and rate constants above are put into theequation governing the mass balance of all chemical components in the system, tobe solved under the boundary condition that the reaction takes place not in thebulk area but only at the interface. In the presentation, the simulation using thismodeling will be demonstrated, compared to the experimental results from spectroscopic measurements. Moreover, the molecular level mechanism of the reactionof boric acid and TMPD, which is proposed by the series of modeling studies, willbe also described.
Date:
Wednesday, Jan. 10 14:30-15:00
830 minutes talks

30 minutes talks

Miho MURATA
Kanagawa University

m-murata@kanagawa-u.ac.jp

Yoshihiro SHIBATA
Waseda University

Title:
Decay estimates of solutions for the Navier-Stokes-Korteweg system
in RN
Abstract:
In this talk, we consider the Navier-Stokes-Korteweg system in RN :in RN , t ∈ (0, T ),
 ∂t ρ + div (ρu) = 0ρ (∂t u + u · ∇u) − Div (S + K) + ∇P (ρ) = 0

in RN , t ∈ (0, T ),(ρ, u)|t=0 = (ρ∗ + ρ0 , u0 )in RN .

Here ρ, u and P denote the density, velocity and pressure of fluid, respectively.
The viscous stress tensor S and the Korteweg stress tensor K are S = 2µD(u) +(ν − µ)div uI, and K = (κρ∆ρ + κ/2|∇ρ|2 )I − κ∇ρ ⊗ ∇ρ with I the N × N identity
matrix and D(u) = {∇u + (∇u)T }/2, µ, ν are viscosity coeﬃcients, and κ is acapillary coeﬃcient. ρ∗ is a positive constant. This system was first introduce byKorteweg D. J.[1].
It is shown that the system admits a unique, global strong solution for small initial
data in the Lp in time and Lq in space setting. For the purpose, the main tools
are the maximal Lp -Lq regularities and Lp -Lq decay properties to the linearized
equations. This talk is based on a joint work with Professor Yoshihiro Shibata in
Waseda University.
References:[1] Korteweg D. J., “Sur la forme que prennent les équations du mouvement desfluides si l’on tient compte des force capillaires causées par des variations
de densité considérables mais continues et sur la théorie de la capillarite dansl’hypothèse d’une variation continue de la densité,” Archives Néelandaises dessciences exactes et naturelles, Ser 2, (6), 1–24 (1901).
Date:
Wednesday, Jan. 10 16:55-17:25

30 minutes talks9

30 minutes talks

Thieu Huy NGUYEN
Hanoi University of Science and Technology
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology
Vien Toan ung dung va Tin hoc, Dai hoc Bach khoa Hanoi,
1 Dai Co Viet, Hanoi, Vietnam
huy.nguyenthieu@hust.edu.vn

Title:
Asymptotic Behavior of Certain Navier-Stokes Flows on Unbounded
Domains
Abstract:
In this talk, we will present some our recent results published in [1, 2, 3] onboundedness, periodicity and stability of solutions to fluid flow problems in unbounded domains. We start from a general framework of evolution equations on
an unbounded domain Ω, namely, we consider the general semi-linear equations
on Ω of the formu + Au = Pdiv(G(u) + F (t))t(1)
u(0) = u0 ,where −A generates a C0 -semigroup (e−tA )t≥0 on Ldσ,w (Ω), P is Helmholtz projection; G is a nonlinear and local Lipschitz operator acting from Ldσ,w (Ω) into
d/2

d/2

2

2

Lσ,w (Ω)d , and F (·) is a time-dependent second-order tensor in Lσ,w (Ω)d . Under relevant assumptions on Lp − Lq smoothing properties of (e−tA )t≥0 and local
Lipschitz properties of G, and using the interpolation functors combined with differential inequalities and fixed point arguments we are able to prove the existenceof bounded (in time t) solutions to (1) for each bounded tensor F (·). Then, we canuse either an ergodic approach in concrete cases (see [2]) or topology arguments in
the general case (see [1]) to prove the existence of periodic solutions to fluid flow
problems. Moreover, our methods can be extended to obtain the stability of suchbounded and/or periodic solutions to such problem (see [3]). Our abstract results
can be applied to certain Navier-Stokes flows such as Navier-Stokes-Oseen flow,
the Navier-Stokes flow past rotating obstacles, and, in the geophysical setting etc.
References:[1] M. Geissert, M. Hieber, Thieu Huy Nguyen, A General Approach to Time
Periodic Incompressible Viscous Fluid Flow Problems, Archive for Rational
Mechanics and Analysis 220 (2016), 1095–1118.
[2] T.H. Nguyen, Periodic motions of Stokes and Navier-Stokes flows around arotating obstacle, Arch. Ration. Mech. Anal., 213 (2014), 689–703.
[3] T.H. Nguyen, T.N.H. Vu, T.X. Pham, Boundedness and stability of solutions
to semi-linear equations and applications to fluid dynamics, Communications
on Pure and Applied Analysis 15 (2016), 2103–2116.
10

30 minutes talks

30 minutes talks

Date:
Thursday, Jan. 11 14:30-15:00

Takahiro OKABE
Hirosaki University
Title:
Remark on the strong solvability of the Navier-Stokes equationsin the weak Ln space
Abstract:
We consider the Cauchy problem of the Navier-Stokes equations in weak Lebesgue
space Ln,∞ (Rn ), n ≥ 3 with non-trivial forces f . The aim of this talk is that weconstruct a strong solution of (N-S) in the framework of Ln,∞ , i.e., the solution
which satisfies the orbit u(t) lies in Ln,∞ (Rn ) and satisfiesdu − ∆u + P[u · ∇u] = Pf
dt

in Ln,∞ (Rn ),t > 0.

(DE)

Borchers-Miyakawa [2] construct such a solution with f ≡ f (x) as a perturbed
equation from the stationary solution for f . However, for non-trivial forces f =f (x, t), we need some qualitative condition for f in order to verify (DE). So introducing the subspace where the Stokes semigroup is strongly continuous we observethat the mild solution of (N-S) becomes a strong solution, i.e., satisfies (DE) whenf in the subspace.
Moreover, by virtue of the subspace, we establish the local well-posedness within
e n3 ,∞ (Rn )), n ≥ 4 and f ∈weak mild solutions of (N-S) under f ∈ BC([0, T ); Le p,∞ (Rn ) = {f ∈ Lp,∞ (Rn ); f ∈ L∞ (Rn )}∥·∥p,∞ . By
BC([0, T ); L1 (R3 )), where L
controlling the singularity of initial data and f , we obtain not only (small) global
solutions, but also a unique (large) local solutions of (N-S). Especially for theuniqueness criterion for a general data, we find Brezis’ argument [1] can be applicable. Therefore, we see that if the initial data has a bad singularity in Ln,∞ (Rn ) thenthe mild solution in the class BC([0, T ); Ln,∞ (Rn )) never stays in C((0, T ); Lr (Rn ))with some r > n.
This is a joint work with Professor Yohei Tsutsui.
References:[1] H. Brezis, Arch. Rational Mech. Anal., 128 (1994), 359–360.
[2] W. Borchers and T. Miyakawa, Acta Math., 174 (1995), 311–382.
[3] H. Kozono and M. Yamazaki, Math. Z., 228 (1998), 751–785.
Date:Thursday, Jan. 11 15:10-15:40

30 minutes talks

11

30 minutes talks

Hirokazu SAITO
Waseda University
Title:
Maximal regularity for the Navier-Stokes-Korteweg system
and its application
Abstract:
In this talk, we would like to consider the Navier-Stokes-Korteweg system on
domains of RN , N ≥ 2. Korteweg introduced in the early 1900s a stress tensor
including the density gradient ∇ρ in order to model fluid capillarity eﬀects. After
that, Dunn and Serrin [1] derived rigorously
K(ρ) =

κ(∆ρ2 − |∇ρ|2 )I − κ∇ρ ⊗ ∇ρ2that is called a Korteweg tensor nowadays, where κ > 0 is a capillary coeﬃcient.
In this approach, the momentum equation is turned intoρ(∂t u + u · ∇u) = Div(S(u) + K(ρ) − P (ρ)I),
with the standard viscous stress S(u) = µD(u) + (ν − µ)divuI and the pressure P (ρ), which is called the Navier-Stokes-Korteweg equation. We discuss inthis talk the maximal regularity for the linearized system and its application to
the local solvability of the Navier-Stokes-Korteweg system with non-slip boundary
condition. Our result of the local solvability especially relaxes the regularity ofsolutions for the previous work due to Kotschote [2].
References:[1] J. E. Dunn and J. Serrin. On the thermomechanics of interstitial working.
Arch. Rational Mech. Anal., 88(2):95–133, 1985.
[2] M. Kotschote. Strong solutions for a compressible fluid model of Korteweg
type. Ann. Inst. H. Poincaré Anal. Non Linéaire, 25(4):679–696, 2008.
Date:
Wednesday, Jan. 10 16:15-16:45

Okihiro SAWADA
Gifu University
Yanagido 1-1, Applied Physics Course, Gifu University, Gifu City, 501-1193, Japan.
okihiro@gifu-u.ac.jp

Title:On the time-local solvability of the primitive equations
with linearly growing initial data

12

30 minutes talks

30 minutes talks

Abstract:
The primitive equations in an infinity layer domain are considered:⃗ H − ∆U
⃗H + U⃗ ·∇⃗U⃗H + ∇⃗ HP = 0∂t U
in Ω × (0, T ),

∂3 P = 0in Ω × (0, T ),
(P)
⃗
⃗

∇·U = 0in Ω × (0, T ),
⃗ H |t=0 = U⃗ H,0 in Ω.
U⃗ = (U⃗ H , U3 ) = (U1 , U2 , U3 ) denotes the velocity, and P is the pressure
Here U
at t ∈ (0, T ) and ⃗x = (⃗xH , x3 ) = (x1 , x2 , x3 ) ∈ Ω = R2 × (−h, h) with h > 0;
⃗ = (∇⃗ H , ∂3 ) = (∂1 , ∂2 , ∂3 ) and ∆ = ∂ 2 + ∂ 2 + ∂ 2 . We impose∂i = ∂/∂xi , ∇1
2
3⃗ H on the boundary ∂Ω = Γ, and somethat U3 = 0 and some conditions for U⃗ H,0 . The horizontal velocity U⃗ H iscompatibility conditions for the initial data U
mainly discussed, since we see
∫ x3
⃗H ·U
⃗ H (x1 , x2 , ζ, t)dζ.
U3 (x1 , x2 , x3 , t) = −∇−h⃗ H,0 (⃗x) = −M⃗xH + ⃗u0 (⃗x). So, ⃗u :=Let M ∈ R2×2 with tr M = 0, and let U⃗ H + M⃗xH might satisfy the intgral equation(u1 , u2 ) := U⃗u(t) = e ⃗u0 −

∫ t

tL

{
}
⃗ H ⃗u + U3 ∂3 ⃗u − 2M⃗u (s)dse(t−s)L P ⃗u · ∇0with some ⃗u0 in certain Sobolev spaces associated with{}∥·∥Lp (Ω)⃗H ·φ⃗ ∈ C ∞ (Ω)2 : ∇⃗=0
Lσp (Ω) := φfor p ∈ (1, ∞), via the Ornstein-Uhlenbeck type semigroup {etL }t≥0 generated by⃗ HφL⃗
φ := ∆⃗
φ + M⃗xH · ∇⃗ − Mφ⃗ − (I − P)trD ∂3 φ
⃗,
{
}
2,p
2
⃗ Hφ
D(L) := φ⃗ ∈ H (Ω) : ∂3 φ
⃗ |ΓN = 0 and φ⃗ |ΓD = 0, M⃗xH · ∇⃗ ∈ Lp (Ω)2 .
∫ h
1φ⃗ (x1 , x2 , ζ)dζ.
2h −h
To obtain smoothing properties of the semigroup, derivatives of the associatedkernel are calculated. To establish the time-local existence and uniqueness of mild
solutions (solutions to the integral equation), the adapted Fujita-Kato scheme isused.
This is a joint work with Amru Hussein and Martin Saal at TU Darmstadt.
Here, P is the projection from Lp (Ω) to Lσp (Ω), and φ⃗ (x1 , x2 ) :=Date:
Friday, Jan. 12 12:15-12:45

30 minutes talks

13

30 minutes talks

Kohei SOGA
Keio University
Title:On convergence of Chorin’s projection method to a Leray-Hopf weaksolution
Abstract:
Finite diﬀerence approximation is an eﬀective approach to prove existence ofLeray-Hopf weak solutions to the incompressible Navier-Stokes equations on abounded domain. Ladyzhenskaya demonstrates a lot about this issue in her book,where she discretizes the equations including the pressure term and the divergencefree constraint. Chorin (’69) presents a discrete version of the Helmholtz decomposition operator, which is applied to a finite diﬀerence scheme similar to Ladyzhenskaya’s in such a way that the non-divergence-free part of a diﬀerence solution
is eliminated at each time step. Although these two methods seem mathematically equivalent, the actual computational process (on a computer) of operating
the discrete Helmholtz decomposition operator is simpler or quicker than solving
the whole equations with the pressure term and the divergence-free constraint.
Hence, Chorin’s approach is widely used as “Chorin’s projection method”. Since
Navier-Stokes equations are nonlinear and the finite diﬀerence schemes are implicit,
convergence of Chorin’s projection method does not follow from Ladyzhenskaya’s
results. Chorin shows convergence and error estimates of his scheme, assuming that
there is an exact C 5 -solution. Temam (’69) announces finite diﬀerence schemes
which are convergent to Leray-Hopf weak solutions, modifying Ladyzhenskaya andChorin’s frameworks to be more abstract and compatible with functional analytic
arguments. Unfortunately, Temam’s schemes do not seem friendly to practicalnumerical simulations as Chorin pointed out.
In this talk, I show slightly modified Chorin’s projection method, which is diﬀerent from Temam’s schemes. The reason for our modification is that the original
Chorin’s scheme fails to give a closed L2 -energy estimate. We slightly change thenonlinear term to obtain nice estimates. Then, adjusting well-known compactness results to our discrete setting, we discuss weak and strong convergence of thescheme.
Date:
Tuesday, Jan. 09 15:10-15:40

Ryo TAKADA
Kyushu University

Faculty of Mathematics, Kyushu University
Fukuoka 819-0395, JAPAN

Title:
Strongly stratified limit for the 3D inviscid Boussinesq equations
14

30 minutes talks

30 minutes talks

Abstract:
We consider the initial value problem for the 3D inviscid Boussinesq equations
with the eﬀect of stable stratification:∂t v + (v · ∇)v = −∇q + θe3
t > 0, x ∈ R3 ,∂ θ + (v · ∇)θ = −N 2 v
t > 0, x ∈ R3 ,
t
3
(BN )

t ⩾ 0, x ∈ R3 ,∇ · v = 0v(0, x) = v (x), θ(0, x) = θ (x)x ∈ R3 .
00Here, N > 0 is the Brunt-Väisälä (buoyancy) frequency for the constant stratification. The system (BN ) is derived from the original Boussinesq equations byconsidering the perturbation about a mean state in hydrostatic balance.
In this talk, we consider the singular limit as N → ∞, and show that theclassical solution v N of (BN ) converges to that of the 2D Euler equations. The key
ingredients of the proof are to introduce the modified linear dispersive solutions,and the inhomogeneous Strichartz estimates for the linear propagator related tothe stable stratification.
Date:
Tuesday, Jan. 09 14:30-15:00

Patrick TOLKSDORF
Technical University of Darmstadt

tolksdorf@mathematik.tu-darmstadt.de

Title:Morrey regularity of solutions to the Stokes equations in three-dimensionalbounded Lipschitz domains
Abstract:Let Ω ⊂ R3 be a bounded Lipschitz domain. Then, we define for 1 < p < ∞ and0 ≤ λ < 3 the Morrey space
Lp,λ (Ω) := {f ∈ Lp (Ω) : ∥f ∥Lp,λ (Ω) < ∞},where∥f ∥pLp,λ (Ω) :=

sup

r−λ

∫

x0 ∈Ω
0<r<diam(Ω)|f (x)|p dx.
B(x0 ,r)∩Ω

For f ∈ Lp,λ (Ω) and p ≥ 2, we consider solutions u to the Stokes equations−∆u + ∇π = fin Ωdiv(u) = 0
in Ω



u=0
on ∂Ω
30 minutes talks

15

30 minutes talks

and ask whether u lies in Lp,λ (Ω) as well, i.e., we ask for the boundedness of theoperator A−1 : Lp,λ (Ω) → Lp,λ (Ω), where A denotes the Stokes operator.
To prove the boundedness, we first investigate the case p = 2 and prove by meansof some classical estimates the boundedness of this operator by using a characterization of Morrey spaces by so-called Morrey–Campanato spaces. Once thisis achieved, we present a generalization of a Calderón–Zygmund theorem, whichallows to extrapolate bounded operators in L(L2 (Ω)) ∩ L(L2,λ (Ω)) to bounded operators in L(Lp (Ω)) ∩ L(Lp,λ (Ω)) for p > 2. Finally, we sketch how to verify therequirements of this Lp,λ -extrapolation theorem in the case of the Stokes system
above.
This talk presents a work in progress and is a joint work with Martin Saal fromTechnische Universität Darmstadt.
Date:
Friday, Jan. 12 14:30-15:00

Kazuyuki TSUDA
Osaka University
Title:
Diﬀusive property and smoothing eﬀect for solution to the compressible
Navier-Stokes-Korteweg equation
Abstract:
Time decay estimate of a solution to the compressible Navier-Stokes-Korteweg
system is studied. It is known that the system describes two phase flow with phasetransition between liquid and vapor as a diﬀuse interface model in a compressible
fluid. Concerning time decay estimate, Wang and Tan ([3]) show convergence
rates of L2 norm of a solution for 3 dimensional case under small initial value
around constant state. The decay rate is same as that of heat kernel. In thistalk concerning the linearized problem, the decay estimates with diﬀusive wave
property (Cf., [1, 2]) for initial date are derived. Furthermore, as an application,
we give time decay estimate of a solution to nonlinear system.
In contrast to the compressible Navier-Stokes system, for linear system regularities of initial dates are lower and independent of the order of derivative of thesolution owing to smoothing eﬀect from the Korteweg tensor. Furthermore, for the
nonlinear system diﬀusive wave properties are obtained with initial dates having
lower regularity than that of studies of the compressible Navier-Stokes system and[3].
References:[1] D. Hoﬀ and K. Zumbrun, Multi-dimensional diﬀusion waves for the NavierStokes equations of compressible flow, Indiana Univ. Math. J., 44, (1995),
pp.603–676.
16

30 minutes talks

30 minutes talks[2] T. Kobayashi and Y. Shibata, Remark on the rate of decay of solutions to linearized compressible Navier-Stokes equations, Pacific Journal of Mathematics,
207 (2002), pp. 199–234.
[3] Y. Wang and Z. Tan, Optimal decay rates for the compressible fluid modelsof Korteweg type, J. Math. Anal. Appl., 379 (2011), pp. 256–271.
Date:
Tuesday, Jan. 09 16:55-17:25

Shinya UCHIUMI
Waseda University

Title:A pressure-stabilized finite element method with higher-order elements
Abstract:
We consider a finite element method for the Navier-Stokes problem and its linearized problem for small viscosity.
One issue in these problems is treatment of the material derivative term. In
convection-dominated flow problems, it is important to put weight on information in the upwind direction to make schemes stable. We here use the LagrangeGalerkin (LG) method, which is a mixed method of the characteristics and thefinite element method. One of the advantages of this method is that the resultant
matrix is symmetric, which allows us to use eﬃcient linear solvers. Recently a
LG scheme with a locally linearized velocity has been developed (Tabata-Uchiumi,
Math. Comp., 2018) to solve discrepancy between the convergence theory andactual computation arising from the diﬃculty in the computation of integration in
a composite function. There the inf-sup stable P2/P1-element was used.
Here we combine this method to a pressure-stabilized method for the flow problems with small viscosity. In the scheme we use the equal-order approximationof order k for both the velocity and pressure, and add a symmetric pressure stabilization term. Symmetry of the LG method is inherited. Note the fact that,apart from the issue of the material derivative in these problems, dependence on
the viscosity appears even in the simpler Stokes problems. We show a theoreticalresult that shows eﬃciency for small viscosity. We also show numerical examples
that reflect the result.
Date:
Wednesday, Jan. 10 15:10-15:40

Erika USHIKOSHI
Yokohama National University

Title:On Hadamard’s formula of the Stokes equations
30 minutes talks

17

30 minutes talks

Abstract:Let Ω ⊂ R3 be a bouded domain with a smooth boundary ∂Ω. For any ε ≥ 0
and for any ρ ∈ C ∞ (∂Ω), we define a perturbed domain by Ωε , whose boundaryis ∂Ωε := {y + ερ(y)νy ; y ∈ ∂Ω}. Here, νy is a unit outer normal vector aty ∈ ∂Ω. Under such a perturbation, we consider the eigenvalue problem of the
Stokes equations with a slip boundary condition;x ∈ Ω(ε),
∆Φ(x) + λ(ε)Φ(x) = ∇q(x),div Φ(x) = 0,x ∈ Ω(ε),
(2e(Φ)ν(x) + κΦ(x)) × ν(x) = 0, Φ(x) · ν(x) = 0, x ∈ ∂Ω(ε),(0.1)

where κ is a positive (friction constant) ，and {eij }i,j=1,2,3 is a deformation tensor
1 ∂Φi ∂Φjdefined by eij (Φ) =+, i, j = 1, 2, 3. We see that the the spectral set
2 ∂xj
∂xi
of the Stokes equations with a slip boundary condition (0.1) consists of a descretesequence of positive number. If we arrange them with counting multiplicity inincreasing order, we have0 < λ1 (ε) ≤ λ2 (ε) ≤ λ3 (ε) ≤ · · · ≤ λm (ε) ≤ · · · → ∞,where the m-th eigenvalue is denoted by λm (ε). In this talk, we consider εdependency of the eigenvalue having the multiplicity, and establish the Hadamard
variational formula for that. This is a joint work with Professor Shuichi Jimbo(Hokkaido
University).
Date:
Thursday, Jan. 11 16:55-17:25

18

30 minutes talks

10 minutes talks

10 minutes talks

Ken FURUKAWAThe University of Tokyo
Title:
Asymptotic stability of the three-dimensional Oseen vortex.
Abstract:
In this talk, we consider asymptotic stability of the three-dimensional Oseen
vortex in the vertically periodic Euclidean space. First, we will construct a weaksolution to the perturbed equation with logarithmic energy estimate. Next, we
will show the decay of the solution by using Fourier expansion in vertical variable.
The main term is the averaged term. We can show the decay of it by using same
argument in 2-dimensional case.
Date:
Wednesday, Jan. 10 17:35-18:15

Mathis GRIESTechnical University of Darmstadt
gries@mathematik.tu-darmstadt.de

Title:The hydrostatic Stokes semigroup and primitive equations
with Dirichlet boundary conditions on spaces of bounded functions
Abstract:
The primitive equations and its linearized form, called the hydrostatic Stokes equations, are presented in spaces with norms of the L∞ -type. The impact of Dirichletboundary conditions is discussed with an emphasis on semigroup estimates, which
are used to construct local solutions to the full problem.
Date:
Tuesday, Jan. 09 17:35-18:15

Mitsuo HIGAKI
Kyoto University
Title:
On fast rotating time-periodic flows in a two-dimensional cylindrical
domainAbstract:
We consider the two-dimensional Navier-Stokes equations in a domain boundedby two coaxial cylinders, where the outer cylinder is fixed and the inner cylinder
rotates fast. It is classical that there is an exact stationary solution for any rotating
10 minutes talks

19

10 minutes talks

speed, which is called the Couette flow. The main aim of this talk is to constructa time-periodic solution around the Couette flow by imposing a suitable external
force. Moreover, in the fast rotation case we describe a boundary layer structure of
the flow near the two boundaries. These results have been obtained in a joint workwith Prof. Isabelle Gallagher (Universite Paris-Diderot and DMA, Ecole Normale
Superieure) and Prof. Yasunori Maekawa (Kyoto University).
Date:
Friday, Jan. 12 16:15-16:55

Naoto KAJIWARA
The University of Tokyokajiwara@ms.u-tokyo.ac.jp

Title:
Well-posedness for the phase-field Navier-Stokes equationsin the maximal regularity space
Abstract:
We study the phase-field Navier-Stokes equations in the maximal regularity space.
This equation models the dynamics of vesicle membranes in incompressible viscous
fluids (e.g. red blood cells in the body). The description of the membrane is given
the terms of a phase field function φ, i.e. φ ≈ +1 inside the vesicle membrane,φ ≈ −1 outside and the thin transition layer of width is characterized by a small
parameter ε. We transform the equations into a quasi-linear parabolic evolution
equation and use the maximal Lp regularity theory to prove local well-posedness
and global well-posedness if the initial data is close to the variational strict stable
solution.
Date:
Tuesday, Jan. 09 17:35-18:15

Klaus KRESSTechnical University of Darmstadt
kkress@mathematik.tu-darmstadt.de

Title:
Strong time-periodic solutions to the bidomain equations subject toarbitrary large forces
Abstract:
We consider the bidomain equations which are describing the electrical propertiesof the heart. First, we show the existence of a weak time-periodic solution to thebidomain system subject to arbitrary large forces. Then, we use a result on global
existence of the initial value problem and a weak-strong uniqueness argument to20

10 minutes talks

10 minutes talks

obtain the existence of a strong time-periodic solution.
This talk is based on a joint work with Prof. Yoshikazu Giga and Naoto Kajiwara(The University of Tokyo).
Date:
Tuesday, Jan. 09 17:35-18:15

Hana MIZEROVÁCzech Academy of Sciences
mizerova@math.cas.cz

Title:
Convergence of entropy stable numerical schemes
for the Euler equations via dissipative measure-valued solutions

Abstract:
We study convergence of entropy stable numerical schemes for the compressible
Euler equations. We establish a suitable consistency formulation and show that afamily of numerical solutions generates Young measure which represents a dissipative measure-valued solution of the limit system. The weak-strong uniquenessprinciple implies that the numerical solution converges strongly to the strong solution of the limit system as long as the latter exists.
The content of this talk results from the collaboration with Eduard Feireisl (Czech
Academy of Sciences) and Mária Lukáčová (University of Mainz).
Date:
Friday, Jan. 12 16:15-16:55

Kohei NAKAO
Shinshu University
Title:
Beale-Kato-Majda type extension criterion of smooth solutions to theNavier-Stokes equations
Abstract:
We shall establish Beale-Kato-Majda type extension criterion of smooth solutions
to the Navier-Stokes equations. It is known that if a smooth solution u to the∫T
Navier-Stokes equations on (0, T ) satisfies 0 ∥rot u(τ )∥L∞ dτ < ∞, then u can becontinued to the smooth solution on (0, T ′ ) for some T ′ > T . In this talk, we shallslightly relax this condition for extension of smooth solutions to the Navier-Stokes
equations.
This is a joint work with Prof. Yasushi Taniuchi (Shinshu university).
Date:
Thursday, Jan. 11 17:35-18:15
10 minutes talks

21

10 minutes talks

Andreas SCHMIDTTechnical University of Darmstadt
Title:
Strong solutions of the Navier-Stokes equations with the Coulomb boundary condition
Abstract:
We will consider the Navier-Stokes equations with the Coulomb friction law
boundary condition. This condition means that the solution can slip at the boundary once the tangential stress exceeds a given threshold. If that is the case, thetangential component of the solution has the opposite direction of the tangential
stress. We will discuss the existence of strong solutions u ∈ W21 (0, T ; L2 (Ω)) ∩L2 (0, T ; H 2 (Ω)) in a bounded domain Ω.
Date:Thursday, Jan. 11 17:35-18:15

Anton SEYFERTTechnical University of Darmstadt
Title:The Helmholtz-Weyl Decomposition in Lp on Exterior Domains
Abstract:
Given a vector field u ∈ Lp (Ω) defined on an arbitrary smooth three-dimensional
exterior domain Ω, we will decompose it into a harmonic vector field, a gradient
field and the curl of a vector field, each of them satisfying suitable boundary conditions. This extends a result of Kozono and Yanagisawa (2009), who consideredanalogous decompositions in the case of bounded domains, where the decompositions always turned out to be unique. In our case, this decomposition is uniquefor a wide range of desired boundary conditions and integration parameters p, butthere are also cases, where it lacks either uniqueness or existence. Our construction
is based on the solvability of the weak Neumann and Dirichlet problems, as wellas the solvability of a system of weak Poisson problems with vanishing tangential
or vanishing normal boundary condition.
Date:
Wednesday, Jan. 10 17:35-18:15

Suma’inna
Waseda University

Yoshihiro SHIBATA Waseda University
Title:
On some nonlinear problem for the thermoelastic plate equations
22

10 minutes talks

10 minutes talks

Abstract:
In this talk, we will discuss a non linear thermoelastic plate equation in generaldomain (Ω). The problem is formulated by the equation system:
utt + ∆2 u + ∆θ + b∆((∆u)3 ) = f1 , in Ω × (0, ∞),.
θt − ∆θ − ∆ut = f2 , in Ω × (0, ∞).

(1)with c ∈ R is a positive constant, subject to the initial conditionu|t=0 = u0 , in Ω,
ut |t=0 = u1 , in Ω,
θ|t=0 = θ0 ,

(2)

in Ω,

and the Dirichlet boundary conditionu|Γ = θ|Γ = ∂ν u|Γ = 0.

(3)

In equations (1), u = u(x, t) denotes a vertical displacement at time t at the pointx = (x1 , · · · , xn ) ∈ Ω while θ = θ(x, t) describes the temperature. The nonlinear
term appearing in (1) represents the nature of magnitoelastic material due to anonlinear dependence between the tension of deformation and stress.
In the previous research, we have proved the maximal Lp -Lq -regularity for the
linearized problem (1) - (3) (b = 0). By using this result, we shall prove the uniquesolutions of the equations (1) - (3) with the help of the fix point theorem.
Date:
Wednesday, Jan. 10 17:35-18:15

Yuka TERAMOTO
Kyushu University
Title:On the spectrum of linear artificial compressible system
Abstract:
We consider the stability of stationary solutions of the incompressible NavierStokes system and the corresponding artificial compressible system. The sets ofstationary solutions of both systems are the same and the incompressible system
is obtained from the artificial compressible one as the limit the artificial Mach
number ϵ to 0 which is a singular limit. It is proved that if a stationary solutionof the incompressible system is asymptotically stable and the velocity field of thestationary solution satisfies an energy-type stability criterion by variational methodwith admissible functions being only potential flow parts of velocity fields, then it
is also stable as a solution of the artificial compressible one for suﬃciently small ϵ.
The result can be applied to the Taylor problem.
10 minutes talks

23

10 minutes talks

Date:
Wednesday, Jan. 10 17:35-18:15

Hiroyuki TSURUMI
Waseda University
Title:
Well-posedness and ill-posedness of the stationary Navier-Stokes equations in Triebel-Lizorkin spaces
Abstract:
We consider the stationary Navier-Stokes equations in Rn for n ≥ 3. First,
we show existence and uniqueness of solutions in homogeneous Triebel-Lizorkin−1+ nspaces Ḟp.q p . We can show this well-posedness by the boundedness of the Riesz
transform, the para-product formula, and the embedding theorem in homogeneous
Triebel-Lizorkin spaces, using a method similar to that of Kaneko-Kozono-Shimizu
in homogeneous Besov spaces. On the other hand, it is also proved that external−3 norms are arbitrary small can produce solutions whose Ḟ −1forces whose Ḟ∞,q∞,q
norms are arbitrary large. Such norm inflation phenomena are shown by constructing the sequence of external forces, as similar to those of initial data proposed by
Bourgain-Pavlović in the non-stationary problem.
Date:
Thursday, Jan. 11 17:35-18:15

Keiichi WATANABE
Waseda University
Title:
Maximal regularity theorem of compressible-incompressible two-phaseflows with phase transitions
Abstract:
We consider the free boundary problem for compressible-incompressible twophase flows with phase transitions in a general domain. Two fluids are separated
by a sharp interface and a surface tension is taken into account. The NavierStokes-Korteweg equation is used for the compressible fluid and the Navier-Stokes
equation is used for the incompressible fluid. We seek the solutions in a neighbourhood of a constant state (u+ , u− , ρ+ , π− , h) = (0, 0, ρ∗+ , 0, 0), where u± arevelocity fields, ρ+ is a density, π− is a pressure field, h is a hight function, and ρ∗+
is a positive constant. In this talk, we show the maximal Lp -Lq regularity of solutions to the linearized equations with the help of Weis’s operator valued Fourier
multiplier theorem. This work extends the result obtained by Shibata [Funkcial.
Ekvac., 2016].
24

10 minutes talks

10 minutes talks

Date:
Tuesday, Jan. 09 17:35-18:15

David WEGMANN
Technical University of Darmstadt
Title:
Maximal Regularity for the Stokes Operator on non Cylindrical Timespace Domains: The Case of an Exterior Domain
Abstract:
For t ∈ [0, ∞) let Ω(t) denote an exterior domain with suﬃcient smooth boundary∪and let Q := t (Ω(t) × {t}). We consider the Stokes equations
ut − ∆u + ∇p = f in Q
div u = 0 in Q(1)u(0) = u0 on Ω(0)in the non cylindrical time-space domain Q with Dirichlet boundary data. Assuming that there exists a suﬃcient smooth diﬀeomorphism from Ω(0) to Ω(t) we
proved that the system (1) has maximal regularity. Note that this improves [2]since our result is globally in time.
The proof is based on ra esult of Giga, Giga and Sohr [1] and a perturbationargument for operators with bounded H ∞ -calculus.
This is a result of a collaboration with Prof. Dr. Reinhard Farwig and Prof. Dr.
Hideo Kozono.
References:[1] Mariko Giga, Yoshikazu Giga, and Hermann Sohr, Lp estimate for abstract
linear parabolic equations, Proc. Japan Acad. Ser. A Math. Sci. 67 (1991), no.
6, 197–202. MR1120516[2] Jürgen Saal, Maximal regularity for the Stokes system on noncylindrical spacetime domains, J. Math. Soc. Japan 58 (2006), no. 3, 617–641. MR2254403
Date:
Thursday, Jan. 11 17:35-18:15

Marc WRONA
Technical University of Darmstadt
Title:
Well-posedness of the Q-tensor model with nontrivial tumbling andaligning eﬀects

10 minutes talks

25

10 minutes talks

Abstract:
In this work on liquid crystals, we use the most comprehensive description ofnematics, namely the Q-tensor, proposed by P.G. de Gennes, which can - otherthan the Ericksen-Leslie model - describe line defects and biaxial configurations.
For a suﬃciently small ratio of tumbling and alignment eﬀects (denoted by ξ), theexistence of a unique strong solution is shown. This solution extends to a global
strong solution, provided the initial data are small enough. To this end we use
a quasilinear approach. More precisely, an abstract local well-posedness property
for a certain class of quasilinear operators and nonlinear terms as well as the
maximal regularity of a tri-diagonal diﬀerential operator plus a small perturbation
are proved. The results are due to a joint work with Matthias Hieber and Miho
Murata.
Date:
Friday, Jan. 12 16:15-16:55

Sebastian ZAIGLER
Technical University of Darmstadt
zaigler@mathematik.tu-darmstadt.de

Title:
Regularity structures for the primitive equations
Abstract:
The new technique of regularity structures introduced by Martin Hairer is amethod to find local solutions for subcritical SPDEs with rough data, which wereformerly inaccessible. Regularity structures are spaces of abstract “Taylor expansions”, in which a solution of a SPDE can be found by solving fixed point equations.
The sequence of these fixed points, which are renormalised solution of the SPDE,
converges to a solution of the SPDE. We will give a short introduction how tomodify them to solve the primitive equations with white noise.
Date:
Friday, Jan. 12 16:15-16:55

26

10 minutes talks