Schedule for Aug 28, 2023
09:50-09:55 Opening
10:00-10:50 Jinxin Xue (Tsinghua University)
Title: Generic dynamics of mean curvature flows.
Abstract: Mean curvature flow (MCF) is a way of evolving a hypersurface in Euclidean space according to a velocity field that is the negative mean curvature at each point of the hypersurface. Singularities always develop under MCF, so it is crucial to analyze singularities. We study mean curvature flow from a perspective of dynamical systems. We show how generic MCF avoids some unstable singularities and how dynamics is related to geometric information of the flow. We will also show that how the dynamical approach can give a way to study the isolatedness of cylindrical singularities. This talk is based a series of joint works with Ao Sun.
11:10-12:00 Yoshie Sugiyama (Osaka University)
Title: On the sharp ε-regularity theorem for the Keller-Segel systems and its application to the analysis and numerical simulation of singular sets
Abstract: We deal with the two-dimensional Keller-Segel system describing chemotaxis in the whole plane under the nonnegative initial data. As for the Keller-Segel system, the $L^1$-norm is the scaling invariant one with respect to the initial data, and so if the initial data are sufficiently small in $L^1$, then the solution exists globally in time. On the other hand, if its $L^1$-norm is large, then the solution blows up in a finite time. The first purpose of this talk is to construct an approximating solution globally in time even though the initial data is large in $L^1$ and to reveal properties of a limiting measure for such a solution beyond the blow-up time. The second purpose is to show the existence of singular points of a limiting measure and its continuity in time variable, while the solution exists in a classical sense until the blow-up time. We also introduce a numerical simulation on the blow-up structure of solution. This is a joint work with M.Miura(Yamato University), J.Choi(Pusan National University) and Y.Chen(Osaka University).
13:30-14:20 Xuecheng Wang (Tsinghua University)
Title: Nonlinear stability of the Vlasov-Poisson system in R^3
Abstract: We consider the stability problem for the 3D Vlasov-Poisson system in the whole space around the spatially homogeneous nontrivial equilibrium. In particular, we give linear stability for a class of general equilibrium and nonlinear stability for a special equilibrium, which is the so-called Poisson equilibrium. This talk is based on joint works with A. Ionescu (Princeton University), B. Pausader (Brown University), and K. Widmayer (University of Zurich and University of Vienna).
14:40-15:30 Naoto Shida (Nagoya University)
Title: Bilinear pseudo-differential operators with $S_{0,0}$ class symbols on Besov spaces
Abstract: We consider bilinear pseudo-differential operators with symbols in the bilinear H\"ormander class $BS^m_{0,0}$. The boundedness of these operators between Lebesgue spaces was studied by Miyachi-Tomita (2013) and Kato-Miyachi-Tomita (2022). In this talk, we generalize these results into the settings of Besov spaces. Furthermore, we mention that some restrictions on the exponents of Besov spaces are necessary to prove the boundedness.
15:50-16:40 Futoshi Takahashi (Osaka Metropolitan University)
Title: On eigenvalue problems involving the critical Hardy potential and Sobolev type inequalities with logarithmic weights in two dimensions
Abstract: We consider the two-dimensional eigenvalue problem for the Laplacian with the Neumann boundary condition involving the critical Hardy potential. We prove the existence of the second eigenfunction and study its asymptotic behavior around the origin. A key tool is the Sobolev type inequality with a logarithmic weight, which is shown as an application of the weighted nonlinear potential theory. This talk is based on a joint work with Megumi Sano (Hiroshima University).