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

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.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

3-minute presentations of all partcipants in the alphabetic order.

3-minute presentations of all partcipants in the alphabetic order.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

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!

https://www.mimuw.edu.pl/~rybka/birs/AbstracBanffMazon.pdf

https://www.mimuw.edu.pl/~rybka/birs/AbstractBERS2018-Manfredi.pdf

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.

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.

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).

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.

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.

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

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.

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.

https://www.mimuw.edu.pl/~rybka/birs/abstract_deckelnick.pdf

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.

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)

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.

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.

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).

We share impression and we sum up the two days.

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.

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?

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.

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.

https://www.mimuw.edu.pl/~rybka/birs/mucha-tv1.pdf

https://www.mimuw.edu.pl/~rybka/birs/Abstract_Moll.pdf

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.

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.

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 = ½. 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.

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.

https://www.mimuw.edu.pl/~rybka/birs/Lasica_abstract.pdf

https://www.mimuw.edu.pl/~rybka/birs/KShirakawa_title-abstract.pdf

https://www.mimuw.edu.pl/~rybka/birs/MBonforte.pdf

This talkwas delivered by popular demand during the wrap-up session.

Talks of those who feel that they could improve their presentation or otherwise feel they talk more. This will be scheduled on site.

The summary of the conference

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.