# Registered Data

## [01272] Interface motion and related topics

**Session Date & Time**:- 01272 (1/2) : 4D (Aug.24, 15:30-17:10)
- 01272 (2/2) : 4E (Aug.24, 17:40-19:20)

**Type**: Proposal of Minisymposium**Abstract**: Understanding the interface dynamics, such as in crystal growth and moving boundaries between two phases, is a topic of great mathematical interest and is also important from engineering, materials science, and other perspectives. This mini-symposium aims to bring together researchers working on interfacial motion and related topics to share recent advances and discuss this domain of interest. Topics to be presented include mathematical analysis of interface behavior, numerical methods specific to interface motion, and applied research such as the shape optimization.**Organizer(s)**: Michal Beneš, Tetsuya Ishiwata**Classification**:__35R35__,__65N99__,__76D27__**Speakers Info**:- Michal Beneš (Czech Technical University in Prague Faculty of Nuclear Sciences and Physical Engineering)
- Daniel Ševčovič (Comenius University Bratislava, Faculty of Mathematics, Physics and Informatics)
- Cyril Izuchukwu Udeani (Comenius University Bratislava, Faculty of Mathematics,Physics and Informatics)
- Miroslav Kolář (Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering)
- Koya Sakakibara (Okayama University of Science / RIKEN)
- Shunsuke Kobayashi (Faculty of Engineering, University of Miyazaki)
- Tomoya Kemmochi (Graduate School of Engineering, Nagoya University)
- Julius Fergy Rabago (Faculty of Mathematics and Physics, Institute of Engineering, Kanazawa University)

**Talks in Minisymposium**:**[02736] Numerical computation of the Plateau problem by the method of fundamental solutions****Author(s)**:**Koya Sakakibara**(Okayama University of Science / RIKEN)- Yuuki Shimizu (The University of Tokyo)

**Abstract**: We propose a numerical scheme for the Plateau problem based on the method of fundamental solutions. After giving the existence of approximate surfaces and convergence analysis, some numerical experiments show the usefulness of the proposed scheme.

**[02751] Motion of Space Curves by Binormal and Normal Curvature****Author(s)**:**Michal BENEŠ**(Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague)- Miroslav KOLÁŘ (Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague)
- Daniel ŠEVČOVIČ (Faculty of Mathematics, Physics and Computer Science, Comenius University in Bratislava)

**Abstract**: We discuss the motion of closed non-intersecting space curves by curvature in binormal and normal directions with application in vortex dynamics. We formulate the general motion law in space by binormal and normal curvature and mention its analytical properties. The finite-volume scheme allows to solve the motion numerically with stabilization by the tangential velocity redistributing discretization nodes. We demonstrate behavior of the solution on several computational studies combining normal and binormal velocity and mutual interactions.

**[02845] Numerical solution to a free boundary problem for the Stokes equation using the coupled complex boundary method in shape optimization settings****Author(s)**:**Julius Fergy Tiongson Rabago**(Kanazawa University)- Hirofumi Notsu (Kanazawa University)

**Abstract**: A new reformulation of a free boundary problem for the Stokes equations governing a viscous flow is proposed. Using the shape derivative of the cost associated with the new cost functional, a Sobolev-gradient based descent method is employed to solve the shape optimization problem. For validation and evaluation of the method, numerical experiments are carried out both in two and three dimensions which are compared with the ones obtained via the classical Dirichlet-data tracking approach.

**[02939] Mathematical modeling of flame/smoldering front-evolution and its application****Author(s)**:**Shunsuke Kobayashi**(University of Miyazaki)- Shigetoshi Yazaki (Meiji University)
- Kazunori Kuwana (Tokyo University of Science)

**Abstract**: The Kuramoto--Sivashinsky equation is well-known as a mathematical model describing the interfacial dynamics of combustion phenomena. In this talk, we focus on flame spreading over thin solid fuels and report the results of applying the Kuramoto--Sivashinsky equation to the following two topics: 1. the behavior of flame/smoldering fronts expanding circle with time. 2. the spreading speed of flame front in a spatially non-uniform region with a bellows shape.

**[03222] Structure-preserving numerical methods for gradient flows of planar closed curves****Author(s)**:**Tomoya Kemmochi**(Nagoya University)- Yuto Miyatake (Osaka University)
- Koya Sakakibara (Okayama University of Science)

**Abstract**: In this talk, we consider numerical approximation of constrained gradient flows of planar closed curves. We will develop structure-preserving methods for these equations that preserve both the dissipation and the constraints. To preserve the energy structures, we introduce the discrete version of gradients according to the discrete gradient method and determine the Lagrange multipliers appropriately. Some numerical examples are presented to verify the efficiency of the proposed schemes.

**[04050] Novel numerical methods for solving nonlinear evolutionary equations with application in mathematical finance optimization problems****Author(s)**:**Cyril Izuchukwu Udeani**(Comenius University in Bratislava)- Daniel Sevcovic (Comenius University)

**Abstract**: This employs physics-informed DeepONet (PI-DeepONet), which incorporates known physics into the neural network via two networks, to approximate the solution operator of a nonlinear Hamilton--Jacobi--Bellman (HJB) equation arising from the stochastic optimization problem, where an investor's goal is to maximize the conditional expected value of the terminal utility. We first transform the nonlinear HJB equation into a quasilinear parabolic equation using the Ricatti transform and then approximate the solution of the transformed equation using PI-DeepONet.

**[04059] Multidimensional partial integro-differential equation in Bessel potential spaces with applications****Author(s)**:**Daniel Sevcovic**(Comenius University)- Cyril Izuchukwu Udeani (Comenies University)

**Abstract**: In the talk we analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. We employ the theory of abstract semilinear parabolic equations in order to prove existence and uniqueness of solutions in the scale of Bessel potential spaces. We prove existence and uniqueness of a solution to the PIDE. As an application to option pricing in the one-dimensional space, we consider a general shift function arising from nonlinear option pricing models.

**[04426] Qualitative and numerical aspects of dynamics of diffusion and transport mechanisms on evolving curves****Author(s)**:**Miroslav Kolar**(Czech Technical University in Prague)

**Abstract**: In this talk we discuss the model of diffusion and transport acting on evolving curves. The model is coupled with the geometrical evolution equation for moving interfaces in form $$ \text{normal velocity} = \text{curvature} + \text{force}.$$ This model is being developed within the context of study of the vortex dynamics. The technique of incorporation an artificial tangential velocity for the stabilisation of numerical calculations is discussed and qualitative and quantitative computational results in 2D and 3D are shown.