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 ■Wed, Mar. 08

Thu, Mar. 09

Fri, Mar. 10

9:20-9:30

Opening

9:30-10:40

Reinhard RACKE ①(Univ. of Konstanz)

9:30-10:40

RACKE ②

9:30-10:40

KAGEI ③

10:50-11:30

Mitsuo HIGAKI(Kyoto University)

10:50-12:00

KAGEI ②

10:50-12:00

RACKE ③

11:40-12:20

Yutaka TERASAWA(Nagoya Univ.)

12:15-12:55

Anupam
PAL CHOUDHURY(TU Darmstadt)

12:15-12:55

Hideyuki MIURA(Tokyo Inst. Tech.)

Lunch break

Lunch BreakLunch Break

14:20-15:00

Patrick TOLKSDORF(TU Darmstadt)

14:40-15:20

Amru HUSSEIN(TU Darmstadt)

14:50-15:30

Giorgio
METAFUNE(Univ. of Salento)

15:30-16:40

Yoshiyuki KAGEI ①(Kyushu Univ.)

15:50-16:30

Nobu KISHIMOTO(Kyoto University)

16:00-17:10

Taiga KUMAGAI(Waseda Univ.)

17:00-18:00

Shota ENOMOTO(Kyushu Univ.)

16:50-18:10

Marc WRONA(TU Darmstadt)

60 min.

80 min.

70 min.

Anton SEYFERT(TU Darmstadt)

SebastianZAIGLER(TU Darmstadt)

Suma'inna(Waseda Univ.)

David WEGMANN(TU Darmstadt)

Mathis GRIES(TU Darmstadt)

Klaus KRESS(TU Darmstadt)

Andreas SCHMIDT(TU Darmstadt)

Shinya UCHIUMI(Waseda Univ.)

Naoto KAJIWARA(Univ. of Tokyo)

Go TAKAHASHI(Waseda Univ.)

Tobias TOLLE(TU Darmstadt)
TomoyaKEMMOCHI(Univ. of Tokyo)●→Main-course●→40min talkClosing

17:15-17:20

18:00-

Closing
Recaption@Takeuchi Lounge●→12min talkYoshiyuki KAGEI

Main-Course

Main-Course

Yoshiyuki KAGEI
Kyushu University, Fukuoka, Japan
Title:
Stability and bifurcation analysis of the compressible Navier-Stokes equations
Abstract:
In this mini-course I will talk about the instability of the plane Poiseuille flow and thebifurcation of traveling wave solutions.

1

The perturbation equations

We consider the compressible Navier-Stokes equations in an two-dimensional infinite
layer Ω
∂t ρ + div (ρv) = 0,(1)
ρ(∂t v + v · ∇v) − ν∆v − (ν + ν ′ )∇div v + ∇P (ρ) = νρe1 .

(2)Here the equations are written in a non-dimensional form;Ω = {x = (x1 , x2 ) : x1 ∈ R, 0 < x2 < 1};ρ = ρ(x, t) and v = ⊤ (v 1 (x, t), v 2 (x, t)) are the density and the velocity, respectively;
P = P (ρ) is the pressure which is a smooth function of ρ satisfying
P ′ (1) > 0;
ν and ν ′ are the non-dimensional parameters that satisfy
ν > 0 and ν + ν ′ ≥ 0;and e1 = ⊤ (1, 0) ∈ R2 .
The system (1)–(2) is considered under the boundary conditionv|x2 =0,1 = 0.

(3)

One can see that (1)–(3) has a stationary solution us = ⊤ (ϕs , vs ), called the Poiseuilleflow:
1
ϕs = 1, vs = x2 (ℓ − x2 )e1 .
2
We consider the stability of the Poiseuille flow us under the perturbation u = (ϕ, w) =(γ 2 (ρ − 1), v − vs ) which is periodic in x1 . The perturbation u is governed by the systemof equations(4)∂t ϕ + vs1 ∂x1 ϕ + γ 2 div w = f 0 (ϕ, w),
ν∂t w − ν∆w − ν̃∇div w + ∇ϕ − 2 ϕe1 + vs1 ∂x1 w + (∂x2 vs1 )w2 e1 = f (ϕ, w),(5)γ
2Main-Course

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Yoshiyuki KAGEI

where f 0 (ϕ, w), f (ϕ, w) are the nonlinearities; γ =system is considered under the boundary conditions:w|x2 =0,1 = 0,

ϕ, w:

√

P ′ (1); and ν̃ = ν + ν ′ . This2π
α -periodic in x1 ,(6)

Here α is a given positive number (wave number). The Reynolds number R and theMach number M are given by R = 1/(16ν) and M = 8/γ, respectively.

2

Instability of the Poiseuille flow

We set Ωα = [− απ , απ ) × (0, 1). We define the linearized operator L on X = L2∗ (Ωα ) ×L2 (Ωα ) defined by{}
1
D(L) = u = ⊤ (ϕ, w) ∈ X : w ∈ H0,per(Ωα ), Lu ∈ L2 (Ωα ) ,
L=vs1 ∂x1γ 2 div





+

∇

0

0

− γν2 e1vs1 ∂x1 + (∂x2 vs1 )e1 ⊤ e2.

−ν∆ − ν̃∇div∫
Here L2∗ (Ωα ) = {ϕ ∈ L2 (Ωα ) : Ωα ϕ(x) dx = 0} and e2 = ⊤ (0, 1) ∈ R2 .
We have the following instability result.

Theorem 1 ([2]) There are positive numbers r0 and η0 such that if α ≤ r0 , then{}σ(−L) ∩ λ ∈ C : |λ| ≤ η0 = {λmα (ν) : |m| = 1, · · · , k}for some k ∈ N, where λmα (ν) is a simple eigenvalue of −L and satisfiesiλmα (ν) = − (αm) + κ0 (αm)2 + O(|αm|3 )6
as |αm| → 0. Here κ0 is the constant given by
(
)
1
1
ν2
ν ν̃
2
κ0 =
−γ −
−
.
12ν 28015γ 2 30γ 2( 1)
1Therefore, if γ 2 < 280
and 2ν 2 + ν ν̃ < 30γ 2 280− γ 2 , then κ0 > 0, and so the
Poiseuille flow us = ⊤ (ϕs , vs ) is unstable.

3

Bifurcating traveling waves

We next consider the bifurcation of the Poiseuille flow. We fix γ in such a way that12280 −γ > 0, and regard ν as a bifurcation parameter. We denote the linearized operator
L by Lν . If ν is suitably large, then κ0 satisfies κ0 < 0. When ν decreases, κ0 satisfies
κ0 = 0 at a certain value of ν, and below such a value of ν, κ0 satisfies κ0 > 0. It thenfollows that for each 0 < α ≪ 1 there exists a positive number ν0 such thatRe λ±α (ν0 ) = 0 ;Re λ±α (ν) < 0 ⇔ ν > ν0 ;Main-Course

Re λ±α (ν) > 0 ⇔ ν < ν0 .
3

Yoshiyuki KAGEI

Main-Course

We assume that the spectrum of −Lν0 satisfies the following condition.
σ(−Lν0 ) ∩ {λ; Re λ = 0} = {λα (ν0 ), λ−α (ν0 )}.

(7)

Under this assumption we will prove the existence of bifurcating traveling wave solutions.
Theorem 2 ([3]) Assume that (7) holds true. Then there exists a solution branch{ν, u} = {νε , uε } (|ε| ≪ 1) satisfying
νε = ν0 + O(ε),
uε = uε (x1 − cε t, x2 ),uε (x1 + 2π
α , x2 ) = uε (x1 , x2 ),


1

√
uε (x1 , x2 ) = ε  2γ1 2 (−x22 + x2 ) 22 cos αx1 (1 + O(α)) + O(ε2 ),0cε = 61 + O(ε).
Remark 3 Iooss–Padula ([1]) showed that for each ν there exists a positive number
Λ such that the set σ(−Lν ) ∩ {λ; Re λ ≥ −Λ} consists of finite number of eigenvalues
with finite multiplicities. Therefore, it seems very unlikely that (7) does not hold for
all 0 < α ≪ 1.
References:[1] G. Iooss and M. Padula, Structure of the linearized problem for compressible parallel fluid flows, Ann. Univ. Ferrara, Sez. VII, 43 (1998), pp. 157–171.
[2] Y. Kagei and T. Nishida, Instability of plane Poiseuille flow in viscous compressible
gas, J. Math. Fluid Mech., 17 (2015), pp. 129–143.
[3] Y. Kagei and T. Nishida, T raveling waves bifurcating from plane Poiseuille flowof the compressible Navier-Stokes equation, preprint, 2015，MI Preprint Series
2015-9, Kyushu University.
Date:⃝
1 Wednesday, Mar. 08 15:30-16:40⃝
3 Friday, Mar. 10 09:30-10:40

4⃝
2 Thursday, Mar. 09 10:50-12:00

Main-Course

Main-Course

Reinhard RACKE

Reinhard RACKE
University of Konstanz, Konstanz
Title:
Hyperbolic Navier-Stokes Equations
Abstract:
The classical incompressible Navier-Stokes equations in the whole space Rn , n = 2, 3,
ut − µ∆u + (u · ∇)u + ∇p = 0

in (0, ∞) × Rn ,(0.1)

div u = 0in (0, ∞) × Rn ,(0.2)

u(0, ·) = u0

in Rn ,

(0.3)

with µ > 0 being the viscosity, for the velocity vector u = u(t, x) : (0, ∞) × Rn → Rn of
a fluid, and p = p(t, x) : (0, ∞) × Rn → R the related pressure, arise from the transportlaw
ut + (u · ∇)u + ∇p = div S(0.4)

and the constitutive law for the tensor S,S=µ(∇u + (∇u)′ ),
2

(0.5)

together with the incompressibility (zero divergence) condition (0.2) and initial conditions (0.3). We replace the Fourier type relation (0.5) by the Cattaneo type relation
τ St + S =µ(∇u + (∇u)′ ).
2

(0.6)

The Fourier type constitutive assumption (0.5) — in addition to the pressure contribution — leads to the well-known parabolic-type classical Navier-Stokes system (0.1)–(0.3); there, in particular, we have the eﬀect of an infinite propagation speed of signals,
as it is well-known as modeling problem/paradox for heat equations, or, more generally,for flux type equations (diﬀusion problems, . . . ) where the flux relation is given by the
Fourier type. There are applications, however, where it is more reasonable to work withhyperbolic models.
Diﬀerentiating the transport equation (0.4) with respect to t, and using the new
relation (0.6), we obtain the new hyperbolicly perturbed Navier-Stokes systemτ utt − µ∆u + ut + ∇p + τ ∇pt = −(u · ∇)u − (τ ut · ∇)u − (τ u · ∇)ut

(0.7)

in (0, ∞) × Rn ,div u = 0u(0, ·) = u0 ,

ut (0, ·) = u1

Main-Coursein (0, ∞) × Rn ,(0.8)

in Rn .

(0.9)
5

Reinhard RACKE

Main-Course

It will turn out that, in this system, at least the vorticity ∇ × u has finite propagation
speed.
There is not only the hyperbolic character of a wave equation for u, that complicatesthings by less regularization properties, but we notice the nonlinearities which are — incontrast to the classical case — of highest order, see the term (τ u · ∇)ut . The equation(0.7) can be regarded as a damped wave equation only for small values of u, since theterm (τ ut · ∇)u might disturb the positive damping term ut for large u. — This is ahint to think about a possible blow-up situation for large data.
Therefore, we will combine and apply techniques known for nonlinear heat equations,where additional diﬃculties will arise through the Helmholtz projections. The result
will be first a – more general – local existence theorem, and then a global existence
theorem for small data. This can be extended to an almost global existence theorem
for large data relating the admissible size of the data to the size of the relaxation
parameter τ .
The compressible Navier-Stokes equations with heat conduction in [0, ∞) × Rn , forn ≥ 1, can be written in the following form∂t ρ + div (ρu) = 0,∂t (ρu) + div (ρu ⊗ u) + ∇p = div S,
1 2
1∂t (ρ(e + u )) + div (ρu(e + u2 ) + up) + div q = div (uS),22(0.10)(0.11)(0.12)

where ρ, u = (u1 , u2 , · · · , un ), p, S, e and q represent fluid density, velocity, pressure, stress tensor, specific internal energy per unit mass and heat flux, respectively.
The equations (0.10), (0.11) and (0.12) are the consequence of conservation of mass,
momentum and energy, respectively.
To complete the system, we need to impose constitutive assumptions on p, S, e andq. First, we assume the fluid to be a Newtonian fluid, that is,
S = µ(∇u + (∇u)T ) + µ′ ∇div u,(0.13)

where µ and µ′ are the coeﬃcient of viscosity and the second coeﬃcient of viscosity,
respectively, satisfyingµ > 0, µ′ +2
µ ≥ 0.
n

The heat flux q is assumed to satisfy
τ ∂t q + q + κ∇θ = 0,(0.14)

which represents Cattaneo’s law (Maxwell’s law, . . . ) and gives rise to heat waves withfinite propagation speed. Here, τ > 0 is the constant relaxation time and κ > 0 is the6Main-Course

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Reinhard RACKE

constant heat conductivity. For small relaxation parameter τ a global well-posedness
result is obtained for small data.
Instead of relaxing in the heat equation, one may think of relaxing the constitutive law
for the stress tensor. Maxwell’s relaxation replaces (0.13) by the diﬀerential equationτ ∂t S + S = µ(∇u + ∇uT −2
divu In ) + λdivu In ,n(0.15)

where λ := µ′ + n2 µ. A splitting of the tensor in the form
S = S1 + S2 In(0.16)and studying∂t ρ + div(ρu) = 0,(0.17)∂t (ρu) + div(ρu ⊗ u) + ∇p = div(S1 ) + ∇S2 ,(0.18)1 2
1 2∂t (ρ(e + u )) + div(ρu(e + u ) + up) − κ△θ = div(u(S1 + S2 In )), (0.19)
22
τ1 ∂t S1 + S1 = µ(∇u + ∇uT −2
divuIn ),nτ2 ∂t S2 + S2 = λdivu,(0.20)(0.21)

is successful in leading to a global well-posedness result for small data.
The plan of the lectures is as follows:
Lecture I:
Survey and motivation for relaxation. Local well-posedness for hyperbolic incompressible Navier-Stokes equations [3].
Lecture II:
Proof of local existence. Global well-posedness for small data [3, 4].
Lecture III:
Almost global existence for large data [5]. Compressible Navier-Stokes equations withhyperbolic heat or with revised Maxwell’s law [1, 2].
References:[1] Hu, Y., Racke, R.: Compressible Navier-Stokes equations with hyperbolic heat
conduction. J. Hyperbolic Diﬀerential Equations. 13 (2016). 233–247.
[2] Hu, Y., Racke, R.: Compressible Navier-Stokes equations with revised Maxwell’slaw. J. Math. Fluid Mech. (2016) (accepted ).
Main-Course

7Reinhard RACKE

Main-Course[3] Racke, R., Saal, J.: Hyperbolic Navier-Stokes equations I: local well-posedness.
Evolution Equations Control Theory 1 (2012), 195–215.
[4] Racke, R., Saal, J.: Hyperbolic Navier-Stokes equations II: global existence ofsmall solutions. Evolution Equations Control Theory 1 (2012), 217–234.
[5] Schöwe, A.: Global strong solution for large data to the hyperbolic Navier-Stokes
equation. arXiv:1409.7797 (2014).
Date:⃝
1 Wednesday, Mar. 08 9:30-10:40⃝
2 Thursday, Mar. 09 09:30-10:40⃝
3 Friday, Mar. 10 10:50-12:00

8

Main-Course40 minutes talks

40 minutes talks

Mitsuo HIGAKI
Kyoto University, Kyoto
Department of Mathematics, Kyoto University, Japan.
mhigaki@math.kyoto-u.ac.jp

Title:
Navier wall law for nonstationary viscous incompressible flows
Abstract:
In fluid mechanics, it is a basic subject to understand the mathematical structureof flows near a solid wall with a rough surface. In the following we consider the
initial-boundary value problem of the Navier-Stokes system in the two-dimensionalrough-boundary domain Ωε = {x = (x1 , x2 ) ∈ R2 | εω( xε1 ) < x2 < ∞}.
∂t uε − ∆uε + uε · ∇uε + ∇pε = 0 , t > 0 , x ∈ Ωε ,





div uε = 0 , t ≥ 0 , x ∈ Ωε ,







uε (x1 , x2 ) is 2π-periodic in x1 , t ≥ 0 ,uε |t=0 = u0 , x ∈ Ωε .

The Dirichlet (no-slip) boundary condition is imposed on the rough boundary ∂Ωε .
uε = 0on ∂Ωε .

The unknown functions uε = uε (t, x) = (uε1 (t, x), uε2 (t, x))⊤ and pε = pε (t, x)
are respectively the velocity field and the pressure field of the fluid. The initial
data u0 is assumed to be given by the zero-extension of some velocity field aon the half-plane R2+ = {x ∈ R2 | x2 > 0}. The boundary function ω : R →(−1, − 12 ) is assumed to be smooth and 2π-periodic. The parameter ε = N1 , N ∈ N,
characterizes the amplitude and the pulse width (namely the “roughness”) of therough boundary ∂Ωε .
The Navier wall law in fluid mechanics is a typical approach to describe the eﬀectof such a rough boundary on the flows, which consists of the next two steps: (i)replacing the original boundary by an artificially flat one, and (ii) imposing a new
condition on the flat boundary, which reflects the averaged eﬀect of the roughness.
However, the derivation of the Navier wall law often relies on formal computations,and it is therefore important to justify the wall law with a mathematical rigor. The
justification of the Navier wall law is directly related to the higher order expansionof the rough domain flow uε in the parameter ε. In order to make such an expansion
rigorous, one naturally needs the high regularity of the flow appearing in the limit
as ε goes to zero. Although recently Mikelić, Nec̆asová, and Neuss-Radu (2013)
discusses the Navier wall law for the nonstationary flow with an external force,
40 minutes talks9

40 minutes talks

its argument is based on the assumptions that the initial data is zero, and thatthe external force is smooth and identically zero near t = 0. Thus the regularity
problem of the ε-zero limit flow is essentially avoided by these special assumptions.
In this talk we verify the Navier wall law for the initial data in C 1 class undernatural compatibility conditions. Our proof relies on the boundary layer analysis
and the L∞ theory of the Navier-Stokes system in the half plane.
Date:
Wednesday, Mar. 08 10:50-11:30

Amru HUSSEIN
Technical University of Darmstadt, Darmstadt

Title:
Strong Lp Well-Posedness of the 3D Primitive Equations
Abstract:
Primitive equations are considered to be a fundamental model for geophysicalflows. Here, the Lp theory for the primitive equations is developed. In this study,
the linearized Stokes type problem plays a prominent role, and it turns out that it
can be treated eﬃciently using perturbation methods for H ∞ -calculus.
Applying maximal Lp regularity and proving H 2 a priori bounds one proves that
this set of equations is globally strongly well-posed for arbitrary large initial datalying in a trace space which is a subspaces of H 2/p,p , 1 < p < ∞. Thus, the general
Lp setting admits rougher data than the usual L2 theory with initial data in H 1 .
Date:Thursday, Mar. 09 14:40-15:20

Nobu KISHIMOTO
Kyoto University, Kyoto

Title:
Remark on global regularity for the rotating Navier-Stokes equations
in a periodic domain
Abstract:
We consider the rotating incompressible Navier-Stokes equations on a threedimensional torus. In the periodic setting, Babin, Mahalov and Nicolaenko (’97,’99) constructed global smooth solutions for large Coriolis parameters (i.e., highspeed rotation) for periodic domains of arbitrary aspect ratios. Their proof wasbased on analysis of the limit equation, in which nonlinear interactions are restricted onto the resonant frequencies. They pointed out that the regularity properties of solutions to the limit equation depend discontinuously on the aspect ratios. In particular, for almost all aspect ratios they obtained a priori estimate
10

40 minutes talks

40 minutes talksof Sobolev norms of solutions independent of the viscosity coeﬃcient (and they
could even show long time existence for inviscid flow), while in the worst case theestimate they obtained depends exponentially on the inverse of the viscosity coefficient. Note that the regular torus (i.e., the same period in all three directions)or torus with rational aspect ratios is included in the worst case.
In this talk, we focus on the case of regular (or rational) torus and give an
improved estimate on the resonant interactions. The key ingredient in our proof
is to count the total number of nontrivial resonant frequencies via combinatorialargument using the divisor bound. Such an argument is a standard tool in the
study of periodic nonlinear dispersive equations, while it seems less developed in
the context of equations of rotating fluids in a periodic domain due to complicateddispersion relations. As an application, we show global regularity for the rotating
Navier-Stokes equations with fractional Laplacian. This talk is based on a joint
work with Tsuyoshi Yoneda (The University of Tokyo).
Date:Thursday, Mar. 09 15:50-16:30

Giorgio METAFUNE
University of Salento, Salento
Department of di Mathematics and Physics “Ennio De Giorgi”,
Università del Salento, Lecce, Italy.
giorgio.metafune@unisalento.it

Title:
Sharp heat kernel bounds for a class of parabolic operators with singularcoeﬃcients
Abstract:
We study parabolic problems associated to the second order elliptic operator in
RN
N
∑
xi xj
x
L = ∆ + (a − 1)
Dij + c 2 · ∇ − b|x|−2
2
|x|
|x|
i,j=1

with a > 0 and b, c real coeﬃcients.
Note that the second order coeﬃcients are discontinuous when a ̸= 1 and singularities appear in the drift and potential terms. The choice a = 1, c = 0 yields the
Schrödinger operator with inverse square potential. The conditionb
D := +a(N −1+c−a2a)2≥0

is necessary and suﬃcient for the existence of a realization of L generating a positive
semigroup and reduces to the classical one in the case of Schrödinger operators.
40 minutes talks11

40 minutes talks
The operator L becomes self-adjoint in a suitable weighted L2 -space which weuse as a tool for construncting the generated semigroup. However, generation in
the unweighted Lp -spaces is also characterized. Letting
N −1+c−a √N −1+c−a √s1 :=− D, s2 :=
+ D
2a
2a
it turns out that there exists a realizazion Lp,int between the minimal and the
maximal operator that generates a semigroup in Lp (RN ) if and only if s1 < N/p <s2 + 2.
We describe the domain and show that the generated semigroup is bounded
analytic of angle π/2 and positive for t > 0. As a consequence, the spectrum of
Lp,int coincides with the half-line (−∞, 0].
We prove that the semigroup is represented by a kernel p(t, x, y) which satisfies
the double side estimates( )− γ [()()]− N +1+√D
)
(
2
2
N
|x|
|x|
|y|
c|x − y|2−2p(z, x, y) ≈ Ct
exp −
1 ∧ 1
1 ∧ 1
|y|
t
t2
t2
where γ = (N − 1 + c)/a − N + 1 and the constants c, C may diﬀer in the upperand lower bounds. Note that γ = 0 if and only if L is self-adjoint.
Integrating the above kernel estimates with respect to t we also obtain precise
kernel bounds of the Green function.
Finally we remark that kernel estimates for |x|α L can be obtained from the results
above via a change of variablels. However, this tranformation does not relate thekernel of |x|α ∆ to that of the Laplacian but rather to the kernel of a suitable L as
above, where discontinuities necessarily appear.
Most of the content of these lecture is based on joint works with Chiara Spina,
Luigi Negro (University of Salento) and Motohiro Sobajima (Tokyo University of
Science).
Date:
Friday, Mar. 10 14:50-15:30

Hideyuki MIURA
Tokyo Institute of Technology, Tokyo

Title:On uniqueness for the harmonic map heat flow in supercritical dimensions
Abstract:
We examine the question of uniqueness for the equivariant reduction of theharmonic map heat flow in the energy supercritical dimension. It is shown that,generically, singular data can give rise to two distinct solutions which are both
stable, and satisfy the local energy inequality. We also discuss how uniqueness canbe retrieved.
12

40 minutes talks

40 minutes talks

Date:
Friday, Mar. 10 12:15-12:55

Anupam PAL CHOUDHURY
Technical University of Darmstadt, Darmstadt

anupampcmath@gmail.com

Title:On the nematic liquid crystal flows
Abstract:
In this talk, we shall briefly review some existing results on the nematic liquid
crystal flows in a bounded domain and discuss some ongoing works in the case of
Dirichlet boundary conditions for the director field. We shall be mainly focusing
on the Lp − Lq maximal regularity approach.
Date:
Thursday, Mar. 09 12:15-12:55

Yutaka TERASAWA
Nagoya University, Nagoya
Graduate School of Mathematics, Nagoya University,
Furocho Chikusaku Nagoya 464-8602, Japan
yutaka@math.nagoya-u.ac.jp

Title:
Finite energy for Navier-Stokes equations and Liouville-type theorems
in two dimensional domains
Abstract:
This talk is based on a joint work with Professor Hideo Kozono (Waseda University) and Professor Yuta Wakasugi (Ehime University).
We consider the Cauchy problem for the Navier-Stokes equations
n
 vt − ∆v + (v · ∇)v + ∇p = 0, (x, t) ∈ R × (0, T ),(1)
div v = 0,(x, t) ∈ Rn × (0, T ),
 v(x, 0) = v (x),
n
x∈R .
0
Here v = v(x, t) = (v1 (x, t), . . . , vn (x, t)) and p = p(x, t) denote the velocity andthe pressure, respectively. Also, v0 (x) = (v01 (x), . . . , v0n (x)) stands for the given
initial velocity.
Serrin, Shinbrot and Taniuchi proved the conditions for the energy identity of
weak solutions of (1). Here we give another condition which ensures the energy
identity of a smooth solution and as its application, we also give a Liouville-type
result. Our first main result is the following.
40 minutes talks13

40 minutes talks
Theorem 1 Let n ≥ 2, v0 ∈ L2σ (Rn ) and let (v, p) be a smooth solution of (1).
Assume that there exist q1 , q2 , r1 , r2 satisfying
(
)
3n
3n
, 3 ≤ r1 ≤ ∞ and (q1 , r1 ) ̸=
, ∞ , (2)3 ≤ q1 ≤
n−1
n−1

(
)
(q2 , r2 ) ̸= 2n , ∞(n ≥ 3),2nn−2
2 ≤ q2 ≤, 2 ≤ r2 ≤ ∞ and(3)q ̸= ∞
n−2
(n = 2)
2

such that v ∈ L3 (0, T ; Lq1 ,r1 (Rn )) ∩ L2 (0, T ; Lq2 ,r2 (Rn )). Assume also that thepressure p satisfies p ∈ L∞ (Rn × (0, T ). Then, we have thatv ∈ L∞ (0, T ; L2σ (Rn )) ∩ L2 (0, T ; Ḣσ1 (Rn ))and that∥v(t)∥2L2 + 2∫ t
0∥∇v(τ )∥2L2 dτ = ∥v0 ∥2L2

for all t ∈ (0, T ).
An immediate consequence of this theorem is the following Liouville-type theorem.
Corollary 2 Let n ≥ 2, and let v0 ≡ 0 in Rn . Suppose that v is a smooth solutionof (1) with its associted pressure p. If v ∈ L3 (0, T ; Lq1 ,r1 (Rn ))∩L2 (0, T ; Lq2 ,r2 (Rn ))for such (q1 , r1 ) and (q2 , r2 ) as in (2) and (3), respectively, and if p ∈ L∞ (Rn ×(0, T )), then it holds that v(x, t) ≡ 0 on Rn × (0, T ).
We also treat the marginal case of (2) and (3) and get the condition for finite
energy of solutions and get the corresponding Liouville-type theorem.
We next handle the equation for vorticity in 2D unbounded domains. Under
some asymptotic behavior at infinity and some conditions at the boundary, we
prove that the vorticity are square integrable.
Date:
Wednesday, Mar. 08 11:40-12:20

Patrick TOLKSDORF
Technical University of Darmstadt, Darmstadt
tolksdorf@mathematik.tu-darmstadt.de

Title:On the Stokes and Navier-Stokes equations in bounded Lipschitz domains
Abstract:
To perform certain iteration or fixed point methods to prove the existence ofsolutions to the Navier-Stokes equations in Lp -spaces, properties like maximal Lq regularity of the Stokes operator, Lp -Lq -estimates of the Stokes semigroup, and14

40 minutes talks

40 minutes talks

gradient estimates of the Stokes semigroup are needed. In this talk, I will give an
overview of how to obtain these properties if the underlying domain is a bounded
Lipschitz domain and I will present the key steps of their proofs.
Date:
Wednesday, Mar. 08 14:20-15:00

40 minutes talks

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12 minutes talks

Shota ENOMOTO
Kyushu University, Fukuoka
s-enomoto@math.kyushu-u.ac.jp

Title:
Asymptotic behavior of solutions to the compressible Navier-Stokes
equation around space-time periodic states
Abstract:
We consider large time behavior of solutions to the compressible Navier-Stokes
equation around space-time periodic states in an infinite layer of Rn (n = 2, 3)
under the action of a space-time periodic external force. If the external force is
suﬃciently small, then the compressible Navier-Stokes system has a space-time
periodic solution. We show that the space-time periodic states is asymptoticallystable if the initial perturbation is suﬃciently small. Furthermore, it is shownthat the asymptotic leading part is given by a product of a solution of the onedimensional viscous Burgers equation and a space-time periodic function whenn = 2, and by a product of a solution of the two-dimensional heat equation and aspace-time periodic function when n = 3.
This talk is based on a joint work with Professor Yoshiyuki Kagei (Kyushu
University) and Mr. Mohamad Nor Azlan.
Date:
Wednesday, Mar. 08 17:00-18:00

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

Title:
A maximum regularity approach to the free boundary value problem
for the primitive equations of the ocean
Abstract:
The free boundary value problem for the primitive equations of the ocean ispresented and the equations are then transformed to yield a set equations on a
fixed domain. We solve the linearised problem via the method of maximal Lp −regularity for parabolic problems and then introduce a fixed-point argument for
the quasilinear problem.
Date:
Thursday, Mar. 09 16:50-18:10

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12 minutes talks

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

Title:
Strong time periodic solution for semilinear parabolic equation on a realinterpolation space
Abstract:
We study the time periodic solution for semilinear parabolic equations. By Da
Prato and Grisvard theorem, a generator of analytic semigroups on X has the maximal regularity property in a real interpolation space DA (θ, p). In the previous talkby Kress, he constructs the linear theory for time periodic solution in this real interpolation space. We prove existence and uniqueness of the time-periodic solution
in the maximal regularity class Lq (0, T ; DA (θ, p)) for the semilinear problem withpolynomial nonlinearities from the linear theory and the fixed-point argument under a condition on p, q, θ.
This talk is based on a joint work with Prof. Matthias Hieber, Dr. Patrick
Tolksdorf and Mr. Klaus Kress (Technical University of Darmstadt). .
Date:
Friday, Mar. 10 16:00-17:10

Tomoya KEMMOCHIThe University of Tokyo, Tokyo
Title:
Error estimate for the finite element semi-discretization of the nonstationary hydrostatic Stokes equation
Abstract:
We consider the finite element method for the non-stationary hydrostatic Stokes
equation. The equation is the linearized problem of the primitive equation, which
describes the model for geophysical flows, such as the ocean and the atmosphere. In
this talk, we present the finite element semi-discretization. That is, we discretize
the space variables only. We provide an optimal error estimate for the velocity
by the semigroup approach. As in the case for the Stokes problems, the inf-sup
Date:condition plays a crucial role.
Thursday, Mar. 09 16:50-18:10

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12 minutes talks

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

Title:
Strong time periodic solutions for linear parabolic equations in a realinterpolation space
Abstract:
In this talk, we study time periodic solutions for linear parabolic equations. By
the Da Prato and Grisvard theorem, a generator of an analytic semigroup on X
has the maximal regularity property in a real interpolation space DA (θ, p). We
prove the existence and uniqueness of a time periodic solution in the maximal regularity class W 1,q (0, T ; DA (θ, p)) ∩ Lq (0, T ; DA (θ, p)) for a given T -periodic righthand side f : R → DA (θ, p) with f|(0,T ) ∈ Lp (0, T ; DA (θ, p). In the following talk,
Kajiwara extends the result to the semilinear problem with polynomial nonlinear
terms.
This talk is based on a joint work with Prof. Matthias Hieber, Dr. Patrick Tolksdorf (Technical University of Darmstadt) and Naoto Kajiwara (The University ofTokyo).
Date:
Friday, Mar. 10 16:00-17:10

Taiga KUMAGAI
Waseda University, Tokyo
Title:
A class of Hamilton-Jacobi equations and ode systems on graphs
Abstract:
We consider the problem
ε
 ε b · Du
+ |Duε | = fu −εuε = 0in Ω,
on ∂Ω,

where ε is a positive parameter, Ω is an open subset of R2 determined through a
Hamiltonian H, uε : Ω → R denotes the unknown, f : Ω → R is a given, continuous, and nonnegative function, and b : R2 → R2 is a Hamiltonian vector field.
We study the asymptotic behavior of solutions as ε goes to zero and showthat the limit of the solutions is described as solutions of a system of odes ona graph. Freidlin-Wentzel, Freidlin-Weber, Sowers, by probabilistic techniques,and Ishii-Souganidis, by pde techniques, studied stochastic perturbation problems
for Hamiltonian flows. Our problem can be seen as perturbation problems for
Hamiltonian flows by optimal control theoretic disturbance.
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12 minutes talks

Date:
Friday, Mar. 10 16:00-17:10

Andreas SCHMIDTTechnical University of Darmstadt, Darmstadt
Title:The Navier-Stokes equations with the Coulomb boundary condition
Abstract:
We will consider the Navier-Stokes equations on manifolds 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 thecase, the tangential component of the solution has the opposite direction of thetangential stress. We will formulate this system, that has already been studied in
the Euclidean case, on manifolds and obtain the existence of weak solutions. We
will also discuss the dependence of the solution on the threshold.
Date:
Wednesday, Mar. 08 17:00-18:00

Anton SEYFERTTechnical University of Darmstadt, Darmstadt
Title:The Poisson Equation in Exterior domains with Hodge Boundary Conditions
Abstract:
We consider a system of Poisson equations with Hodge boundary conditions (alsocalled perfectly conducting wall condition), which is a boundary condition of zerothand first order. Using an analogous method as in the case of the Stokes equations
with Dirichlet boundary conditions, we show the existence of unique solutions inexterior domains with gradients in Lorentz spaces. The analogous treatment ismainly possible due to the availability of Sobolev embeddings in Lebesgue andLorentz spaces on exterior domains despite the lack of zero boundary conditions.
Date:
Friday, Mar. 10 16:00-17:10

12 minutes talks

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12 minutes talks

Suma’inna
Waseda University, Tokyosuma.inna@fuji.waseda.jp / suma.inna@uinjkt.ac.id

Title:The Existence of R-bounded Solution Operators of the Thermoelastic
Plate Equation with Dirichlet Boundary Conditions
Abstract:
We consider the linearized thermoelastic plate equation with the Dirichlet boundary condition in general domain Ω, given by
utt + ∆2 u + ∆θ = f1

in (0, ∞) × Ω,θt − ∆θ − ∆ut = f2

in (0, ∞) × Ω,with the initial conditions u|t=0 = u0 , ut |t=0 = u1 , θ|t=0 = θ0 in Ω and theboundary condition u = θ = 0, ∂ν u = g on (0, ∞) × Γ. Here, u = u(x, t) denotes
a vertical displacement at time t and at the point x = (x1 , · · · , xn ) ∈ Ω, whileθ = θ(x, t) describes the temperature. This work extends the result obtained by
Naito and Shibata that study the problem in the half space case. We prove theexistence of R-bounded solution operators for the corresponding resolvent problem
on general domains. Then, the operator-valued Fourier multiplier theorem gives
the maximal Lp -Lq regularity of the above time-dependent problem. For thismatter, we use the ideas due to Denk and Shibata in the free boundary condition
case.
Date:
Friday, Mar. 10 16:00-17:10

Go TAKAHASHI
Waseda University, Tokyo
Title:
Partial regularity and extension of solutions to the Navier-Stokes equations
Abstract:
In this talk, we work on local-in-time classical solutions to the incompressible
Navier-Stokes equations. The objective of this talk is to establish a time-extension
criterion to the local-in-time solutions. There are quite a few time-extension criteria for local-in-time strong solutions, some of which were obtained by improving
upon some inequalities such as critical Sobolev inequalities. Now we take a slightly
diﬀerent viewpoint and focus on partial regularity of the solutions. First we discuss
partial regularity of solutions, which is an essential argument to obtain the local
boundedness of the solutions. Then we will establish a time-extension criterion to
the local-in-time solutions as an application of an ϵ-regularity theorem.
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12 minutes talks

Date:
Wednesday, Mar. 08 17:00-18:00

Tobias TOLLE
Technical University of Darmstadt, Darmstadt
tolle@mathematik.tu-darmstadt.de

Title:
Numerical modelling of jumps in two-phase flows in the context of onefield formulated Navier-Stokes equations
Abstract:
Methods for the numerical simulation of two-phase flows can be distinguished bytheir formulation of the Navier-Stokes equations. The first group uses two separate sets of equations which are coupled at the interface by jump conditions. The
other group employs a one-field formulation of the Navier-Stokes equations which
introduces additional terms acting at the interface. While the latter is superior in
handling large interfacial deformations, it is challenging to maintain the propertychanges at the interface as sharp as possible without deteriorating the numericalaccuracy.
This talk discusses several approaches on how to deal with interfacial discontinuities in the context of a hybrid Level Set / Front Tracking method on unstructured
meshes. For illustration, these approaches are tested within a hydrodynamic test
case. To provide some context, the results are compared to simulations performed
with a state-of-the-art volume of fluid solver.
This talk is based on joint work with Tomislav Maric and Prof. Dieter Bothe
from the Mathematical Modelling and Analysis group, TU Darmstadt.
Date:
Thursday, Mar. 09 16:50-18:10

Shinya UCHIUMI
Waseda University, Tokyo
Title:A Lagrange-Galerkin scheme for high-Reynolds-number flow problems
Abstract:
The Lagrange-Galerkin scheme is a powerful numerical method for flow problems.
Thanks to introduction of the locally linearized velocity, it is possible to implement
the scheme exactly without numerical quadrature [Tabata and Uchiumi, Math.
Comp., to appear]. There, convergence of the numerical solution has been proved
for widely used Taylor-Hood element. Here we present a variant of this scheme
which is more robust for high-Reynolds-number problems.
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Date:
Thursday, Mar. 09 16:50-18:10

David WEGMANN
Technical University of Darmstadt, Darmstadt
Title:
Existence of Strong Solutions and Decay of Turbulent Solutions of NavierStokes Flow with Nonzero Dirichlet Boundary Data
Abstract:
We consider the Navier-Stokes equations in a domain with compact boundaryand nonzero Dirichlet boundary data β. A solution is constructed as a sum of avery weak solution b to the instationary Stokes equations with nonzero boundary
data and a weak solution v to a system of Navier-Stokes type with zero boundary
data.
Assuming that β(t) → 0 as t → ∞, we proved in [2] that there exists a solutionv which fulfills ∥v(t)∥2 → 0 as t → ∞. Furthermore, in a bounded domain thesolution v tends exponentially to 0 if the corresponding data is exponentially decreasing.
As a last result, we calculated a lower polynomial bound for the decay rate if Ω
is unbounded. Therefore, we used a suitable spectral decomposition of the Stokes
operator as introduced in [1] and Duhamel’s formula.
Recently, we proved the same decay result for an arbitrary turbulent solution.
The main tool for the proof is to show the existence of a strong solution after sometime T > 0 and to identify the turbulent solution with the strong solution in thetime interval [T, ∞).
This is a result of a collaboration with Prof. Dr. Reinhard Farwig and Prof. Dr.
Hideo Kozono.
References:[1] Borchers, W., Miyakawa, T., “Algebraic L2 decay for Navier-Stokes flows inexterior domains. II,” Hiroshima Math. J., 3, 621–640 (1991).
[2] Farwig, R., Kozono, H., Wegmann, D.,“Decay of non-stationary Navier-Stokes
flow with nonzero Dirichlet boundary data,” Indiana Univ. Math. J., to appear(2015)
Date:
Wednesday, Mar. 08 17:00-18:00

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12 minutes talks

Marc WRONA
Technical University of Darmstadt, Darmstadt
Title:
Well-posedness of the primitive equations with only horizontal viscosity
for several classes of inital data
Abstract:
The focus of this talk are the primitive equations for planetary atmospheric andoceanic dynamics with only horizontal viscosity. Several classes of initial data
are investigated including W 1,p , p > 2, as well as the anisotripoc spaces H n ∩L2xy Hzk , k ≥ n. Global well-posedness for these classes is shown via a uniform apriori bound for systems with full viscosity. After establishing well-posedness forthese regularized systems, a system version of the classical Grönwall inequality anda logarithmic Sobolev inequality are used to obtain the desired a priori bound.
Date:
Thursday, Mar. 09 16:50-18:10

Sebastian ZAIGLER
Technical University of Darmstadt, 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:
Thursday, Mar. 09 16:50-18:10

12 minutes talks

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