International Workshop
on the Multi-Phase Flow;
Analysis, Modeling and Numerics
Waseda, November 7 – 11, 2016

Abstracts

Thomas Richter
Faculty of Mathematics, University of Magdeburg, Magdeburg, Germany

Finite Elements for Fluid-Structure Interactions
In this lecture, we will present numerical tools for the simulation of coupled fluidstructure interaction problems. Fluid-structure interactions are part of many application
problems in engineering, biology and medicine. We focus on the strongly coupled interaction of an incompressible fluid with a nonlinear elastic structure. Usually the coupling
is be considered to be strong, if fluid and solid mass are comparable (like the coupling ofblood with tissue and unlike the coupling of air and steel). Modeling of such problems
leads to stiﬄy coupled nonlinear systems of partial diﬀerential equations. Many standard
simulation tools will fail.
Part 1. Monolithic schemes in Arbitrary Lagrangian Eulerian Coordinates
We start by describing the ALE scheme for fluid-structure interactions, which is basedon the mapping of the moving fluid domain onto a fixed reference domain. This framework
allows to present the fully coupled system in one closed variational formulation, making
it accessible for many well-established tools like Newton’s method, multigrid, a posteriori
error estimation and gradient based optimization are available.
Part 2. Large deformation problems in the Fully Eulerian Formulation
ALE formulations for fluid-structure interactions experience a limit in terms of large
movement and deformation. They usually do not allow for very large deformation and they
forbid topology change, e.g. by contact or by material failure. We present an alternative
approach, the Fully Eulerian framework, that is based on a mapping of all equations into
the moving Eulerian coordinate framework. The resulting system looks similar to multiphase flows and brings along comparable diﬃculties. The major numerical problem is themoving-interface character with an interface line (or surface) between fluid and structure,
which must be captured and which must be resolved be the discretization. The main
focus of this lecture is to present spatial and temporal discretization schemes for interface
problems, that are also useful for comparable interface problems with moving interfaces.
— — —Giovanni Paolo Galdi
Department of Mechanical Engineering and Materials Science,
Department of Mathematics, University of Pittsburgh, Pittsburgh, USA

Flow of a Viscous Liquid Past an Obstacle at Arbitrary Reynolds
NumberIn this short two-hour course, I shall perform a quick ride through one of the oldest
and most investigated problems in fluid mechanics, namely, the motion of a viscous liquid
past a fixed rigid obstacle. Assuming the flow is uniform at large spatial distance (infinity)from the body, I will mainly focus on the determination of the motion characteristics at
Reynolds numbers, Re, of arbitrary size. Thus, besides basic issue like existence and1regularity, this study will include the occurrence of (multiple) steady and time-periodic
bifurcating phenomena, along with a quick (and, unfortunately, to date still incomplete)glance at the behavior in the limit of very large Re. Certain significant open questions
will also be identified and discussed.
— — —Franco Flandoli
Department of Mathematics, University of Pisa, Pisa, Italy

Mean field results for particles dispersed in fluids
The talk will present two attempts to model a large family of small particles dispersed
in a fluid. Both models aim to be simple and leave open diﬃcult questions about a very
accurate modelling of the two phase flow.
The first model considers the case of interacting particles embedded into a ”turbulent”
fluid, in the very simplified regime when the particles are passive and the fluid is represented by a space-time Gaussian field, white in time, in the spirit of the theory of passive
scalars in turbulent fluids. We show a mean field result, in the limit when the number ofparticles goes to infinity and their interaction is of mean field type, which give rise to astochastic nonlinear PDE.
The second model removes both idealizations, namely the Gaussian field is replaced by
the solution of the Navier-Stokes equations and the particles act on the fluid. However, weidealize the interaction between particles and fluid, having in mind particles of infinitesimal
size and Stokes drag force. The limit PDE is of Vlasov-Navier-Stokes type. We compare
methodologies and models and address open questions.
— — —Yoshihiro Shibata
Department of Mathematics, Waseda University, Tokyo, Japan

On some free boudary problem for the Navier-Stokes equations
I will talk about some free boundary problem for the Navier-Stokes equations. Contents
is the following.
1. Modelling of the free boundary problem for the Navier-Stokes equations.
2. Maximal Lp -Lq regularity and Local well-posedness.
3. Global well-posedness in the bounded domain with surface tension.
4. Global well-posedness in the exterior domain case without surface tension.
— — —2

Suncica Canic
Department of Mathematics, University of Houston, Texas, USA

An Introduction to Fluid-Solid Interaction in Biofluids
Fluid-solid interaction (FSI) problems arise in many applications. They include multiphysics problems in engineering such as aeroelasticity and propeller turbines, as well as
biofluidic application such as self-propulsion organisms, fluid-cell interactions, and theinteraction between blood flow and cardiovascular tissue. A comprehensive study of these
problems remains to be a challenge due to their strong nonlinearity and multi-physics
nature. To make things worse, in many biological applications the solid is composedof several layers, each with diﬀerent mechanical characteristics. This is, for example,
the case with arterial walls whose physiology and pathophysiology are aﬀected by theirinteraction with blood flow, and by the interaction between the diﬀerent arterial walllayers. In the two introductory lectures, the speaker will present an overview of the main
problems in FSI in blood flow motivated by cardiovascular applications. In particular,mathematical modeling, analysis, and design of partitioned numerical algorithsm for FSI
in hemodynamics will be discussed.
Lecture 1: Modeling of Fluid-Solid Interaction in Hemodynamics
In this introductory lecture, models for blood flow and arterial walls will be presented.
The coupling between an incompressible, viscous fluid modeling blood flow, and an elastic/viscoelastic solid modeling arterial walls will be described. The geometric nonlinearities in the coupled problem will be discussed, and the reasons for the instabilities in theclassical partitioned (loosely-coupled) FSI schemes will be explained. Issues related to theuse ofʠ correct ʡboundary conditions will be touched. Modeling of vascular devices suchas stents using the 1D hyperbolic net approaches will be presented.
Lecture 2: Analysis and a Partitioned Numerical Scheme for Fluid-Solid
Interaction in Hemodynamics
The speaker will present an overview of the mathematical analysis results for fluidsolid interaction with application to blood flow, and an example of a partitioned numerical
scheme, which inspired a constructive existence proof for this class of problems, will be
discussed.
— — —

3

Elena Frolova
St. Petersburg Electrotechnical University, St. Petersburg State
University, St. Petersburg, Russia

Free boundary problem of magnetohydrodynamics for two liquids
We consider the free boundary problem governing the motion of two viscous incompressible electrically conducting capillary fluids separated by a closed interface. Media ismoving under the action of magnetic field. We assume that the first fluid is contained ina bounded variable domain Ω1t , which is surrounded by a bounded variable domain Ω2t ,filled with the second fluid. The boundary of Ω2t consists of two disjoint components: thefree boundary Γt = ∂Ω1t and the fixed surface S. Both Γ0 and S are homeomorphic to asphere, dist{Γ0 , S} ≥ δ > 0. The surface S is assumed to be a perfect conductor.
We prove local in time solvability of this problem in Sobolev-Slobodetskii spaces
2+l,1+l/2
W2, 1/2 < l < 1. To reduce the free boundary problem for a problem in a fixed
domain, we use Hanzawa coordinate transform. The proof is carried out by the successive approximations method and based on the careful analysis of the corresponding linear
problem and estimates of nonlinear terms.
— — —Tadahisa Funaki
Graduate School of Mathematical Sciences, University of Tokyo, Tokyo,
Japan

Sharp interface limit for a stochastically perturbed mass conserving
Allen-Cahn equation
We will discuss the sharp interface limit for a mass conserving Allen-Cahn equation
added an external mild stochastic noise. In the limit, a stochastically perturbed volume
preserving mean curvature flow is derived. Our approach is the asymptotic expansion
method, in which powers of derivatives of the noise appear and these terms are diverging.
However, making the convergence speed of our mild noise to the white noise slow enough,we can control them. This is joint work with Satoshi Yokoyama. The paper is posted on
the arXiv.
— — —Matthias Hieber
Department of Mathematics, Technical University Darmstadt, Darmstadt,
Germany

Strong Solutions to Two-Phase Free Boundary Value Problems for
Newtonian and Non-Newtonian Fluids

4In this talk we discuss one- and two-phase free boundary value problems subject tosurface tension and gravitational forces for various classes of Newtonian and non- Newtonian fluids and discuss their well-posedness properties within the Lp -setting. The fluids
considered are special classes of so called Stokesian fluids. We show in particular that forgiven T > 0 this type of problems admit a unique, strong solution on (0, T ) provided theinitial data are suﬃciently small in their natural norms. This is partly joint work with
Hirokazu Saito (Waseda University).
— — —Toshiaki Hishida
Graduate School of Mathematics, Nagoya University, Nagoya, Japan

Navier-Stokes flow past a rigid body: attainability of steady solutions as
the limit of unsteady weak solutions, starting and landing cases
Consider the Navier-Stokes flow in 3D exterior domains, where a rigid body is translating with prescribed translational velocity −h(t)u∞ with constant vector u∞ ∈ R3 \ {0}.
Finn raised the question whether his steady solutions are attainable as the limit for t → ∞of unsteady solutions starting from motionless state when h(t) = 1 after some finite timeand h(0) = 0 (starting problem). This was aﬃrmatively solved by Galdi, Heywood andShibata for small u∞ . In this talk, we study some generalized situation in which unsteadysolutions start from large motions being in L3 . We then conclude that the steady solutions
for small u∞ are still attainable as the limit of evolution of those fluid motions which arefound as a sort of weak solutions. The opposite situation, in which h(t) = 0 after somefinite time and h(0) = 1 (landing problem), is also discussed. In this latter case, the trivial
solution is attainable in the same way as above no matter how large u∞ is. This talk isbased on a joint work with Paolo Maremonti (Seconda Universita degli Studi di Napoli).
— — —Yoshifumi Kimura
Graduate School of Mathematics, Nagoya University, Nagoya, Japan

Search for a standard model of vortex reconnection
As a fundamental process in the whole vortex problems, vortex reconnection has beenstudied intensively and providing renewed interest in both theoretical and computational
fluid mechanics. In this talk, inspired by recent novel experiment by Kleckner and Irvineon the dynamics of a trefoil knot vortex [1], we first present a linearized model in which two
Burgers-type vortices are driven together by an axisymmetric straining velocity field [2].
Extending this linearized model, we then consider the nonlinear vortex interaction using
the Biot-Savart integral. It will be demonstrated that there are clear scaling properties
in time for the minimum distance of anti-parallel vortices, the maximum velocity and the
maximum axial strain rate before reconnection, and that specific shapes of vortices areformed near during the reconnection.
5

References[1] Kleckner, D. and Irvine, W.T.M. (2013) Creation and dynamics of knotted vortices.
Nature Physics 9, 253258. doi: 10.1038/nphys2560.
[2] Kimura, Y and Moﬀatt, H.K. (2014) Reconnection of skewed vortices. J. Fluid Mech.
751, 329–345.
— — —Yasunori Maekawa
Department of Mathematics, Kyoto University, Kyoto, Japan

On some stability results for two-dimensional exterior flows
In this talk recent progress will be introduced on the stability theory for some typicalviscous incompressible flows in a two-dimensional exterior domain. Particular interests are:time-dependent weak L2 flows including forward self-similar type such as the Lamb-Oseen
flows, stationary circular flows decaying in a scale-critical order, and Oseen flows.
— — —Anne Robertson
Department of Mechanical Engineering and Materials Science,
Department of Bioengineering, University of Pittsburgh, Pittsburgh, USA

Modeling the mechanobiology of the arterial wall: Our current
understanding and future research directions
The mechanical strength of the artery wall is vital to the health of the individual. This
integrity is in turn dependent on the state of the central load bearing components of thewall: collagen fibers, elastin fibers and vascular smooth muscle cells. In healthy arteries,
the physical organization and mechanical properties the elastin and collagen fibers enable
the artery wall to withstand the mechanical loads created by the time periodic pulsation of
the blood, without excessive energy dissipation and without exposing the cells in the wall toinordinate strains. Furthermore, these fibers must be able to withstand on the order of 109
loading cycles over the course of a seventy-five year old lifetime without excessive fatigue.
Fortunately, collagen fibers, which are responsible for the wall strength, are not simply
created at birth and then left to slowly wear out over time. Rather, the cells in the arterywall are designed to continually renew these collagen fibers. In fact, is it now understoodthat the healthy arterial wall is an active structure- with collagen fibers undergoing growth
and remodeling in response to blood flow (hemodynamic) derived chemical and mechanicalstimuli. This replacement process also adds an element of flexibility to the wall, enabling
it to change the collagen fiber organization in response to altered flow and mechanicalloads.
6

In this talk, we discuss mathematical models that have been developed to describe
this coupled response between the mechanics and the biology (mechanobiology) as wellas the somewhat universal factors that appear to dominate the inherent optimization ofcollagen fiber organization. Of great interest are theoretical frameworks that are capableof modeling the response of the artery wall to mechanical and chemical perturbations to
the flow and mechanical loading. In this talk, we discuss some of these mathematicalframeworks with particular focus on the i) prototypical healthy artery wall, ii) aneurysmsof brain arteries, and iii) in vivo tissue engineering of neoarteries. Recent advances inbioimaging enable unprecedented data on the collagen fiber architecture. For example,
multi-photon microscopy (MPM) can be used to visualize both elastin and collagen inartery walls by exploiting auto-fluorescence and second harmonic generation, respectively.
In this talk, we discuss bioimaging results obtained in our laboratory using MPM thatare used to drive these theoretical models. The talk will also include a discussion of open
questions, future directions and the need for further collaborations with mathematicians,
biologist and engineers.
This multi-institutional work has been supported by the NIH- NIH/ 1R21NS08003101A1, NIH 1R21NS088256:, NIH R01 1R01NS097457-01 and the University of Pittsburgh
ADRC Brain Bank.
— — —Kiyoshi Saito
Department of Applied Mechanics and Aerospace Engineering, Waseda
University, Tokyo, Japan

Experimental Study on Two-Phase Flow in Vapor Compression HeatPump
In recent years, heat-pump technology attracts attention as an innovative technology
to realize energy saving in various fields, both of civilian and industrial sectors. A vapor compression heat pump is a system which follows a reverse Carnot cycle as the basic
principle. Since the system eﬃciency is high and also the configuration is comparatively
simple, the vapor compression heat pump technology is spreading all over the world. Various two-phase flow phenomena occur inside the heat pump; evaporation, condensation,
expansion, phase separation, wetting, high speed flow and so on. From the engineering
viewpoint, various researches on these problems have been carried out over many years. In
this report, we would like to introduce some parts of these eﬀorts and to share these problems with you all. First of all, flow boiling heat transfer inside small channels is discussed
by showing some experimental results. A small channel is defined as a channel with adiameter smaller than 1 mm, and heat transfer inside these channels has recently become
a hot topic in the engineering field. This is because their heat transfer performance issignificantly higher than that of conventional channels, and this characteristics can reduce
the required charge of working fluid. A small channel shows heat transfer characteristics
diﬀerent from a conventional channel. Secondly, we discuss experimental results of phaseseparation in a T-shaped junction. Separating technique of gas-liquid two-phase flow into
each phase is very important to enhance the system eﬃciency. However, as you know,complete separation of two-phase flow is extremely diﬃcult. Some visualization resultswith high-speed camera are also shown. We hope that this discussion will contribute7

establishing collaborations beyond this specific research field and to realizing an energy
saving society.
— — —Franck Sueur
Institute of Matematics, University of Bordeaux, Bordeaux, France

Point vortex dynamics as zero-radius limit of the motion of rigid bodies
in a perfect incompressible fluid
The vortex point system is usually considered as an idealized model where the vorticity
of an ideal incompressible two-dimensional fluid is concentrated in a finite number ofmoving points. In this talk I will present some results obtained in collaboration with O.
Glass, C. Lacave and A. Munnier which establish that point vortex dynamics can also
be obtained as zero-radius limit of rigid bodies motions under the influence of the force
exerted by the fluid pressure on its boundary.
— — —Yoshie Sugiyama
Department of Mathematics, Kyushu University, Fukuoka, Japan

Time global existence and finite time blow-up criterion for solutions to
the Keller-Segel system coupled with Navier-Stokes fluid
We will deal with the chemotaxis model under the eﬀect of the Navier-Stokes fluid,i.e., the incompressible viscous fluid. We shall show the existence of a local mild solution
for large initial data and a global mild solution for small initial data in the scale invariant
class demonstrating that n0 ∈ L1 (R2 ) and u0 ∈ L2σ (R2 ). Our method is based on theperturbation of linearization together with the Lp − Lq -estimates of the heat semigroup.
As a by-product of our method, we shall prove the smoothing eﬀect and uniqueness of
our mild solution. In addition, we shall show a blow-up criterion which almost covers
the well-known threshold number 8π of the size ||n0 ||L1 (R2 ) under the rest state of thefluid motion. Furthermore, the blow-up rate will be also discussed. This is based on a
joint work with Professor Hideo KOZONO(Waseda university) and Mr. Masanari MIURA(Kyushu university).
— — —Hiroshi Suito
Graduate School of Environmental and Life Science, Okayama University,
Okayama, Japan

Fluid dynamical and statistical analyses for morphologicalcharacteristics of large arteries8We present numerical experiments related to cardiovascular problems such as the aortic aneurysm, which is one of the life-threatening diseases. In this study, morphologicalcharacterization of blood vessels, which has wide variety among individuals, is introduced.
Diﬀerences in the morphology of the vessels bring about diﬀerences in the flow characteristics, which lead to diﬀerent outcomes.
We consider a number of patient-specific models of the aorta as constructed from CT
scans. We compute the flow fields with Space–Time Variational Multiscale (ST-VMS)
method[1–4].
This work was supported by JST-CREST mathematics program.

References[1] K. Takizawa and T.E. Tezduyar, “Multiscale space–time fluid–structure interaction
techniques”, Computational Mechanics, 48 (2011) 247–267, 10.1007/s00466-011-0571z[2] K. Takizawa, T.E. Tezduyar, A. Buscher and S. Asada, “Space–time interfacetracking with topology change (ST-TC)”, Computational Mechanics, 54(4), pp.955–
971 (2013), doi: 10.1007/s00466-013-0935-7.
[3] K. Takizawa, T.E. Tezduyar, A. Buscher and S. Asada, “Space–time fluid mechanics
computation of heart valve models”, Computational Mechanics, 54(4), pp.973–986(2014), doi: 10.1007/s00466-014-1046-9.
[4] H. Suito, K. Takizawa, V.Q. Huynh, D. Sze, and T. Ueda, “FSI analysis of the bloodflow and geometrical characteristics in the thoracic aorta”, Computational Mechanics,
54(4), pp.1035–1045 (2014), doi: 10.1007/s00466-014-1017-1.
— — —Kenji Takizawa
Department of Modern Mechanical Engineering, Waseda University,
Tokyo, Japan

Space-Time Computational Analysis with NURBS in Space and TimeSpace-Time (ST) Variational Multiscale (ST-VMS) method [1] and its predecessor STSUPS [2] have a good track record in computational analysis of complex fluid-structureinteractions (FSI) and flows with moving boundaries and interfaces (MBI). When an FSI or
MBI problem requires high-resolution representation of boundary layers near solid surfaces,
ALE and ST methods, where the mesh moves to follow the fluid-solid interface, meet that
requirement. Moving-mesh methods have been practical in more classes of complex FSI
and MBI problems than commonly thought of. With a number of complementary methods
introduced recently, the ST methods can now do even more than that. They can deal withcontact between solid surfaces or other topology changes, as enabled by the ST-TC method[3], or a spinning solid surface that is in contact with a solid surface, as enabled by the
ST Slip Interface TC (ST-SI-TC) method [4]. Using NURBS as basis functions is further
9increasing the accuracy and scope of the ST methods [5]. In the ST context, the options forusing NURBS as basis functions are in space (ST-NS), in time (ST-NT), and in space andtime (ST-NST). A general-purpose NURBS mesh generation method introduced recently
makes the use of ST-NS and ST-NST options more practical in computations involving
complex geometries. That practicality is further increased by the ST-SI method [6]. We
present an overview of all these methods and some test computations.

References[1] K. Takizawa and T.E. Tezduyar, Multiscale space-time fluid-structure interaction
techniques, Computational Mechanics, 48, 247-267, 2011.
[2] T.E. Tezduyar, Computation of moving boundaries and interfaces and stabilization
parameters, International Journal for Numerical Methods in Fluids, 43, 555-575,
2003.
[3] K. Takizawa, T.E. Tezduyar, A. Buscher and S. Asada, Space-time interface-tracking
with topology change (ST-TC), Computational Mechanics, 54, 955-971, 2014.
[4] K. Takizawa, T.E. Tezduyar, S. Asada and T. Kuraishi, Space-time method for flow
computations with slip interfaces and topology changes (ST-SI-TC), Computers &Fluids, DOI: 10.1016/j.compfluid.2016.05.006.
[5] K. Takizawa, T.E. Tezduyar, Y. Otoguro, T. Terahara, T. Kuraishi and H. Hattori,
Turbocharger flow computations with the Space-Time Isogeometric Analysis (STIGA), Computers & Fluids, DOI: 10.1016/j.compfluid.2016.02.021.
[6] K. Takizawa, T.E. Tezduyar, H. Mochizuki, H. Hattori, S. Mei, L. Pan and K. Montel,
Space-time VMS method for flow computations with slip interfaces (ST-SI), Mathematical Models and Methods in Applied Sciences, 25, 2377-2406, 2015.
— — —Akifumi Yamaji
Cooperative Major in Nuclear Energy, Graduate School of Advanced
Science and Engineering, Waseda University, Tokyo, Japan

Application of Moving Particle Semi-implicit Method (MPS) for
Analysis of Molten Core Behavior in Severe Accident of Nuclear
Reactors
Moving Particle Semi-implicit Method (MPS) is a Lagrangian based particle method
for incompressible flows. The basic governing equations are Navier-Stokes equation withmass and energy conservation equations and these equations are discretized by the following particle interaction models with a weight function which determines interaction
distance of the particles (calculation points):10

d ! φj − φi(rj − ri )w(|rj − ri |)n0|rj − ri |2j̸=i"#d ! (ϕj − ϕi )(rj − ri )Divergence model : ⟨∇ · ϕ⟩i = 0
w(|rj − ri |)n|rj − ri |2
Gradient model : ⟨∇φ⟩i =

j̸=i

%
2d !
Laplacian model : ∇2 φ i =(φj − φi )w(|rj − ri |)
λn0
$

j̸=i

This method is suitable for tracking evolution of free surfaces involving solid-liquid
phase changes such as melting and solidification. The phase change is currently modeledby ʠ solid fraction ʡ as a function of enthalpy. Some additional models incorporated in
the current MPS method for analysis of melting/solidification includes models for: surface tension, heat conduction, convection (buoyancy), radiation, turbulence (Large Eddy
Simulation), viscosity change, etc. The developed method has been extensively tested
and applied for predicting molten core behavior during severe accident of a nuclear reactor. There is particularly a great need to reveal the precise progression of the Fukushima
accident and the current status of the damaged reactors for decommissioning planning(debris removal from the damaged reactors). Further development and improvement of
the method is desired for mechanistic simulations, which do not rely on parameter adjustments (tuning). Some topics for further model improvements may include (miscellaneous):chemical reaction, eutectic formation, mixing, dissolution, gas generation.
KEYWORDS
Moving Particle Semi-implicit Method (MPS), particle method, Lagrangian method,
incompressible flow, phase change, mechanistic analysis.
— — —Hiroaki Yoshimura
Department of Applied Mechanics and Aerospace Engineering, Waseda
University, Tokyo, JapanThe Lagrangian formulation of nonholonomic mechanical systems
In this talk, we study the Lagrangian formulation of nonholonomic mechanical systems
together with Lagrange-d ʟAlembert-Pontryagin principle. Specifically, we consider thecase in which a configuration manifold is given by a Lie group together with the socalled Lie-Dirac reduction theory. Then, the Lagrange-Dirac systems for nonholonomic
mechanical systems on Lie groups with broken symmetry will be studied to develop implicit
Euler-Poincare-Suslov equations with advected parameters with some illustrative examples
such as the Chaplygin ball and the second-order Rivlin-Ericksen Navier-Stokes fluids. We
will also show some extensions of the Lagrangian variational formulation to the cases inwhich nonequilibrium thermodynamics is included by considering nonlinear nonholonomic
constraints.
— — —
11

Wojciech Zajaczkowski
Polish Academy od Science, Warsaw, Poland

On some free boundary problems to magnetohydrodynamics
We consider the magnetohydrodynamics motion in a domain bounded by a free boundary. In the exterior domain to this motion is generated electromagnetic field by some given
magnetic field. On the free boundary we have the transmision conditions. First we show
existence of local solutions. Next, under some smallness restrictions we prove global existence. We consider incompressible and compressible cases. We use the energy method, so
L2 approach. However we mention on some results in the Lp approach.
— — —

12

Sebastian Elias Graiff Zurita
Balserio Institute, Rio Negro, Argentina

Discrete mechanical systems: existence of trajectories
In this talk we are going to see that discrete mechanical systems, viewed as discretetime dynamical systems, behave in three diﬀerent ways: given an initial condition such asystem can have no solution, a unique solution, or multiple solutions. But, given a mechanical system, we can construct a one-parameter family of discrete mechanical systems
that have a unique solution, in a neighborhood of a given initial condition, for all valuesof the parameter. Our construction uses the alternative formulation of discrete mechanical system given in [1], which is diﬀerent from the usual construction (see, for example,[2]), but equivalent in some sense. Finally, we will discuss shortly the error analysis forthese systems, and we will show that it is possible to extend all the analysis to discretemechanical systems with forces.

References[1] G. W. Patrick, C. Cuell. Error analysis of variational integrators of unconstrained
Lagragian systems. Numer. Math. 113 (2009) 243-264.
[2] J. E. Marsden, M. West. Discrete mechanics and variational integrators. Numerica
Acta 10 (2001) 357-514.
— — —Kai Lee
Graduate School of Mathematical Sciences, University of Tokyo, Tokyo,
Japan

Sharp interface limite for the stochastic Allen-Cahn equations
In this talk, we consider the stochastic Allen-Cahn equation with the Diriclet boundary
condition;⎧⎨u̇ε (t, x) = ∆uε (t, x) + 1 f (uε (t, x)) + Ẇ ε (x), t > 0, x ∈ [−1, 1],tε⎩uε (0, x) = uε (x), x ∈ R, uε (t, ±1) = ±1, t > 0,0with a small parameter ε > 0. This equation describes a behavior of interface and theparameter ε corresponds to a width of interface. We are interested in the behavior of thesolution uε when ε → 0, and we call it the sharp interface limit. In this case, we can expectthat the interface motion at the limit is described by a Brownian motion reflected at theboundary of [−1, 1]. We prove it from the Mosco convergence of Diriclet form which is
associated with the L2 [−1, 1]-valued Markov process uε (t). If we have more time, I would
like to present about my previous work of ”generation of interface”.
13

— — —

Hideki Murakawa
Faculty of Mathematics, Kyushu University, Fukuoka, Japan

Mathematics of cell-cell adhesion: modeling and analysis
Cell adhesion and cell sorting processes are essential in organ formation during embryonic development and in maintaining multicellular structure. Armstrong, Painter andSherratt [J. Theor. Biol. 243 (2006), pp. 98–113] proposed a nonlocal advection-diﬀusion
system as a possible continuous mathematical model for these phenomena. Although thesystem is attractive and challenging, it gives biologically unrealistic numerical solutions.
We identify the problems and change underlying idea of cell movement from “cells move
randomly” to “cells move from high to low pressure regions”. Then we provide a modified
continuous model. Numerical experiments illustrate that the modified model is able toreplicate not only the Steinberg’s cell sorting experiments but also some phenomena which
can not be captured at all by Armstrong-Painter-Sherratt model. Furthermore, we givetheoretical results about the modified model.
— — —Issei Oikawa
Department of Mathematics, Waseda University, Tokyo, Japan

A finite element method for the Stokes equations with a penalized slip
boundary condition
In this talk, we present a penalty method for the Stokes equations with the slip boundary condition and its finite element approximation. When the domain is smooth, it is
known that a variational crime occurs if the slip boundary condition is directly imposed.
The penalty method enables us to avoid the variational crime and is easy to implementby popular finite element softwares, such as FreeFem and FEniCS. The error analysis of
our method and numerical results will be presented. This talk is based on joint work withTakahito Kashiwabara and Guanyu Zhou.
— — —Kaori Onozaki
Department of Applied Mechanics and Aerospace Engineering, Waseda
University, Tokyo, Japan

Trajectory design for space missions in the 4-body system
Low energy trajectories are required for space missions since resources are limited in thespace. We will show the trajectory from the Earth to the Moon under the condition that14

a spacecraft is influenced by gravities of the Sun, the Earth and the Moon. The 4-bodysystem will be decoupled into two diﬀerent 3-body systems with perturbations. Respective
trajectory will be obtained in each perturbed system, and then the Earth-Moon trajectory
will be constructed by connecting these trajectories based on the structures of the stableand unstable manifolds, which exist in both perturbed systems.
— — —Hirokazu Saito
Department of Mathematics, Waseda University, Tokyo, JapanFree boundary problem of a compressible fluid model of Korteweg typeIn this talk, we would like to consider a free boundary problem of a compressible
fluid model of Korteweg type. Korteweg introduced in the early 1900s a new stress tensor
T = S+K with S = µD(v)+(ν−µ) div vI and K = (κ/2)(∆ρ2 −|∇ρ|2 )I−κ∇ρ⊗∇ρ, where
K is called a Korteweg tensor nowadays, in order to treat phase transition phenomena.
We in this talk deal with a model problem on RN+ , N ≥ 2, arising from the free boundary
problem. If there is time, then we will discuss the case of general domains.
— — —Koya Sakakibara
Graduate School of Mathematical Sciences, University of Tokyo, Tokyo,
Japan

Structure-preserving numerical scheme for Hele-Shaw problems by themethod of fundamental solutions
The classical Hele-Shaw problems is description of a motion of viscous fluid in a quasi
two-dimensional space, which was starting from a short paper [1] in 1898 by Henry Selby
Hele-Shaw (1854-1941). In his experiment, viscous fluid is sandwiched between two parallel
plates with a narrow gap, and the apparatus is called Hele-Shaw cell. He succeeded
to visualize streamlines by means of colored water in the cell. The classical Hele-Shaw
problem is stated as follows:15⎧in D(t), t ∈ [0, T ),⎪⎨△p(·, t) = 0p(·, t) = γk(·, t)
on C (t), t ∈ [0, T ),⎪⎩V (·, t) = −∇p(·, t) · N (·, t) on C (t), t ∈ [0, T ),

(0.1)
N
T

C (t)
where △ and ∇ are the Laplace operator and the gradient in R2 ,2xrespectively, D(t) ⊂ R is a bounded region occupied by fluid, C (t)is the boundary of D(t) (positively oriented closed curve), p(·, t) isD(t)the pressure function in D(t), γ is the surface tension coeﬃcient,k(·, t) is the curvature (sign convention is the way that k = 1 if
D(t) is a unit disk), N (·, t) is the unit outward normal vector defined by N (·, t) = −T ⊥ (·, t), and V (·, t) is the normal velocity, on
C (t). See the right figure (in the figure, x is the position veck>0 k<0
tor and T is the unit tangent vector). Here, a · b represents theusual two-dimensional Euclidean inner product for a, b ∈ R2 , anda⊥ = (−b, a)T for a = (a, b)T .
We have three marked properties of the classical Hele-Shaw problem (0.1): the total
length of C (t) is decreasing in time, the enclosed area of D(t) is preserving and thebarycenter of D(t) is being fixed. These properties are called curve-shortening (CS), areapreserving (AP) and barycenter-fixed (BF), respectively. Thus it is natural to consider
that the numerical solution should satisfy these geometrical variational structures in somediscrete sense, however, there does not exist such numerical scheme as far as we know.
The purpose of this talk is that for (0.1) we propose a simple numerical scheme by meansof the method of fundamental solutions combined with the uniform distribution method.
As a result, we have succeeded to construct numerical scheme which satisfies CS-, AP- and
BF-properties exactly, in which we have to solve system of “quadratic” equations, and theone which satisfies CS-, AP- and BF-properties asymptotically, in which we have to solve
system of “linear” equations. Our scheme can also be applied for other vatiarions: theone-phase exterior problem, the one-phase interior Hele-Shaw problem with sink/source
points, two-phase Hele-Shaw problem, and so on.
The contents of this talk are based on the joint work with Professor Shigetoshi Yazaki
of Meiji University [2].

References[1] H. S. Hele-Shaw, The flow of water, Nature 58 (1898), 34–36.
[2] K. Sakakibara and S. Yazaki, Structure-preserving numerical scheme for the one-phase
Hele-Shaw problems by the method of fundamental solutions, submitted revised version.
— — —Yukihito Suzuki
Department of Mathematics, Waseda University, Tokyo, JapanA bracket formulation and structure preserving numerical method for
diﬀuse interface models

16In this talk, a bracket formulation for the Navier-Stokes/Cahn-Hilliard equations,which employs a skew-symmetric Poisson bracket and a symmetric negative semidefinite dissipative bracket, will be proposed. We will also discuss a numerical method which
properly inherits the conservation laws of energy, helicity and enstrophy derived from thebracket formulation, as well as the de Rham complex on a domain in thee dimensional
Euclidean space. Some numerical examples will be shown.
— — —17