# Registered Data

## [00072] Evolution equations in materials science: Multiscale modeling, analysis, and simulation

**Session Date & Time**:- 00072 (1/3) : 3C (Aug.23, 13:20-15:00)
- 00072 (2/3) : 3D (Aug.23, 15:30-17:10)
- 00072 (3/3) : 3E (Aug.23, 17:40-19:20)

**Type**: Proposal of Minisymposium**Abstract**: Materials science has become increasingly efficient and contributes with new products. The increased material functionality relies on good experimental grip on microstructure evolution. Mathematics plays a crucial role in using experimental understanding to shed light where experiments are inaccessible. Mathematical challenges are though unsolved. Elastic porous materials have many practical applications, however the mathematical treatment of elasticity equations for realistic media is underdeveloped as the small-strains-hypothesis needs to be adopted while the porosity of real materials (e. g. when biology is involved) disagrees. Our symposium focuses on the development of advanced mathematical methodologies applicable to materials having complex microstructures.**Organizer(s)**: Toyohiko Aiki, Adrian Muntean**Classification**:__35Exx__,__74-XX__**Speakers Info**:- Adrian Muntean (Karlstad University)
- Chiharu Kosugi (Japan Women's University)
- Akiko Morimura (Japan Women's University)
- Kota Kumazaki (Nagasaki University)
- Yusuke Murasee (Meijo University)
- Yoshiho Akagawa (National Institute of Technology Gifu College)
- Grigor Nika (Karlstad University)
- Michael Eden (Karlstad University)
- Shuji Yoshikawa (Oita University)
- David Wiedemann (University of Augsburg)
- Rainey Lyons (Karlstad University)

**Talks in Minisymposium**:**[03965] Solvability of a dynamical model for the elastic curves****Author(s)**:**Chiharu Kosugi**(Yamaguchi University)- Toyohiko Aiki (Japan Women's Univeristy)

**Abstract**: To establish a mathematical model for stretching and shrinking motions of the compressible elastic curves like rubber bands, we discuss problems whose feature is that the strain function is nonlinear and non-smooth, and the stress function has singularity. For the problem, thanks to a priori estimates for the strain from below and center of mass globally in time we can show results on existence, uniqueness, and large time behavior of solutions.

**[04211] Effective hydromechanic models for fibre-reinforced hydrogels****Author(s)**:**Michael Eden**(Karlstad University)- Hari Shankar Mahato (IIT Kharagpur)

**Abstract**: We consider highly heterogeneous two-component media composed of a connected fibre-scaffold with periodically distributed inclusions of hydrogel. While the fibres are assumed to be elastic, the hydromechanical response of hydrogel is modeled via Biot’s poroelasticity. We show that the resulting mathematical problem admits a unique weak solution and investigate the limit behavior of the solutions with respect to a scale parameter characterizing the heterogeneity of the medium.

**[04217] Morphology formation in ternary mixtures: A continuum model****Author(s)**:**Rainey Lyons**(Karlstad University)

**Abstract**: We study the ability of a coupled nonlocal system of two quasilinear parabolic partial differential equations to produce phase separation patterns. This system is derived in the literature as the rigorous hydrodynamic limit of a suitably scaled interacting particle system of Blume–Capel–type driven by Kawasaki dynamics. In this talk, we will discuss the potential of the model to produce morphologies, the growth of these patterns, well-posedness, and regularity of solutions.

**[04230] A two-scale model describing swelling phenomenon in porous materials****Author(s)**:**Kota Kumazaki**(Kyoto University of Education)- Adrian Muntean (Karlstad University)

**Abstract**: In this talk, we propose a two-scale model describing the swelling phenomenon in porous materials. This model consists of a diffusion equation for the relative humidity distributed in materials and a free boundary problem describing the swelling process in microscopic pores. We consider each microscopic pore as a one-dimensional interval and correspond the interval to each point of materials. In this talk, we discuss the global solvability of this model.

**[04239] Partial differential equations for moisture transport in porous materials****Author(s)**:**Akiko Morimura**(Japan Women's University)- Toyohiko Aiki (Japan Women's Univeristy)

**Abstract**: We consider the initial-boundary value problems for nonlinear parabolic equations describing moisture transport in a porous material. As a first step in this research, we suppose that the mass distribution in air is given and propose the problem for equation called ellipitic-parabolic type. The purpose of this talk is to establish existence and uniqueness of solutions to the approximate problem by applying the evolution equation theory and the standard fixed-point argument.

**[04388] Decay estimates for a unit cell model of composite materials****Author(s)**:**Shuji Yoshikawa**(Oita University)

**Abstract**: We shall introduce decay estimates for a unit cell model of composite materials. The model we study here is the one of toy models, but the result corresponds to the first step of a mixture of transmission and waveguide problems from the viewpoint of PDEs.

**[04483] Improved corrector regularity in homogenization with non-smooth coefficients****Author(s)**:**Grigor Nika**(Karlstad University)

**Abstract**: We propose an advanced model of microscopic heat conduction, capable of addressing size effects in heterogeneous media. By leveraging sound scaling arguments, we enhance the differentiability of the corrector in the classical problem of periodic homogenization of linear elliptic equations in three dimensions. This enables us to elucidate the crucial role that correctors play in quantifying the differences between the microscopic and macroscopic solutions in heterogeneous media. Furthermore, if the data are of the form $f={\rm div}~{\rm \bf F}$ with ${\rm \bf F} \in {\rm L}^{3}(\Omega, \mathbb{R}^3)$, then we recover a stronger version of the classical corrector convergence theorem.

**[04592] Numerical simulations and analysis for mathematical modeling of adsorption phenomena****Author(s)**:**Yusuke Murase**(Meijo University)

**Abstract**: In this talk, we discuss numerical simulations ans mathematical properties of mathematical modeling for adsorption phenomena and modeling for moisture transport in concrete material. The adsorption model is a free boundary problem composed by heat equation and free boundary equation, and the moisture transport model is combined with adsorption model and diffusion equations. Which is proposed by T. Aiki, K. Kumazaki, N. Sato, and Y. Murase.

**[04680] An elastoplastic model with a time-dependent threshold function****Author(s)**:**Yoshiho Akagawa**- Takeshi Fukao (Ryukoku University)
- Risei Kano (Kochi University)

**Abstract**: We investigate the well-posedness of an elastoplastic model described by quasi-variational inequalities, applying the abstract theory of evolution equation. The prototype model is introduced by Duvau-Lions. It is characterized by the constraint for the deviatoric part of the stress tensor. In the case when the constraints depend on time and some unknown strain history, then the model represents more realistic phenomena. This is joint work with Risei Kano (Kochi University), and Takeshi Fukao (Ryukoku University).

**[04859] Diffusion in the presence of microstructures: Does vesicle micro-dynamics enhance the signalling among plants macro-transport?****Author(s)**:**Adrian Muntean**(Karlstad University)- Sander Hille (Leiden University)
- Omar Richardson (Simula Consulting)

**Abstract**: We study a diffusion-drift problem for signalling among plants in the context of measure-valued equations. We show preliminary results concerning the modelling and mathematical analysis of scenarios involving the macroscopic diffusion of signalling molecules enhanced by a finite number of microvesicles, with own dynamics able to capture and release signals as a relay system. The macro-micro coupling relies on a two-scale transmission condition. Mild solutions will turn to exist and will be positive weak solutions.

**[05044] Homogenisation of an advection–reaction–diffusion process in a porous medium with coupled evolving microstructure****Author(s)**:- Markus Gahn (Heidelberg University)
- Malte A. Peter (University of Augsburg)
- Iuliu Sorin Pop (Hasselt University)
**David Wiedemann**(University of Ausgburg)

**Abstract**: We consider the homogenization of an advection–reaction–diffusion equation in an evolving porous medium. The microstructure's evolution is coupled with the unknown concentration of a substance, resulting in a free boundary value problem. To rigorously pass to the homogenization limit, we transform the problem into a periodic fixed domain, which results in a highly non-linear problem. We then pass to the homogenization limit.