9:00-9:10 | 邱荣（西南科技大学理学院副院长）致欢迎辞 |
9:10-9:20 | 郑克龙（西南科技大学）西南科技大学数学学科情况介绍 |
Chair | Yongzhong SUN （孙永忠）（南京大学） |
9:20-10:00 | Shibing CHEN（陈世炳）（中国科技大学) Smoothness of free boundary in optimal transport problem Abstract: Free boundary arises in optimal transport problem when only a portion of mass is transported. The problem was first studied by Caffarelli and McCann for disjoint domains and then by Figalli, Indrei for domains having overlap. The $C^1, C^{1,\alpha}$ regularity were established in these works, left open the question whether we have smoothness of free boundary for smooth data. We will discuss our recent resolution to this problem, and explain some of the key ingredients in the proof. This is based on a joint work with Jiakun Liu and Xu-Jia Wang. |
10:00-10:40 | Jun GENG（耿俊）（兰州大学) Periodic parabolic homogenization of Green functions Abstract: For a family of second-order parabolic systems with rapidly oscillating and time-dependent periodic coefficients, we investigate the asymptotic behavior of Green functions. Also, I will introduce some recent results on quantitative homogenization of second-order parabolic systems. |
10:40-10:50 | Tea break |
Chair | Guilong GUI （桂贵龙）（西北大学） |
10:50-11:30 | Yong LIU（刘勇）（中国科技大学) Classification and Morse indices of multiple-end solutions to the elliptic sine-Gordon equation Abstract：In this talk, we discuss the multiple-end entire solutions of elliptic sine-Gordon equation in the plane. We show that the moduli space of all the 2n-end solutions is a smooth manifold of dimension 2n. We also prove that any 2n-end solution is nondegenerated and of Morse index n(n-1)/2. Our method is based on the inverse scattering transform. |
11:30-12:10 | Lin HE（何躏）（四川大学） Global solutions of the compressible Euler-Poisson equations with large initial data of spherical symmetry Abstract：We are concerned with a global existence theory for finite-energy solutions of the multidimensional Euler-Poisson equations for both compressible gaseous stars and plasmas with large initial data of spherical symmetry. One of the main challenges is the strengthening of waves as they move radially inward towards the origin, especially under the self-consistent gravitational field for gaseous stars. A fundamental unsolved problem is whether the density of the global solution forms concentration to become a delta measure at the origin. To solve this problem, we develop a new approach for the construction of approximate solutions as the solutions of an appropriately formulated free boundary problem for the compressible Navier Stokes-Poisson equations with a carefully adapted class of degenerate density-dependent viscosity terms, so that a rigorous convergence proof of the approximate solutions to the corresponding global solution of the compressible Euler-Poisson equations with large initial data of spherical symmetry can be obtained. Even though the density may blow up near the origin at certain time, it is proved that no concentration (delta measure) is formed in the vanishing viscosity limit for the finite-energy solutions of the compressible EulerPoisson equations for both gaseous stars and plasmas in the physical regimes under consideration. |
