Schedule for Aug 30, 2023

10:00-10:50  Ikkei Shimizu (Osaka University)

Title: Local well-posedness of the Lanadu-Lifshitz equation with helicity term

Abstract: I will talk about my previous result on the initial-value problem for the Landau-Lifshitz equation with helicity term. By a kind of transformation, the equation is transformed into a system of nonlinear Schr\"odinger equation with electromagnetic potential, where the potential depends on the unknown function. We obtain nonlinear estimate via energy method applied to this, but the difficulty is the quadratic derivative nonlinearity of helicity term. In my talk, I will explain how to avoid the above difficulty.

11:10-12:00  Wenjia Jing (Tsinghua University)

Title: On the quantitative homogenization of elliptic problems with periodic high contrast coefficients

Abstract: We consider elliptic equations with periodic high contrast coefficients and study the asymptotic analysis when the periodicity is sent to zero and/or the contrast parameters are sent to extreme values. Those coefficients model small inclusions that have very different physical properties compared to the surrounding environment. Homogenization captures the macroscopic effects of those inclusions. We report some quantitative results such as the convergence rates of the homogenization (with proper correctors), uniform regularity for the solutions of the heterogeneous equations, and so on.  The talk is based on joint works with Mr. Xin Fu.

13:30-13:55  Shun Tsuhara (Tohoku University) (25 minutes talk)

Title: The initial boundary value problem for the Schr\"odinger equation with a nonlinear Neumann boundary condition on the two-dimensional half-plane

Abstract: We consider the initial boundary value problem of the Schr\"odinger equation on the half-plane with a nonlinear Neumann boundary condition. In the previous works, Hayashi--Ogawa--Sato considered such a problem for the one-dimensional case. In this talk, we show the boundary Strichartz estimate for an inhomogeneous Neumann boundary data in $L^2(\mathbb{R}^2_+)$ via the Fresnel integral. We also prove that the problem with a nonlinear Neumann boundary condition is locally well-posed in the $L^2$-subcritical and $L^2$-critical case. This talk is based on a joint work with Prof. Takayoshi Ogawa (Tohoku University) and Prof. Takuya Sato (Kumamoto University).

14:10-15:00  Hiroshi Wakui (University of Fukui)

Title: Existence and boundedness of a forward self-similar solution to a minimal Keller-Segel model

Abstract: In this talk, we consider existence of a bounded forward self-similar solution to the initial value problem of a minimal Keller-Segel model. It is well known that the mass conservation law plays an important role to classify its large time behavior of solutions to Keller-Segel models. On the other hand, we could not expect existence of self-similar solutions to our problem with the mass conservation law except for the two dimensional case due to the scaling invariance of our problem. We will show existence of a forward self-similar solution to our problem.

15:05-15:10  Closing