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, Mar. 1
9:20
-9:30
9:30
-10:30
10:45-11:45

Wed, Mar. 2

Fri, Mar. 4

Thu, Mar. 3
9:30

Opening-10:30
10:45

Stefan ULBRICH①(TU Darmstadt)

ULBRICH ③

Ryo TAKADA ①(Tohoku Univ.)-11:45
12:00

ULBRICH ②ULBRICH ④-12:40

TAKADA ②

12:40
12:00

Jonas SAUER(TU Darmstadt)-12:40
12:40-15:10
15:40-16:20
16:30-16:45
16:45-17:00
17:00-17:15-14:10
14:10-15:10Lunch Break-14:10
14:10

Jun KATO(Nagoya Univ.)

15:40
Robert DENK ①(Univ. of Konstanz)
Amru HUSSEIN(TU Darmstadt)

Mathis GRIES(TU Darmstadt)

DENK ②-15:55DENK ③

15:55
Matthias
KOTSCHOTE(Univ. ofKonstanz)Andreas KLAIBER(Univ. ofKonstanz)-16:10
Yohei TSUTSUI(Shinshu Uni.)

16:10-16:25

Martin BOLKART(TU Darmstadt)

16:45-17:00
17:00

SebastianZAIGLER(TU Darmstadt)

Michael FISCHER(TU Darmstadt)

Florian STEINBERG(TU Darmstadt)

Shinya UCHIUMI(Waseda Univ.)

Go TAKAHASHI(Waseda Univ.)-17:15
17:15

David WEGMANN(TU Darmstadt)●→Main-course ●→40min talk

TAKADA ③

TAKADA ④
Tomasz PIASECKI(Polish Acad. ofSci.)
Lunch BreakDENK ④
Tobias SEITZ(TU Darmstadt)
Alexander
DALINGER(TU Darmstadt)
Dai NOBORIGUCHI(Waseda Univ.)
Anton SEYFERT(TU Darmstadt)
Shota ENOMOTO(Kyushu Univ.)-17:30

Naofumi MORI
(Kyushu Univ.)-17:40

Closing

18:00-

Closing Reception
at Cafeteria●→15min talk

Robert DENK

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Main-Course

Robert DENK
University of Konstanz, Konstanz, Germany
Title:
Maximal regularity for parabolic evolution equations
Abstract:
Lecture 1: Maximal regularity and R-boundedness
Maximal regularity is one of the standard approaches to investigate semilinear andquasilinear parabolic evolution equations. A prototype example is the graphical mean
curvature ﬂow given by∂t u − ∆u = −

n
∑
i,j=1

∂i u∂j u∂i ∂j u (t ∈ (0, T )),1 + |∇u|2

u(0) = u0 .

The main idea of maximal regularity is show that the linearized equation induces an isomorphism between suitably chosen function spaces and to reduce the nonlinear equation
to a ﬁxed-point equation in these function spaces.
In our lectures, we will consider maximal regularity in Lp -Sobolev spaces in timeand space with p ∈ (1, ∞). For the initial value u|t=0 or for boundary values, one has
to describe the trace spaces of Lp -Sobolev spaces. It turns out that these are Besov
spaces of non-integer order. To deal with the nonlinear terms in the equation, one canapply Sobolev embedding results for suﬃciently large p. If the linearized operator has
maximal regularity, the nonlinear equation has a unique solution at least for small timeintervals or for small data.
A closed operator A : X ⊃ D(A) → X acting in a Banach space has maximal Lp regularity if the Cauchy problem∂t u − Au = f(t ∈ (0, T )),u(0) = u0

has a unique solution u ∈ Hp1 ((0, T ); X) ∩ Lp ((0, T ); D(A)) depending continuously on
the data f and u0 . Taking Fourier (or Laplace) transform Ft in time, we see that thisis equivalent to the condition that
Ft−1 iτ (iτ − A)−1 Ft
deﬁnes a continuous operator in Lp (; X). In this case, the (operator-valued) symbolτ 7→ iτ (iτ − A)−1 is said to be an Lp -multiplier. A suﬃcient condition for symbols tobe an Lp -multiplier is given by a vector-valued version of Michlin’s theorem which isbased on the so-called R-boundedness of the symbol.
Lecture 2: Maximal regularity for boundary value problems
For (linearized) equations in the whole space n , R-boundedness of the resolvent andtherefore maximal Lp -regularity can be shown by vector-valued Michlin theorems. For

2

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Robert DENK

boundary value problems in domains, however, one has to study model problems in thehalf space n+ := {x ∈n : xn > 0} of the form
λu − Au = f
γ0 u = g

in n+ ,
on n−1 ,n =n−1 .
where γ0 u := u|n−1 stands for the trace of the function u on the boundary ∂+There is a standard method to solve such boundary value problems, using partial Fouriertransform F ′ in the tangential variables. This reduces the problem to an ordinary
diﬀerential equation in the half-line (0, ∞). Taking inverse Fourier transform, we candescribe the solution operator as a sum of the whole space resolvent and an operatorof the form
∫ ∞(Kϕ)(x′ , xn ) =(F ′ )−1 w(ξ ′ , xn + yn , λ)(F ′ ϕ)(ξ ′ , yn )dyn .
0Such operators are called singular Green operators. By results on the R-boundednessof integral operators, it can be shown that parabolic boundary value problems havemaximal Lp -regularity. For general boundary conditions, the Shapiro-Lopatinskii condition has to be satisﬁed. This condition can equivalently be written as the invertibilityof the so-called Lopatinskii matrix.
Lecture 3: Parabolic mixed-order systems
The analysis of the Stefan problem with Gibbs-Thomson correction leads to the
Lopatinskii matrix)(1−|ξ ′ |2′
(ξ ′ ∈n−1 , ℜλ ≥ 0).
L(ξ , λ) = √ ′ 2|ξ | + λλThis is an example of a mixed-order system (Douglis-Nirenberg system). For maximal
regularity results, one has to determine Sobolev type spaces in which the corresponding
operator L(D′ , ∂t ) induces an isomorphism.
A general approach to this is given by the Newton polygon method. Here one ﬁrst
considers the determinant of the matrix and describes the diﬀerent inhomogeneities(with respect to ξ ′ and λ) in a geometrical way. By this, one can deﬁne mixed-order(anisotropic) Sobolev spaces in time t and space x which are adapted to the equation
and which are the appropriate function spaces for the data and for the solution.
It is possible to deﬁne a generalization of classical parabolicity (so-called N-parabolicity)which is the essential condition for maximal Lp -regularity for mixed-order systems. As
applications, we mention Stefan problems and the thermoelastic plate equation.
Lecture 4: Maximal regularity for pseudodiﬀerential operators
Let X be a Banach space.

The symbol of an operator-valued symbol of classS µ (n ; L(X)) is a C ∞ -function a : n ×n → L(X) satisfying{}
sup ⟨ξ⟩|α|−µ ∥∂ξα a(x, ξ)∥L(X) : x ∈n , ξ ∈n < ∞Main-Course(α ∈n0 ).
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Robert DENK

Main-Course

For such a symbol, the corresponding pseudodiﬀerential operator [a] is given by∫−n/2([a]u)(x) = (2π)
a(x, ξ)(F u)(ξ)dξ.
n

If the operator norm in the deﬁnition of S µ is replaced by the R-bound, the resultingµ n( ; L(X)) yields R-bounded operator families. Therefore, parabolic
symbol class SR
pseudodiﬀerential operators in the whole space have maximal Lp -regularity for everyp ∈ (1, ∞).
The situation for pseudodiﬀerential (non-local) boundary value problems is more
complicated. It can be shown that singular Green operators are R-bounded, too, which
implies maximal Lp -regularity. As an example, we consider the Stokes equation in an
inﬁnite cylinder with no-slip boundary conditions. Here, the elimination of the pressure
leads to a non-local boundary value problem for the velocity.
Date:⃝
1 Tuesday, Mar. 1 14:10-15:10⃝
3 Thursday, Mar. 3 14:10-15:10

4⃝
2 Wednesday, Mar. 2 14:10-15:10⃝
4 Friday, Mar. 4 14:10-15:10

Main-Course

Main-Course

Ryo TAKADA

Ryo TAKADA
Tohoku University, Sendai
Mathematical Institute, Tohoku University
Sendai 980-8578, JAPAN

Title:
Dispersive estimates for rotating fluids and stably stratified fluids
Abstract:
The most important two features in the geophysical ﬂuid dynamics are rotation of
the ﬂuid and stable stratification, and it is known that these eﬀects exhibit dispersive
natures. In this mini-course, we shall present recent mathematical results on the dispersive estimates for the linear propagators associated with the rotation of the ﬂuid
and the stable stratiﬁcation.

1. Derivation of the equations
We consider the initial value problems for the rotating Navier-Stokes equations:
t > 0, x ∈ R3 ,∂t u + (u · ∇)u = ν∆u − 2Ωe3 × u − ∇p
(NSΩ )∇·u=0
t > 0, x ∈ R3 ,
u(0, x) = u0 (x)x ∈ R3 ,and the stably stratiﬁed Boussinesq equations:∂t u + (u · ∇)u = ν∆u − ∇p + θe3∂ θ + (u · ∇)θ = κ∆θ − N 2 u
t


∇·u=0




u(0, x) = u (x),0

3t > 0, x ∈ R3 ,t > 0, x ∈ R3 ,t > 0, x ∈ R3 ,

θ(0, x) = θ0 (x)(BN )x ∈ R3 .

Here, u = (u1 (t, x), u2 (t, x), u3 (t, x))T , p = p(t, x) and θ = θ(t, x) are the unknown functions, representing the velocity ﬁeld, the scalar pressure and the thermal disturbanceabout a mean state in hydrostatic balance, respectively, while u0 = (u0,1 (x), u0,2 (x), u0,3 (x))T
is the given initial velocity ﬁeld and θ0 = θ0 (x) is the given initial thermal disturbance.
Ω ∈ R \ {0} is the angular frequency of the background rotation, and N > 0 is theBrunt-Väisälä (buoyancy) frequency for the constant stratiﬁcation.
In the ﬁrst lecture, we shall derive the system (NSΩ ) from the original Navier-Stokes
equations with the boundary condition that u behaves like Ωe3 × x + O(1) as |x| →∞, and the system (BN ) from the original Boussinesq equations by considering the
perturbation about a mean state in hydrostatic balance. Moreover, we give the explicit
representations of the linear semigroups generated by the linear operators in (NSΩ ) and(BN ).

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5Ryo TAKADA

Main-Course

2. Dispersive estimates for linear propagators
In the second and third lectures, we shall consider the dispersive estimates (the L1 -L∞
temporal decay estimates) for the linear propagators associated with the Coriolis forceΩe3 × u in (NSΩ ) and the stable stratiﬁcation −N 2 u3 in (BN ):
∫
ξ3
D3
1
ix·ξ±iΩt |ξ|±iΩt |D|eef (x) =
fb(ξ) dξ,(2π)3 R3
∫
|ξh |
|Dh |
1
ix·ξ±iN t |ξ|±iN t |D|f (x) =efb(ξ) dξ,e(2π)3 R3√
respectively. Here ξh = (ξ1 , ξ2 ) and |ξh | = ξ12 + ξ22 . We give the decay estimates
±iΩt

D3

|D| , andfor the above linear propagators by applying the Littman theorem to e
by making use of the stability of the stationary phase estimates under the small C mperturbations with a large m ∈ N for eof our dispersive estimates.

|D |h±iN t |D|

. Also, we shall mention the sharpness

3. Global existence of solutions to (NSΩ ) and (BN )In the fourth lecture, we apply our dispersive estimates to the study of the existence
and uniqueness of global in time solutions for (NSΩ ) and (BN ). In particular, we give
explicit relations between the size of given data and the rotation speed |Ω|, and thebuoyancy frequency N :11∥u0 ∥Ḣ s 6 C|Ω| 2 (s− 2 ) ,1

1∥θ0 ∥Ḣ s 6 CN 2 (s− 2 )+1

with some s > 1/2, which ensure the unique existence of global solutions to (NSΩ ) and(BN ), respectively. Consequently, it is shown that the size of the initial velocity u0 andthe thermal disturbance θ0 can be taken large in proportion to the speed of rotation|Ω| and the strength of stratiﬁcation N , respectively.
Date:⃝
1 Thursday, Mar. 3 9:30-10:30⃝
3 Friday, Mar. 4 9:30-10:30

6⃝2 Thursday, Mar. 3 10:45-11:45⃝
4 Friday, Mar. 4 10:45-11:45

Main-Course

Main-Course

Stefan ULBRICH

Stefan ULBRICH
Technical University of Darmstadt, Darmstadt
Title:
Theory and methods for optimization problems governed by the NavierStokes equations
Abstract:
The modelling and numerical simulation of ﬂows plays an important role in physics,
engineering, chemistry, medicine, and in other disciplines. In most applications, theultimate goal is not only the mathematical modelling and numerical simulation of theﬂow, but rather the optimization or optimal control of the considered process. Typicalexamples are the optimal control of ﬂows by actuators to reduce skin friction or the
optimal shape design of an aircraft. The resulting optimization problems are very
complex and a thorough mathematical analysis is necessary to design eﬃcient solution
methods.
In this short course, we will focus on the optimal control of the unsteady incompressible
Navier-Stokes equations. We will consider optimization problems of the form [5, 6, 7,
9, 11]min J(v, u)

subject tovt − ν∆v + (v · ∇)v + ∇p − F (u) = 0

on ΩT := (0, T ) × Ω,

div v = 0

on ΩT ,v(0, ·) − v0 = 0

on Ω,

suitable boundary conditions(1)

on I × ∂Ω,v ∈ Vad , u ∈ Uad ,where Ω ⊂d is a bounded suﬃciently smooth domain, v ∈ W ⊂ L2 (0, T ; H 1 (Ω)d ) is
the velocity ﬁeld and p : ΩT → is the pressure. Moreover, u ∈ U is a (usually timedependent) control, F : U → Z is a control operator and J : W × U → is the objectivefunction. Finally, Uad ⊂ U and Vad ⊂ W are closed convex admissible sets for thecontrol and the state. We will give examples from ﬂow control applications leading to
problems of this type.
We will consider the following topics.
(1) Existence of optimal solutions,(2) Optimality conditions,(3) Optimization methods for the eﬃcient solution of (1).
To allow for a complete theory, we will focus on the case d = 2.
The problem (1) is of the general form
min J(y, u)

subject toE(y, u) = 0,y ∈ Yad ,u ∈ Uad(2)

with state space Y , control space U and state equation E(y, u) = 0, E : Y × U 7→ Z.
We will see that with an appropriate choice of Y and Z the operator E : Y × U 7→ Z
induced by the Navier-Stokes equations is continuously Féchet-diﬀerentiable, admits
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Stefan ULBRICH

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for each u ∈ U a unique solution y = y(u) of E(y, u) = 0 and the linearized operator
Ey′ (y, u) has a bounded inverse.
After studying the existence of optimal solutions of (1), we will derive ﬁrst order
optimality conditions by introducing and applying optimality theory for optimization
problems (2) in Banach space [2, 9, 10].
We will ﬁrst consider the case without state constraints, i.e., Vad = W . This will allow
us to work with standard weak solutions of (1). The optimality conditions consist of
the original constraints, an adjoint equation and a stationarity condition. The adjoint
equation involves the dual operator of the linearized Navier-Stokes and determines the
Lagrange multiplier (adjoint state) for the state equation E(y, u) = 0. We will showthat the adjoint state enjoys additional regularity [6, 9, 11], which is essential for theconvergence theory of eﬃcient optimization methods for the solution of (1).
Next, we will discuss optimization methods for (1). In particular, we will show that the
optimality system can for the most relevant cases of admissible sets Uad be rewritten as
a nonsmooth system of equations that can be solved eﬃciently by semismooth Newton
methods [7, 9, 11]. After a theoretical justiﬁcation we will show numerical results.
In a next step we will consider the case of state constraints, where Vad is a closed,
convex and bounded subset of W . Besides pointwise bounds on the velocity (e.g. in
gas transport) the case of pointwise bounds on the stresses are of particular interest in
application (e.g., to avoid damage of particles in ﬂuids within pumps or mixers).
We will derive ﬁrst order optimality conditions for pointwise constraints on the velocity
and partial derivatives of the velocity, respectively. This will require to apply Lp - theory
for strong solutions of the Navier-Stokes equations, since L∞ -regularity of the velocity(or its derivative) is necessary to handle pointwise state constraints (or state gradientconstraints) [3, 12, 13].
To solve state constrained problems eﬃciently, penalty approaches by Moreau-Yosida
regularization in connection with semismooth Newton methods have been proven tobe eﬃcient [3, 8]. We will introduce the underlying ideas and discuss the convergence
theory of this approach. Moreover, we will show numerical results.
Finally, if time permits, we will give an introduction to selected further topics, in particular to the method of mappings for shape optimization problems [1] and to reducedorder model techniques for optimal online control [4, 5] governed by the Navier-Stokes
equations.
References:[1] J. A. Bello, E. Fernández-Cara, J. Lemoine, and J. Simon. The diﬀerentiability ofthe drag with respect to the variations of a Lipschitz domain in a Navier-Stokes
ﬂow. SIAM J. Control Optim., 35(2):626–640, 1997.
[2] J. F. Bonnans and A. Shapiro. Perturbation analysis of optimization problems.
Springer Series in Operations Research. Springer-Verlag, New York, 2000.
[3] S. Bott. Adaptive SQP Method with Reduced Order Models for Optimal Control
Problems with Constraints on the State applied to the Navier-Stokes Equations.
Dissertation, Technische Universität Darmstadt, 2015.

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Stefan ULBRICH[4] J. Ghiglieri and S. Ulbrich. Optimal ﬂow control based on POD and MPC andan application to the cancellation of Tollmien-Schlichting waves. Optim. Methods
Softw., 29(5):1042–1074, 2014.
[5] M. D. Gunzburger. Perspectives in ﬂow control and optimization, volume 5 ofAdvances in Design and Control. Society for Industrial and Applied Mathematics(SIAM), Philadelphia, PA, 2003.
[6] M. D. Gunzburger and S. Manservisi. The velocity tracking problem for NavierStokes ﬂows with bounded distributed controls. SIAM J. Control Optim.,
37(6):1913–1945, 1999.
[7] M. Hintermüller and M. Hinze. A SQP-semismooth Newton-type algorithm applied
to control of the instationary Navier-Stokes system subject to control constraints.
SIAM J. Optim., 16(4):1177–1200, 2006.
[8] M. Hintermüller and K. Kunisch. PDE-constrained optimization subject to pointwise constraints on the control, the state, and its derivative. SIAM J. Optim.,
20(3):1133–1156, 2009.
[9] M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich. Optimization with PDE constraints, volume 23 of Mathematical Modelling: Theory and Applications. Springer,
New York, 2009.
[10] F. Tröltzsch. Optimal control of partial diﬀerential equations, volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2010.
Theory, methods and applications, Translated from the 2005 German original byJürgen Sprekels.
[11] M. Ulbrich. Constrained optimal control of Navier-Stokes ﬂow by semismooth
Newton methods. Systems Control Lett., 48(3-4):297–311, 2003. Optimization andcontrol of distributed systems.
[12] D. Wachsmuth. Regularity and stability of optimal controls of nonstationary
Navier-Stokes equations. Control Cybernet., 34(2):387–409, 2005.
[13] D. Wachsmuth. Analysis of the SQP-method for optimal control problems governedby the nonstationary Navier-Stokes equations based on Lp -theory. SIAM J. Control
Optim., 46(3):1133–1153 (electronic), 2007.
Date:⃝
1 Tuesday, Mar. 1 9:30-10:30⃝
3 Wednesday, Mar. 2 9:30-10:30⃝
2 Tuesday, Mar. 1 10:45-11:45⃝
4 Wednesday, Mar. 2 10:45-11:45

Main-Course940 minutes talks

40 minutes talks

Amru HUSSEIN
Technical University of Darmstadt, Darmstadt
Title:On the primitive Equations in Lp with Temperature: Part I
Abstract:
Primitive equations of the ocean and the atmosphere are considered to be afundamental model for many geophysical ﬂows. In this talk primitive equations
with temperature and salinity are considered in Lp . Existence and uniqueness ofstrong solutions is discussed for arbitrarily large data where initial values lie incertain interpolation spaces contained in H 2/p,p . Local solutions are constructedusing an iteration scheme, and proving global a priori estimates allow one then toextend local solutions globally.
Date:
Tuesday, Mar. 1 15:40-16:20

Jun KATO
Nagoya University, Nagoya
Title:A priori estimate for the Navier-Stokes equations with the Coriolis force
and some applications
Abstract:
In this talk, we consider the Navier-Stokes equations with the Coriolis force,{∂t u − ν∆u + Ωe3 × u + (u, ∇)u + ∇p = 0, in (0, ∞) × R3 ,in (0, ∞) × R3 ,div u = 0,

with initial data u|t=0 = u0 , where the constant ν > 0 denotes the viscosity
coeﬃcient of the ﬂuid, and Ω ∈ R represents the speed of rotation around thevertical unit vector e3 = (0, 0, 1). We prove that a priori estimate,∫ t∥u(t)∥χ−1 + (ν − (2π)−3 ∥u0 ∥χ−1 )∥u(τ )∥χ1 dτ ≤ ∥u0 ∥χ−1 ,0which is ﬁrst derived for the Navier-Stokes equations by Lei and Lin (2011), holds∫
under the eﬀect of the Coriolis force, where ∥f ∥χm = |ξ|m |fb(ξ)| dξ. As an application, existence of a unique global solution for arbitrary speed of rotation isproved, provided that the initial data satisﬁes ∥u0 ∥χ−1 < (2π)3 ν.
This talk is based on a joint work with Hiroki Ito (Nagoya University).
Date:
Thursday, Mar. 3 12:00-12:40
10

40 minutes talks

40 minutes talks

Matthias KOTSCHOTE
University of Konstanz, Konstanz
Department Mathematics and Statistics
matthias.kotschote@uni-konstanz.de

Title:
Spectral analysis for steady states of compressible two-phase fluids ofNavier-Stokes-Allen-Cahn type
Abstract:
This talk is concerned with the so-called “Navier-Stokes-Allen-Cahn” system(NSAC) which is a combination of the Navier-Stokes equations for compressible ﬂuids with a phase ﬁeld description of Allen-Cahn type. The NSAC system admits ofdescribing two-phase patterns in a ﬂowing liquid including phase transformations.
This model ﬁnds its counterpart in the “Navier-Stokes-Cahn-Hilliard” system that
imitates again a ﬂow of a binary mixture being macroscopically immiscible but
without permitting phase transformation.
The ﬁrst purpose of this talk consists in presenting a derivation of NSAC and
NSCH from ﬁrst principles, in the spirit of rational mechanics, for ﬂuids of very general constitutive laws. For NSAC, this deduction conﬁrms and extends a proposalof Blesgen [1]. Regarding NSCH, it continues work of Lowengrub and Truskinovsky[2] and provides the apparently ﬁrst justiﬁed formulation in the non-isothermalcase.
The second part of the talk deals with the linear system obtained by linearizing
around non-constant stationary solutions. For the associated linear operator aquite detailed description of the point spectrum can be accomplished, e.g. instableeigenvalues are characterized in terms of eigenvalues of an elliptic problem.
References:[1]

T. Blesgen: A generalisation of the Navier-Stokes equations to two-phaseﬂows. J. Phys. D: Appl. Phys. 32 (1999), 1119–1123.

[2]

J. Lowengrub, L. Truskinovsky: Quasi-incompressible Cahn-Hilliard ﬂuids and topological transitions. Proc. R. Soc. Lond. A 454 (1998), 2617–2654.

Date:
Wednesday, Mar. 2 15:40-16:20

Tomasz PIASECKI
Polish Academy of Sciences, Warsaw

Institute of Mathematics

Title:
Stationary compressible Navier-Stokes equations with inflow boundary
conditions
40 minutes talks11

40 minutes talks

Abstract:
In my talk I will present results concerning existence of regular solutions in a
vicinity of given laminar ﬂows to the Navier-Stokes system describing stationary
ﬂow of a compressible ﬂuid:
ρv · ∇v − div T(∇v, π) = ρf(1)

div (ρv) = 0
div (ρEv) = ρf · v + div (T(∇v)v) − div q,

supplied with boundary conditions which will be precised, in a bounded domain.
Here v denotes the velocity of the ﬂuid, ρ the density and θ the absolute temperature, the other quantities will be explained in the talk. The mathematical theoryof system (1) is quite well developed in case of homogeneous boundary conditions
understood as conditions when normal component of the velocity vanishes at theboundary. However, admission of ﬂow accross the boundary leads to substantial
mathematical diﬃculties related to the hyperbolicity of the continuity equation.
In particular, in such case we have no large data existence results, the fact which
gives additional motivation to studying regular solutions for small data.
It is well known that in framework of regular solutions the existence results are
subject to certain constraints on the boundary in the neighbourhood of the points
where characteristics of the continuity equation become tangent to the boundary.
Such situation is avoided when we consider a problem in cylindrical domain. Therefore, in the ﬁrst part of my talk I will focus on results in such class of domains.
Here the regular solutions are understood as functions in Sobolev spaces with weakderivatives satisfying the equations a.e. Even though we do not have to tackle thesingularity of the boundary, we still have to deal with lack of compactness in thecontinuity equation. I will present several ways of solving this diﬃculty.
In the second part of my talk I will move to problem in a domain where wehave to deal with boundary singularity. Here we generalize slightly the deﬁnitionof solution which leads to working in fractional Sobolev-Slobodetskii spaces. Suchapproach allows to relax considerably the assumptions of boundary singularity
compared to previous results.
Date:
Friday, Mar. 4 12:00-12:40

Jonas SAUER
Technical University of Darmstadt, Darmstadt
Fachbereich Mathematik, Technische Universität Darmstadt, 64283 Darmstadt,
jsauer@mathematik.tu-darmstadt.de

Title:
Maximal Regularity: An Easy Approach12

40 minutes talks

40 minutes talks

Abstract:
Throughout the last ﬁfteen years or so, the concept of maximal Lp regularity
has been examined and applied successfully to various evolution equations, in particular to quasilinear parabolic equations. The celebrated theorem of Weis statesthat in Banach spaces which fulﬁll the geometric UMD property, maximal regularity of an operator −A generating a strongly continuous semigroup is equivalent
to the R-boundedness of the family {it(it + A)−1 |t ∈ R}. This characterization
has been used extensively to obtain maximal regularity for a variety of diﬀerential
operators. In my talk, I will show that maximal regularity for the initial value
problem is equivalent to a time-periodic variant of maximal regularity, regardlessof the geometric properties of the underlying Banach space. As time-periodic Lp
estimates can be obtained in a fast and straight-forward way for the Laplace andthe Stokes operator on diﬀerent domains such as the whole space, the half spaceand suﬃciently regular bounded domains, this constitutes a slim and elegant proofof maximal Lp regularity of these operators which completely circumvents thesomewhat complicated notion of R-boundedness. In proving the correspondingtime-periodic estimates, the key idea is to replace the time axis with a torus andreformulate the problem on a locally compact abelian group. Subsequently, the Lp
estimates are obtained via Fourier multipliers in a group setting, which gives theresult in the whole space. Other domains can then be treated by similar methods
as are known from the corresponding elliptic problems.
The talk is based on joint works with Mads Kyed [1] and Yasunori Maekawa [2].
References:[1] M. Kyed and J. Sauer, A Method for Obtaining Time-Periodic Lp Estimates,
preprint.
[2] Y. Maekawa and J. Sauer, Maximal Regularity of the Time-Periodic Stokes
Operator on Unbounded and Bounded Domains, preprint.
Date:
Wednesday, Mar. 2 12:00-12:40

Yohei TSUTSUI
Shinshu University, Nagano
Title:
Applications of weighted Hardy spaces to the incompressible NavierStokes equations
Abstract:
Applying the theory of weighted Hardy spaces Hαp (Rn ), we consider time decay
and asymptotic expansion of solutions.

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For φ ∈ C0∞ (B(0, 1)) with φ̂(0) = 1, p ∈ (0, ∞) and α ∈ (−n/p, ∞), Hαp isdenoted by a space of tempered distributions with ﬁnite semi-norm∥f ∥Hαp := ∥Mφ [f ]∥Lpα < ∞,where
Mφ [f ](x) := sup |f ∗ φt (x)|, φt (x) := t−n φ(x/t) and ∥g∥Lpα := ∥|x|α g∥Lp .
t>0

Firstly, we establish the existence of global solutions with initial data a ∈ Ln ∩ Hαphaving small Ln norm. L2 -energy of our solutions decays as t−(n+2)/4−ε , (ε > 0).
Wiegner (‘87) showed the existence of weak solutions decaying as t−(n+2)/4 . In our
approach , to remove ε in our decay, div - curl estimates with critical power weight
αp = n(1 − 1/p) + 1 seem to be necessary. These estimates are constructed by thereal interpolation theory.
To study asymptotic expansion and rapid decay, the previous papers assume
pointwise estimates on the initial data; |a(x)| . ⟨x⟩−γ . We show similar resultswith non-decaying initial data, but belongs to weighted Hardy spaces.
This talk is based on a joint work with Takahiro Okabe in Hirosaki University.
Date:
Thursday, Mar. 3 15:40-16:20

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Martin BOLKART
Technical University of Darmstadt, Darmstadt
Title:The Stokes equations in spaces of BM O-type
Abstract:
We introduce a space BM Ob with norm measuring the BM O-seminorm in theinterior of the domain and the mean value near the boundary. We will then provethat the Stokes semigroup is analytic in these BM O-type spaces. As an application
we can then prove that in some domains the Stokes semigroup on Lp is analytic
even if the Helmholtz projection in Lp does not exist.
Date:
Thursday, Mar. 3 16:30-16:45

Alexander DALINGER
Technical University of Darmstadt, Darmstadt
Title:On the hydrodynamic limit and equilibrium ﬂuctuations of a particle
system with nearest neighbor interactions
Abstract:
For a one-dimensional particle system with nearest neighbor interactions westudy the particle density. Starting with an arbitrary conﬁguration of particles,
we describe the time evolution of the system. First, we show that the particle
density converges in the hydrodynamic limit to a solution of a nonlinear heat
equation. Furthermore, we consider the random ﬂuctuations around its averagestate, the so called equilibrium ﬂuctuations.
Date:
Friday, Mar. 4 15:55-16:10

Shota ENOMOTO
Kyushu University, Fukuoka
Title:On asymptotic behavior of solutions to the compressible Navier-Stokes
equations in an inﬁnite layer under slip boundary condition
Abstract:
We consider large time behavior of solutions to the compressible Navier-Stokes
equation around the motionless state in a two-dimensional inﬁnite layer under the
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slip boundary condition. We show that the asymptotic leading part of perturbations is given by a superposition of self-similar solutions of one-dimensional viscous
Burgers equations if initial perturbations are suﬃciently small.
Date:
Friday, Mar. 4 17:00-17:15

Michael FISCHERTechnical University of Darmstadt, Darmstadt
Title:
Shape optimization with the Boussinesq equations
Abstract:
In this talk we consider the perturbation of identity method for shape optimization
based on the works of Murat and Simon, to solve shape optimization problems
governed by the Boussinesq equations. To this end we give an introduction toshape optimization in general and introduce the perturbation of identity methodby Murat and Simon, see [1].
After the introduction we present a general framework to show Fréchet diﬀerentiability of the control to state operator of nonlinear optimal control problems.
Then we derive the shape optimization problem governed by the Boussinesq
equations and apply the abstract Fréchet diﬀerentiability theorem to prove theshape diﬀerentiability of the solution operator. This extends the work by Lindeman
et al., see [2], from the Navier-Stokes to the Boussinesq equations.
References:[1] F. Murat and J. Simon, Etude de problemes d’optimal design, Lecture Notes
in Computer Science, 41(1976).
[2] F. Lindemann and M. Fischer and M. Ulbrich and S. Ulbrich, Fréchet diﬀerentiability of time-dependent incompressible Navier-Stokes ﬂow with respect
to domain variations, Preprint, (2015)[3] G. Wachsmuth, Diﬀerentiability of implicit functions: Beyond the implicit
function theorem, Journal of Mathematical Analysis and Applications, Vol.
414, 259 - 272, (2014)
Date:
Wednesday, Mar. 2 16:45-17:00

Mathis GRIESTechnical University of Darmstadt, Darmstadt
Title:On the primitive equations in Lp -spaces with temperature, Part II

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Abstract:
In this talk, we report on an ongoing work concerning the primitive equationsin the non-isothermal setting. Our aim is to extend a result on the existence oftime periodic solutions for large forces in the isothermal case to the non-isothermal
setting.
Date:
Tuesday, Mar. 1 16:30-16:45

Andreas KLAIBER
University of Konstanz, Konstanz
Title:
Spectral stability of internal solitary waves in continuously stratiﬁed
ﬂuids
Abstract:
Internal solitary waves (ISWs), which travel in the interior of lakes and oceans,present a fascinating observable wave phenomenon with large amplitudes and aremarkable robustness. These waves are ecologically important since they are
involved in mixing and energy transport.
From the mathematical point of view, ISWs are exact solutions of the 2D Euler
equations for incompressible, inviscid ﬂuids with non-constant density. This talk isconcerned with the study of their spectral stability, which has received only little
attention at a rigorous mathematical level.
After introduction of the mathematical model, the ﬁrst part of the talk deals withan Evans-function approach to spectral stability of ISWs. This approach starts
from the Euler eigenvalue problem and is based on ﬁnite-dimensional truncationsof a formal spatial-dynamics formulation. For small-amplitude ISWs, this yields a
proof of the absence of small unstable modes by comparison with solitons in the
Korteweg-deVries equation.
The second part of the talk deals with the deﬁnition of a “moment of instability”
for ISWs (a classical tool in the stability theory of solitary waves), which presents
a complementary approach to stability of ISWs.
Date:
Wednesday, Mar. 2 16:30-16:45

Naofumi MORI
Kyushu University, Fukuoka
Title:
Global existence and decay property of the Timoshenko-Cattaneo system
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Abstract:
Our main focus in this talk is to show the asymptotic behavior of a nonlinear
version of the Timoshenko system with heat conduction of Cattaneo’s law. As
it has been already proved in Said-Houari and Kasimov (2012) and Mori and
Kawashima (2015), the linear version of this system is of regularity-loss type. It
is well known (Hosono and Kawashima (2006), Ide and Kawashima (2008), Kubo
and Kawashima (2009)) that the regularity-loss property of the linear problem
creates diﬃculties when dealing with the nonlinear problem. In fact, the dissipative
property of the problem becomes very weak in the high frequency region and as aresult the classical energy method fails. To overcome this diﬃculty and following
Ide and Kawashima (2008) and Ikehata (2002), Racke and Said-Houari (2012) use
an energy method with negative weights to create an artiﬁcial damping whichallows us to control the nonlinearity. They prove that for 0 ≤ k ≤ [s/2] − 2 withs ≥ 8, the solutiuon of the problem is global in time and decays as ∥∂xk U (t)∥L2 ≤C(1 + t)−1/4−k/2 , provided that the initial datum U0 ∈ H s (R) ∩ L1 (R). On theother hand, in this talk, we show the global existence result of the same problem
with the critical sobolev space s = 2 by using just an energy method without
any negative weights. Besides, we also show the optimal decay result with theminimal regularity assumptions on the initial data U0 ∈ H 2 (R) ∩ L1 (R) by using atime decay inequarity of L2 (R)-Lq (R)-Lr (R) type ﬁrst developed in Ju, Mori and
Kawashima (2015). This talk is based on a joint work with Prof. Reinhard Rackeof Univ. Konstanz.
Date:
Friday, Mar. 4 17:15-17:30

Dai NOBORIGUCHI
Waseda University, Tokyo
Title:
On a time-splitting method for degenerate parabolic stochastic partial
diﬀerential equations
Abstract:
In this talk we consider degenerate parabolic stochastic partial diﬀerential equations of the following type
du + div(B(u))dt = div(A(u)∇u) + Φ(u)dβ(t)in (0, T ) × Td ,with the initial conditionu(0, ·) = u0 (·)

in Td .

Here Td is the d-dimensional torus, T > 0, and β is a one-dimensional Brownian
motion deﬁned on a stochastic basis (Ω, F , {Ft }, P).
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We introduce the deﬁnition of kinetic solutions characterized by kinetic formulations. To prove the existence of such solutions, we will use a time-splitting
approach.
Date:
Friday, Mar. 4 16:10-16:25

Tobias SEITZ
Technical University of Darmstadt, Darmstadt
Title:
Enhancement of ﬂow measurements using ﬂuid-dynamic constraints
Abstract:
Novel experimental modalities acquire spatially resolved velocity measurements
for steady state and transient ﬂows which are of interest for engineering and biological applications. One of the drawbacks of such high resolution velocity datais their susceptibility to measurement errors. We propose a novel ﬁltering strategy that allows enhancement of noisy measurements to obtain reconstruction ofsmooth divergence free velocity and corresponding pressure ﬁelds, which together
approximately comply to a prescribed ﬂow model. The main step in our approachconsists of the appropriate use of the velocity measurements in the design of alinearized ﬂow model which can be shown to be well-posed and consistent with thetrue velocity and pressure ﬁelds up to measurement and modeling errors. The reconstruction procedure is formulated as a linear quadratic optimal control problem
and the resulting ﬁlter has analyzable smoothing and approximation properties.
Date:
Friday, Mar. 4 15:40-15:55

Anton SEYFERTTechnical University of Darmstadt, Darmstadt
Title:
Almost periodic solutions to incompressible ﬂuid ﬂow problems
Abstract:
In this talk we consider almost periodic solutions in time to various problems
arising in ﬂuid mechanics such as the Navier-Stokes ﬂow in an aperture domainor the ﬂow past a rotating obstacle. Our approach is based on an extension of a
recent result on periodic solutions of this kind of problems.
Date:
Friday, Mar. 4 16:45-17:00

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Florian STEINBERG
Technical University of Darmstadt, Darmstadt
Title:
Complexity Theory in Sobolev Spaces
Abstract:
The framework of second order representations was introduced by Kawamura
and Cook in 2010 and provides a rigorous notion of computation over continuous
structures. It also provides a notion of time and space restricted computation.
Choosing a representation means to specify what information a program computing
an element of the structure has to provide. We relate the existence of a reasonablechoice of information on a compact metric space to a purely metric property of thespace: The metric entropy of a compact metric space measures fast the numberof balls needed to cover the space grows with the radius of the balls decreasing.
We show that the optimal running time of the metric is inherently connected tothe metric entropy. These results are applied to show optimality of some concrete
representations of function spaces.
Date:
Tuesday, Mar. 1 17:00-17:15

Go TAKAHASHI
Waseda University, Tokyo
Title:
On partial regularity and extension of solutions to the Navier-Stokes
equations
Abstract:
In this talk, we work on the local-in-time classical solution to the incompressible
Navier-Stokes equations. The dimension n is 3 or 4. First, we discuss the partial
regularity of suitable weak solutions, which is an essential argument to obtainthe local boundedness of the solutions. Next, we will establish a time extension
criterion to the local-in-time solutions as an application of ϵ regularity theorem.
Date:Thursday, Mar. 3 16:45-17:00

Shinya UCHIUMI
Waseda University, Tokyo
Title:Exactly computable Lagrange–Galerkin schemes and their numericalresults

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Abstract:
We present exactly computable Lagrange–Galerkin schemes for convection-diﬀusion
problems and the Navier–Stokes problems, which we recently developed [1, 2].
Lagrange–Galerkin schemes are powerful numerical methods for ﬂow problems.
In conventional schemes, we have to compute integration of terms which include
composite functions and it is diﬃcult to compute the values exactly in general.
Although numerical quadrature is usually employed to the terms in real computations, the error caused by the quadrature may destroy the stability. We introduce a
locally linearized velocity and the backward Euler method to solve ordinary diﬀerential equations which describe the position of a ﬂuid particle. Then, the schemes
become exactly computable and theoretical stability and convergence are assured.
In this talk, we introduce these exactly computable schemes and show error estimates. We also present some numerical results, which reﬂect these estimates andalso show robustness for high Péclet number and high Reynolds number problems.
This talk is a joint work with Prof. M. Tabata at Waseda University.
References:[1] M. Tabata and S. Uchiumi. A genuinely stable Lagrange–Galerkin scheme
for convection-diﬀusion problems. Japan Journal of Industrial and Applied
Mathematics, 2015. doi:10.1007/s13160-015-0196-2[2] M. Tabata and S. Uchiumi. A Lagrange–Galerkin scheme with a locallylinearized velocity for the Navier–Stokes equations. 2015. arXiv:1505.06681[math.NA]
Date:
Wednesday, Mar. 2 17:00-17:15

David WEGMANN
Technical University of Darmstadt, Darmstadt
Title:
Decay of Non-stationary Navier-Stokes Flow with Nonzero Dirichlet
Boundary Data
Abstract:
We consider the Navier-Stokes equations in a domain with compact boundary andnonzero Dirichlet boundary data β. Recently, in the case of an exterior domain
the existence of a global in time weak solution which satisﬁes the strong energy
inequality was shown in [1]. This solution is constructed as a sum of a very weaksolution b to the instationary Stokes equation with nonzero boundary data and aweak solution v to a system of Navier-Stokes type with zero boundary data.
Assuming that β(t) → 0 as t → ∞, we prove in [2] that the corresponding solution
v fulﬁlls ∥v(t)∥2 → 0 as t → ∞. Furthermore, in a bounded domain the solution v
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tends exponentially to 0 if the corresponding data are exponentially decreasing.
As a last result, we calculate a lower polynomial bound for the decay rate if Ω
is unbounded. Therefore, we use a suitable spectral decomposition of the Stokes
operator as introduced in [3] and Duhamel’s formula.
This is a result of a collaboration with Prof. Dr. Reinhard Farwig and Prof. Dr.
Hideo Kozono.
References:[1] Farwig, R., Kozono, H., “Weak solutions of the Navier-Stokes equations with
non-zero boundary values in an exterior domain satisfying the strong energy
inequality,” J. Diﬀerential Equations, 7, No. 10, 2633–2658 (2014).
[2] Farwig, R., Kozono, H., Wegmann, D., “Decay of non-stationary NavierStokes ﬂow with nonzero Dirichlet boundary data,” Indiana Univ. Math. J.,
accepted for publication (2015)[3] Borchers, W., Miyakawa, T., “Algebraic L2 decay for Navier-Stokes ﬂows inexterior domains. II,” Hiroshima Math. J., 3, 621–640 (1991).
Date:
Thursday, Mar. 3 17:00-17:15

Sebastian ZAIGLER
Technical University of Darmstadt, Darmstadt
Title:
Regularity Structures
Abstract:
Regularity structures are a new method introduced by Martin Hairer to solve
subcritical SPDEs with rough data, which were formerly inaccessible. They build
vector spaces of abstract “Taylor expansions” of functions and distributions leading
to a ﬁxed point equation that solves a SPDE in these abstract spaces. Then it
is possible to ﬁnd a sequence of renormalised solutions of the SPDE so that itconverges to the solution of the original problem. We will give a short introduction
to this technique and outline our goal to use them on the primitive equations withwhite noise.
Date:
Tuesday, Mar. 1 16:45-17:00

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