数学研究所

学术报告

偏微分方程研讨班

报告人：翁上昆 教授（武汉大学）   

题  目：Existence and stability of cylindrical transonic shock solutions under three dimensional perturbations

时  间：2023.11.28（星期二）上午09:00-10:00

地  点：腾讯会议：375-246-471密码：2358

摘 要：In this talk, we discuss the existence and stability of cylindrical transonic shock solutions under three dimensional perturbations of the incoming flows and the exit pressure without any further restrictions on the background transonic shock solutions. The strength and position of the perturbed transonic shock are completely determined by the incoming flows and the exit pressure. The optimal regularity is obtained for all physical quantities, and the velocity, the Bernoulli's quantity, the entropy and the pressure share the same regularity. The problem is reduced to solve a nonlinear free boundary value problem for a hyperbolic-elliptic mixed system. There are two main ingredients in our analysis. One is to use the deformation-curl decomposition to the steady Euler system introduced by Weng and Xin to effectively decouple the hyperbolic and elliptic modes. Another one is the reformulation of the Rankine-Hugoniot conditions, which determines the shock front by an algebraic equation and also gives an unusual second order differential boundary conditions on the shock front for the deformation-curl system. After homogenizing the curl system and introducing a potential function, the solvability of the boundary value problem of the deformation-curl system for the velocity field is reduced to a second order elliptic equation for the potential function with a nonlocal term involving only the trace of the potential function on the shock front. This simplification follows essentially from an oblique boundary condition for the potential function on the shock front which is obtained by solving the Poisson equation on the shock front with the homogeneous Neumann boundary conditions on the intersection of the shock front and the cylinder walls. This is a joint work with Prof. Zhouping Xin.