偏微分方程研讨班

中科院数学与系统科学研究院

数学研究所

偏微分方程研讨班

时  间：2022.11.21（星期一）

地  点：腾讯会议：335-605-109 会议密码：1121

n  下午14 :00-15 :00

报告人：杨雄锋 教授 （上海交通大学）

题  目：The long wave approximation of the Green-Naghdi equations with the Coriolis effect

摘 要：This talk study the long wave asymptotic behavior of Green-Naghdi equations, which could be used to describe the propagation of long-crested shallow-water waves in the equatorial ocean regions with the Coriolis effect due to Earth's rotation. This model equation is called the rotation-Green-Naghdi (R-GN) equations modeling the propagation of wave allowing large amplitude in shallow water. 1. We demonstrate the lift span of the solution to the R-GN model equations in a Sobolev space by the refined energy estimates. 2.We provide a rigorous justification from the solutions of the R-GN equations to the associated solution of the right-left R-BBM or KdV equation in the KdV regime with the small amplitude and the large wavelength. This is the first result on the issue that the solution of R-GN equations are well approximated by the bi-directional R-BBM in the general regular initial data.It is a jointed work with Prof. Yue Liu.

n  下午15 :00- 16 :00

报告人：冯跃红 副教授 （北京工业大学）

题 目：Nonlinear structural stability and linear dynamic instability of transonic steady-states to a hydrodynamic model for semiconductors

摘 要：For unipolar hydrodynamic model of semiconductor device represented by Euler-Poisson equations, when the doping profile is supersonic, the existence of steady transonic shock solutions and -smooth steady transonic solutions for Euler-Poisson Equations were established in Li-Mei-Zhang-Zhang SIMA2018 and Wei-Mei-Zhang-Zhang SIMA2021, respectively. In this talk, we further study the nonlinear structural stability and the linear dynamic instability of these steady transonic solutions. When the -smooth transonic steady-states pass through the sonic line, they produce singularities for the system, and cause some essential difficulty in the proof of structural stability. For any relaxation time: , by means of elaborate singularity analysis, we first investigate the structural stability of the -smooth transonic steady-states, once the perturbations of the initial data and the doping profiles are small enough. Moreover, when the relaxation time is large enough, under the condition that the electric field is positive at the shock location, we prove that the transonic shock steady-states are structurally stable with respect to small perturbations of the supersonic doping profile. Furthermore, we show the linearly dynamic instability for these transonic shock steady-states provided that the electric field is suitable negative. The proofs for the structural stability results are based on singularity analysis, a monotonicity argument on the shock position and the downstream density, and the stability analysis of supersonic and subsonic solutions. The linear dynamic instability of the steady transonic shock for Euler-Poisson equations can be transformed to the ill-posedness of a free boundary problem for the Klein-Gordon equation. By using a nontrivial transformation and the shooting method, we prove that the linearized problem has a transonic shock solution with exponential growths. These results enrich and develop the existing studies.