报告题目：Analytic smoothing effect for the nonlinear Landau equation
报告人： 徐超江 教授
摘要：We consider the Cauchy problem of the full nonlinear Landau equation of Maxwellian molecules, under the perturbation frame work to global equilibrium. We show that if the initial perturbation is small enough in Sobolev space, then the Cauchy problem of the nonlinear Landau equation admits a unique solution which becomes analytic with respectto both position and velocity variables for any positive time. This is the first result of analytic smoothing effect for the spatially inhomogeneous nonlinear kinetic equation.
报告题目：Sharp fundamental gap estimate on convex domains of sphere,
摘要：B. Andrews and J. Clutterbuck proved the fundamental gap (the difference between the first two eigenvalues) conjecture for convex domains in the Euclidean space  and conjectured similar results hold for spaces with constant sectional curvature. We prove the conjecture for the sphere. Namely when D, the diameter of a convex domain in the unit S^n sphere, is ≤ π/2 , the gap is greater than the gap of the corresponding 1-dim sphere model. We also prove the gap is ≥ 3 π^2/D^2 when n ≥ 3, giving a sharp bound. As in , the key is to prove a super log-concavity of the first eigenfunction. This is a joint work with Dr. Shoo Seto and Prof. Guofang Wei.