# Schedule for: 18w5033 - Advanced Developments for Surface and Interface Dynamics - Analysis and Computation

Beginning on Sunday, June 17 and ending Friday June 22, 2018

All times in Banff, Alberta time, MDT (UTC-6).

Sunday, June 17 | |
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16:00 - 17:30 | Check-in begins at 16:00 on Sunday and is open 24 hours (Front Desk - Professional Development Centre) |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

20:00 - 22:00 | Informal gathering (Corbett Hall Lounge (CH 2110)) |

Monday, June 18 | |
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07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

08:45 - 09:00 |
Introduction and Welcome by BIRS Staff ↓ A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions. (TCPL 201) |

09:00 - 10:00 |
short introductions, Bonforte -- Margetis ↓ 3-minute presentations of all partcipants in the alphabetic order. (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:30 |
short introductions, Mazon -- Zahedi ↓ 3-minute presentations of all partcipants in the alphabetic order. (TCPL 201) |

11:30 - 13:00 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

13:00 - 14:00 |
Guided Tour of The Banff Centre ↓ Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus. (Corbett Hall Lounge (CH 2110)) |

14:00 - 14:20 |
Group Photo ↓ Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo! (TCPL 201) |

14:20 - 15:00 |
Jose Mazon: Heat Flow on Metric Random Walk Spaces ↓ https://www.mimuw.edu.pl/~rybka/birs/AbstracBanffMazon.pdf (TCPL 201) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

15:30 - 16:00 |
Juan J. Manfredi: A DISCRETE STOCHASTIC INTERPRETATION OF THE DOMINATIVE p -LAPLACIAN ↓ https://www.mimuw.edu.pl/~rybka/birs/AbstractBERS2018-Manfredi.pdf (TCPL 201) |

16:00 - 16:30 |
Bjoern Stinner: On a diffuse interface approach to PDEs on surfaces and networks ↓ Diffuse interface models based on the phase field methodology have been developed for various free boundary problems. They involve representing the interfaces by thin layers. Some applications feature phenomena on the interfaces described by PDEs for interface resident fields. We will address the questions of how to model such phenomena in the diffuse interface setting and how to numerically approximate the solutions, which may require special consideration due to degeneracies. The approach can be generalised to networks and bubble clusters. One key challenge then is to correctly recover the conditions in the triple junctions formed by three interfaces. The research is motivated by surface active agents (surfactants) in multi-phase flow which can be effectively modelled by Cahn-Hilliard-Navier-Stokes systems. Some preliminary results for a novel numerical scheme will be presented, as well as some simulations to support the theoretical findings. (TCPL 201) |

16:30 - 17:00 |
Catherine Kublik: Applications of signed distance functions to motions of curves in the plane and integration over curves and surfaces ↓ We present two simple and efficient algorithmic frameworks using signed distance functions. The first one is a numerical scheme for computing certain area reserving geometric motion of curves in the plane, such as area preserving motion by curvature. The second is a general technique for integrating over curves and surfaces that are not defined by any explicit parameterization. This method is based on rewriting the curve or surface integral as a volumetric integral formulation over the ambient space. We show applications of both methods. (TCPL 201) |

17:00 - 17:30 |
Nao Hamamuki: A dynamic boundary value problem of the level-set mean curvature flow equation ↓ A dynamic boundary value problem of the level-set mean curvature flow equation is discussed.
We first give a unique existence result of viscosity solutions for more general singular degenerate parabolic equations. The
comparison principle is established by employing a so-called flattening argument to avoid a singularity of the equation, while we
prove existence of solutions by Perron's method.
We also provide a deterministic discrete game interpretation for this problem. The original version of this game was introduced by
Kohn and Serfaty (2006) for the case with no boundary, and we propose a modified game including a kind of reflection near the
boundary so that the corresponding value functions converge to a viscosity sub-/supersolution satisfying a dynamic boundary
condition.
This talk is based on a joint work with Y. Giga (The University of Tokyo) and Q. Liu (Fukuoka University). (TCPL 201) |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

Tuesday, June 19 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 09:30 |
Inwon Kim: Evolution of star-shaped sets in Mean curvature flow with forcing ↓ I will discuss on the evolution of sets in mean curvature flows with forcing, where the forcing term encourages the sets
to have finite, non-zero volume. We will focus on preservation of a strong version of star-shaped geometry throughout the evolution,
which enables us to discuss stability and long-time behavior of the flow. This is joint work with Dohyun Kwon. (TCPL 201) |

09:30 - 10:00 |
Chia-Chieh Chu: Numerical methods for energy minimization problems on surfaces ↓ Partial differential equations on surfaces have wild applications in many areas, such as material science, surfactant problems,
image processing and biology. These PDEs usually originate from minimizing energy functions defined on surfaces. This work
targets applications that use implicit or non-parametric representations of closed surfaces or curves and require numerical
solution for minimization problems defined on the surfaces.
In the previous work, we showed that the energy function defined on surfaces can be extended to the energy function defined on
the nearby tubular neighborhood that gives the same energy when input the constant-along-normal extension. Furthermore, the
extended energy function gives the same minimizer as which the original energy function gives in the sense of restriction on the
surface. This new approach connects the original energy function to an extended energy function and provides a good framework to
solve PEDs numerically on Cartesian grids.
In this talk, we continue the results in previous work to develop a framework to solve more challenging problems defined on
surfaces, such as nonlinear and non-convex energy problems. We propose a direct approach to solve the minimization problems.
Instead of deriving Euler-Lagrange, we discretize the energy functions on Cartesian grids, then find the minimizer of the
discrete energy function. The advantages of this approach include: more compact stencils, easy to implement and be able to
combine with modern techniques on nonlinear/non-convex optimizations. We demonstrate that our method can be applied successfully
on the TV-denoising and obstacle problems defined on surfaces.
This is a joint work with Richard Tsai and Chun-Chieh Lin (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:00 |
John R. King: Moving boundary problems for biological tissue growth ↓ Macroscale (PDE) formulations for tissue growth lead to a variety of novel moving boundary problems. The application of asymptotic
methods (such as thin-rim limits) in analysing their properties will be described. (TCPL 201) |

11:00 - 11:30 |
Nestor Guillen: Free boundary problems as parabolic Integro-differential equations ↓ We demonstrate that a class of one and two phase free boundary problems can be
recast as nonlocal parabolic equations on a codimension one submanifold. The
canonical examples would be one-phase Hele-Shaw and Laplacian growth. In the
special class of free boundaries that are graphs over $\mathbb{R}^d$, we give a
precise characterization that shows their motion is equivalent to that of a
solution of a nonlocal (fractional) and nonlinear parabolic equation in
Euclidean space. Our main observation is that the free boundary condition
defines a nonlocal operator having what we call the Global Comparison Property.
A consequence of the connection with nonlocal parabolic equations is that for
free boundary problems arising from translation invariant elliptic operators in
the positive and negative phases, one obtains, in a uniform treatment for all of
the problems, a propagation of modulus of continuity for weak solutions of the
free boundary flow. This is based on joint works with Hector Chang-Lara and
Russell Schwab. (TCPL 201) |

11:30 - 13:30 | Lunch (Vistas Dining Room) |

13:30 - 14:00 |
Klaus Deckelnick: Title: Error analysis for a diffuse interface approach to an advection–diffusion equation on a moving surface ↓ https://www.mimuw.edu.pl/~rybka/birs/abstract_deckelnick.pdf (TCPL 201) |

14:00 - 14:30 |
Selim Esedoglu: Algorithms for multiphase, anisotropic motion by mean curvature ↓ I will discuss extensions of Merriman, Bence, and Osher's threshold
dynamics algorithm to anisotropic motion by mean curvature of networks.
This class of algorithms aims to generate the desired evolution simply
by alternating convolution and thresholding operations. In its full
generality, the N-phase evolution specifies potentially distinct, normal
dependent mobilities and surface tensions for each interface in the
network, for a total of N-choose-2 pairs. One of the novelties I will
report is how to "bake" the desired anisotropic mobility and anisotropic
surface tension into the convolution kernel of the algorithm. This is
new even in the simplest two-phase setting, and builds upon previous
work with Felix Otto on the variational formulation of threshold
dynamics schemes. (TCPL 201) |

14:30 - 15:00 |
Robert Nürnberg: Numerical approximation of axisymmetric geometric evolution equations ↓ We present variational formulations of mean curvature flow and
surface diffusion for axisymmetric hypersurfaces in R^3.
On recalling important properties of the schemes introduced by the authors
for the corresponding geometric evolution equations for closed curves in
the plane, we introduce suitable finite element approximations, and
investigate their stability and vertex distribution properties.
(joint work with John W. Barrett and Harald Garcke) (TCPL 201) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

15:30 - 16:00 |
Antonin Chambolle: Recent results on crystalline curvature flows ↓ We review our recent strategy to prove existence, uniqueness as well as the convergence of a time discrete scheme to the
curvature flow of a perimeter defined by a non-smooth (crystalline) surface tension. This is joint work with M. Morini, M.
Novaga, M. Ponsiglione. (TCPL 201) |

16:00 - 16:30 |
Sara Zahedi: Cut Finite Element Methods ↓ I will present a new type of finite element methods that we refer to as Cut Finite Element Methods (CutFEM). CutFEM provides
an efficient strategy for solving partial differential equations in evolving geometries. We will consider simulations of
two-phase flow problems. CutFEM allows interfaces separating immiscible fluids to cut through a background mesh in an
arbitrary fashion and hence re-meshing processes are avoided while giving optimal convergence order. Stabilization terms are
added to the weak formulation to guarantee that the linear systems resulting from the method have bounded condition numbers
independently of how the geometry cuts through the background mesh. (TCPL 201) |

16:30 - 17:00 |
Monika Muszkieta: The forward-backward scheme for the minimizing total variation flow in $H^{-s}$. ↓ In the talk, we consider a gradient flow of the total variation in the negative Sobolev space $H^{-s}$, $s\in[0,1]$, under the
periodic boundary condition. We derive a dual formulation of a convex variational problem associated with a semi-implicit time
discretization of this flow. Based on the forward-backward scheme, we construct a minimizing sequence of a given functional and
discuss issues concerning its convergence. We also show and compare results of numerical experiments for simple initial data and
different values of the index s.
This is joint work with Y. Giga (University of Tokyo) and P. Rybka (University of Warsaw). (TCPL 201) |

17:00 - 17:30 |
wrap up ↓ We share impression and we sum up the two days. (TCPL 201) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Wednesday, June 20 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 09:30 |
Jack Xin: Curvature effect in shear flow: slowdown of flame speeds with Markstein number ↓ A well-known folklore in combustion community is that curvature effect in general slows down flame propagation speeds because it
smooths out wrinkled flames. As the first theoretical result in this direction, we prove that the effective flame speed is
decreasing with respect to curvature diffusivity (Markstein number) for shear flows in the level-set G-equation model. The proof
involves several novel and rather sophisticated inequalities arising from the nonlinear structure of the equation. We also show
similar phenomenon in non-shear flows numerically. This is joint work with Jiancheng Lyu and Yifeng Yu. (TCPL 201) |

09:30 - 10:00 |
Dionisios Margetis: On the mathematical modeling of crystal facets: A PDE approach with a touch of discreteness ↓ Advances in materials science have enabled the observation and control of microstructures such as nanoscale defects with remarkable precision. In this talk, I will discuss recent progress and open challenges in understanding how small-scale details in the kinetics of crystal surfaces can macroscopically influence the surface morphological evolution. In particular, the evolution of crystal surface plateaus, facets, is characterized by an effective behavior at the macroscale that is not necessarily only the outcome of averaging, but instead may be dominated by certain discrete (microscopic) events. This "idiosyncrasy" of facet evolution raises
challenging but interesting mathematical questions. My talk will explore via selected examples and methods how the kinetics of microscale defects near facets can plausibly leave their imprints to continuum-scale problems, at larger scales. The main tool is a fourth-order nonlinear PDE for the surface height profile. I will address and provide an answer to the question: Can this continuum description be reconciled with the motion of line defects (steps) at the nanoscale, and how? (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:00 |
Jian-Guo Liu: Dynamics of a degenerate PDE model of epitaxial crystal growth ↓ Epitaxial growth is an important physical process for forming solid films or
other nano-structures. It occurs as atoms, deposited from above, adsorb and
diffuse on a crystal surface. Modeling the rates that atoms hop and break
bonds leads in the continuum limit to degenerate 4th-order PDE that involve exponential
nonlinearity and the p-Laplacian with p=1, for example. We discuss a number of
analytical results for such models, some of which involve subgradient dynamics
for Radon measure solutions. (TCPL 201) |

11:00 - 11:30 |
Yuan Gao: Analytic solution to nonlocal Peirtls-Nabarro models ↓ The Peierls-Nabarro model was first introduced by Peierls Nabarro to describe the continuum model of dislocation in
materials.
It incorporates the atomic effect into the continuum framework and give us an understanding of dislocation core structure. Our main
goal is to determine the displacement in the whole space using Perierls-Nabarro (PN) model, which incorporates the atomic effect
into the continuum framework. We focus on the existence of analytic solution to Peierls-Nabarro model including stationary model
and dynamic model. Since we incorporate the misfit surface energy into total energy of the system, which leads to a nonlinear
boundary condition, the strategy is to first decouple the displacement field and then to uniquely solve a nonlocal equation on
boundary. (TCPL 201) |

11:30 - 13:30 | Lunch (Vistas Dining Room) |

11:30 - 11:40 | Kazutoshi Taguchi: a student talk (TCPL 201) |

13:30 - 17:30 | Free Afternoon (Banff National Park) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Thursday, June 21 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 09:30 |
Piotr Mucha: Anisotropic TV flow in 2D case ↓ https://www.mimuw.edu.pl/~rybka/birs/mucha-tv1.pdf (TCPL 201) |

09:30 - 10:00 |
Salvador Moll: The 1 -dimensional relativistic heat equation: some new result ↓ https://www.mimuw.edu.pl/~rybka/birs/Abstract_Moll.pdf (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:00 |
Takeshi Ohtsuka: Minimizing movement approach using general level set functions for evolving spirals by crystalline curvature flow ↓ In this talk we consider evolving spirals by crystalline
eikonal-curvature flow in the plane. Our focus on this issue is to
propose a framework to treat the evolution of several spirals with
singular diffusion by L^1 type regularization, and merging with each
other. For this purpose, we consider this motion with the minimizing
movement approach based on the algorithm proposed by Chambolle in
2004. This approach considers the motion of interfaces as the
minimizing movement of the singular surface energy and distance from
the original interface. Note that the distance is measured by the
signed distance function of the interface. However, the signed
distance is not well-defined since a spiral curve does not divide the
domain into inside and outside of the curve. To overcome this issue,
we introduce two approaches; construct a signed distance function only
around the curve, or using general level set function for spirals
instead of the signed distance. We present several numerical results,
which is established with the split Bregman iteration.
This is a joint work with Y.-H. R. Tsai. (TCPL 201) |

11:00 - 11:30 |
Norbert Pozar: Viscosity solutions for the crystalline mean curvature flow ↓ In this talk I will present recent results concerning the level set formulation of the crystalline mean curvature flow. In a joint
work with Yoshikazu Giga, we introduce a new notion of viscosity solutions for this problem and establish its well-posedness for
compact crystals in an arbitrary dimension as well as the stability with respect to an approximation by a smooth anisotropic mean
curvature flow. Since the crystalline mean curvature might not be defined even for smooth surfaces, we consider a restricted class
of "faceted" test functions and show that it is sufficiently large to yield a general comparison principle. (TCPL 201) |

11:30 - 13:30 | Lunch (Vistas Dining Room) |

13:30 - 14:00 |
Vaughan Voller: Anomalous Infiltration into Heterogeneous Porous Media: Simulations and Fractional Calculus Models ↓ There has been some recent interest in exploring applications of fractal calculus in transport models. One of the
motivations for this is that such models are able to generate anomalous transport signals. For example, when
fractional calculus is employed to define diffusion transport fluxes (heat, mass etc.) the exponent n in the
space-time scaling differs from the classical value of n = Â˝. In this talk we have two objectives. The first
objective is to identify physically realizable systems that exhibit anomalous transport behaviors. The second is to
arrive at suitable fractional governing equations that can model these systems. To these ends we will build direct
simulations of the infiltration of moisture into a porous media containing a distribution of flow obstacles. When
the obstacles form a repeating pattern, this problem can be viewed as a limit case of the classical one-phase Stefan
melting problem and the measure of the advance of the infiltration length changes with the square root of time.
When the obstacles are distributed in a fractal pattern, however, the infiltration shows a sub-diffusive behavior,
where the time exponent is less that the square root. Through considering the time scaling of Brownian motion in a
fractal obstacle filed we are bale to directly associate this sub-diffusive time exponent to the fractal dimension
of the obstacle filed. This in turn, allows us to develop fractional calculus based governing equations, with a
closed particular solution, for moisture infiltration into a fractal obstacle field. The talk will close with
considerations as to how these findings can be associated with more general Stefan problem that incorporated
fractional calculus treatments. (TCPL 201) |

14:00 - 14:30 |
Tokinaga Namba: Well-posedness of fully nonlinear PDEs with Caputo time fractional derivatives ↓ We will introduce an extended notion of viscosity solutions for initial-boundary value problems of second order fully
nonlinear PDEs that include Caputo time fractional derivatives of order less than one. As is the integer-order case, the unique
existence is established by the comparison principle and Perron's method. Stability with respect to the order of time derivative
is provided by the half-relaxed limit method. (TCPL 201) |

14:30 - 15:00 |
Michał Łasica: Existence of 1-harmonic flow ↓ https://www.mimuw.edu.pl/~rybka/birs/Lasica_abstract.pdf (TCPL 201) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

15:30 - 16:00 |
Ken Shirakawa: Phase field models of grain boundary motions with dynamic boundary conditions ↓ https://www.mimuw.edu.pl/~rybka/birs/KShirakawa_title-abstract.pdf (TCPL 201) |

16:00 - 16:30 |
Matteo Bonforte: Nonlinear and Nonlocal Degenerate Diffusions on Bounded Dom ains ↓ https://www.mimuw.edu.pl/~rybka/birs/MBonforte.pdf (TCPL 201) |

17:00 - 17:30 |
Yoshikazu Giga: On large time behavior of growth by birth and spread ↓ This talkwas delivered by popular demand during the wrap-up session. (TCPL 201) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Friday, June 22 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 10:00 |
Second chance talks ↓ Talks of those who feel that they could improve their presentation or otherwise feel they talk more. This will be scheduled on site. (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:00 |
wrap up ↓ The summary of the conference (TCPL 201) |

11:30 - 12:00 |
Checkout by Noon ↓ 5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon. (Front Desk - Professional Development Centre) |

12:00 - 13:30 | Lunch from 11:30 to 13:30 (Vistas Dining Room) |