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Masaaki Uesaka

Publications and source records attributed to Masaaki Uesaka.

7 recordsLinked to original sources
arXiv2023

Fractional time differential equations as a singular limit of the Kobayashi-Warren-Carter system

This paper is concerned with a singular limit of the Kobayashi-Warren-Carter system, a phase field system modelling the evolutions of structures of grains. Under a suitable scaling, the limit system is formally derived when the interface thickness parameter tends to zero. Different from many other problems, it turns out that the limit system is a system involving fractional time derivatives, although the original system is a simple gradient flow. A rigorous derivation is given when the problem is reduced to a gradient flow of a single-well Modica-Mortola functional in a one-dimensional setting.

math.AP↗
arXiv2022

On a singular limit of the Kobayashi--Warren--Carter energy

By introducing a new topology, a representation formula of the Gamma limit of the Kobayashi-Warren-Carter energy is given in a multi-dimensional domain. A key step is to study the Gamma limit of a single-well Modica-Mortola functional. The convergence introduced here is called the sliced graph convergence, which is finer than conventional $L^1$ convergence, and the problem is reduced to a one-dimensional setting by a slicing argument.

math.AP↗
arXiv2020

A new numerical scheme for constrained total variation flows and its convergence

In this paper, we propose a new numerical scheme for a spatially discrete model of constrained total variation flows, which are total variation flows whose values are constrained in a Riemannian manifold. The difficulty of this problem is that the underlying function space is not convex and it is hard to calculate the minimizer of the functional with the manifold constraint. We overcome this difficulty by "localization technique" using the exponential map and prove the finite-time error estimate in general situation. Finally, we show a few numerical results for the cases that the target manifolds are $S^2$ and $SO(3)$.

math.AP↗
arXiv2020

A finer singular limit of a single-well Modica--Mortola functional and its applications to the Kobayashi--Warren--Carter energy

An explicit representation of the Gamma limit of a single-well Modica--Mortola functional is given for one-dimensional space under the graph convergence which is finer than conventional $L^1$-convergence or convergence in measure. As an application, an explicit representation of a singular limit of the Kobayashi-Warren-Carter energy, which is popular in materials science, is given. Some compactness under the graph convergence is also established. Such formulas, as well as compactness, is useful to characterize the limit of minimizers the Kobayashi-Warren-Carter energy. To characterize the Gamma limit under the graph convergence, a new idea which is especially useful for one-dimensional problem is introduced. It is a change of parameter of the variable by arc-length parameter of its graph, which is called unfolding by the arc-length parameter in this paper.

math.AP↗
arXiv2020

Short-time approximate solutions of an equation modeling a camphor motion

As a profound example of spontaneous motion, we analyze the motion of a camphor particle on a water surface. The motion is modeled as an initial-boundary value problem for a coupled nonlinear system of a diffusion equation and an ordinary differential equation in a two-dimensional domain. Since it seems that the well-posedness of this initial boundary value problem is missing, we provided its proof. Then, by constructing an approximate solution to this initial boundary value problem, we gave a mathematically rigorous interpretation of a camphor motion. That is we showed that the motion of camphor locally in time has a self-avoiding orbit. We also gave the numerical performance of the approximate solution.

math.AP↗
arXiv2018

An algebraic description of screw dislocations in SC and BCC crystal lattices

We give an algebraic description of screw dislocations in a crystal, especially simple cubic (SC) and body centered cubic (BCC) crystals, using free abelian groups and fibering structures. We also show that the strain energy of a screw dislocation based on the spring model is expressed by the Epstein-Hurwitz zeta function approximately.

math-ph↗