|Colloquium, 中央研究院數學研究所 鄭日新研究員|
Thursday, June 21, 16:10—17:00 數學系3174
Title: Complex/CR duality (?)—Volume Renormalization and Invariant Geometric Equations
Abstract: Inspired by submanifold observables in AdS/CFT correspondence, we consider volume renormalization of surfaces or hypersurfaces with boundary curves or surfaces in possible complex/CR duality. Precisely I will talk about biholomorphically invariant curves and surfaces on the boundary of a strongly pseudoconvex domain in C^2. A distinguished class of such invariant curves satisfies a system of 2nd order ODEs, called chains in CR geometry. We interpret chains as geodesics of a Kropina metric in Finsler geometry. The associated energy functional of a curve on the boundary can be recovered as the log term coefficient in a weighted renormalized area expansion of a minimal surface that it bounds inside the domain. For surfaces on the boundary, we express two CR invariant surface area elements in terms of quantities in pseudohermitian geometry. We deduce the Euler-Lagrange equations of the associated energy functionals. In relation to the singular CR Yamabe problem, we show that one of the energy functionals appears as the coefficient (up to a constant multiple) of the log term in the associated volume renormalization. We ask how these “CR Willmore” surfaces are related to geometric quantities inside the domain.
|PDE Seminar, 山東理工大學理學院 孫俊濤教授|
Tuesday, July 3, 16:10—17:00 數學系3174
Title: The Filtration of Nehari Manifold and Its Application in Some Nonlocal Problems
Abstract:Schrodinger-Poisson system, also known as the nonlinear Schrodinger-Maxwell equations, is suggested as a model describing the interaction of a charged particle with the electrostatic field in quantum mechanics. In this talk, by introducing a new set, which is regarded as the filtration of the Nehari manifold, and together with variational methods, we are concerned with the existence of positive solution for a class of non-autonomous Schrodinger-Poisson systems without any symmetry assumptions. Furthermore，the existence of ground state solution is also obtained.
|PDE Seminar, The University of British Columbia 蔡岱朋教授|
Wednesday, July 25, 11:00—12:15 數學系3174
Title: On Global Weak Solutions of Navier-Stokes Equations with Non-Decaying Initial Data
Abstract:We consider the Cauchy problem of 3D incompressible Navier-Stokes equations with uniformly locally square integrable initial data. If the square integral of the initial data on a ball vanishes as the ball goes to infinity, the existence of global weak solutions has been known. However, such data do not include constants, and the only results for non-decaying data are either for perturbations of constants, or when the velocity gradients are in L^p. We construct global weak solutions for non-decaying initial data whose local oscillations decay.