國立清華大學 數學系

Workshop on Dynamical Systems and Applied Mathematics Date: 2017.11.30 – 2017.12.1 1. Date: November 30 (Thu.), 2017 Time: 14:20 Speaker: Takashi Teramoto (Asahikawa Medical University, Japan) Title: Localized solutions in reaction-diffusion systems on curved surface Venue: Lecture Room B, 4F, General Building III Abstract We consider the motion of spot solutions on two-dimensional curved surface. In the framework of weak interaction theory by Ei, the dynamics of spot position is governed by ordinary differential equations (ODEs). The spot motion is determined by the curvature of surface. In this talk, I will start to talk about the results of numerical simulations of localized spot solutions in reaction-diffusion systems and their dynamics in heterogeneous media. The key of our analysis is that the curved surface can be regarded as a spatial heterogeneity in two dimension. This is a joint with S.-I. Ei and A. Sekisaka. -- 2. Date: November 30 (Thu.), 2017 Time: 16:30 Speaker: Petrus van Heijster (Queensland University of Technology, Australia) Title: The stability of travelling wave solutions in a Keller–Segel model with logarithmic chemosensitivity Venue: Lecture Room B, 4F, General Building III Abstract In this talk, I will discuss some recent results we obtained regarding the spectral stability of travelling wave solutions in a Keller-Segel model of bacterial chemotaxis with a logarithmic chemosensitivity function. Upon linearising around the travelling wave solutions, we locate the essential, absolute, and point spectrum of the associated linear operators. We find that all travelling wave solutions have parts of the essential spectrum in the right half plane in the standard unweighted Banach space. However, we show that there exists a range of parameters such that the absolute spectrum is contained in the open left half plane and the essential spectrum can thus be weighted into the open left half plane. In addition, we determine critical parameter values for which the absolute spectrum crosses into the right half plane, indicating the onset of an absolute instability of the travelling wave solution. Most notably, we observe that this crossing always occurs off of the real axis. To complete the story, we (numerically) locate the point spectrum of the travelling wave solutions. This is joint work with P.N. Davis and R. Marangell. -- 3. Date: December 1 (Fri.), 2017 Time: 10:10 Speaker: Yasumasa Nishiura (Tohoku University, Japan) Title: Interplay between Internal and External Interactions for Morphological Transitions arising in Block Copolymer Nanoparticles Venue: Lecture Room B, 4F, General Building III Abstract: Annealing of copolymers has become a tool of great importance to reconfigure nanoparticles. Herein, we present experimental results of annealing copolymer nanoparticles and a theoretical model to describe the morphological transformation from lamellae to onion. A good correspondence between experimental findings and predictions of the model is observed. The model based on finding the steepest direction of decent of an appropriate free energy leads to a set of Cahn-Hilliard equations that correctly describes the dynamical transformation from lamellae particles to onion and reverse onion-like particles, regardless of the nature of the annealing process. This universality makes possible to describe a variety of experimental conditions involving nanoparticles underlying a heating process. A notable advantage of the proposed approach is that it makes possible to selectively control the interaction between the confined copolymer and the surrounding media. This feature endows the model with great versatility to reproduce several combined effects of surfactants in diverse conditions, including cases with reverse affinities for the copolymer segments. A phase diagram to describe a variety of morphologies is presented. We employ relations between the temperature-dependent Flory-Huggins parameter and the width of the interfaces to account for changes in temperature due to the heating process. Simulation results correctly show how the transformation evolves as the temperature increases. This increment in temperature corresponds to progressively smaller values of the interfacial width. We anticipate that the proposed approach will facilitate the design and a more precise control of experiments involving various kinds of annealing processes. This is a joint work with E. Avalos, T. Teramoto, H. Komiyama, and H. Yabu.