Abstract

We introduce rigorous mathematical results concerning the qualitative behavior of nonlinear PDE models describing the mechanistic and chemical properties of vasculogenesis which is the early stage of the formation of a vascular network on a substrate. The first PDE model consists of the compressible Euler equations and a reaction-diffusion equation through nonlocal coupling. Depending on the parametric and boundary conditions, different steady state solutions are constructed on bounded domains, which are reasonable consistent with experimental observations, and the steady states are shown to be locally exponentially stable provided a particular parameter is sufficiently large. The second model contains the incompressible and inhomogeneous Navier-Stokes equations nonlocally coupled with a reaction-diffusion equation. In this case, global stability of two-dimensional, large-data classical solutions on bounded domains subject to no-flow or dynamic Couette flow boundary condition is established. This talk is based on recent joint works with Hongyun Peng (Guangdong University of Technology) and Zhian Wang (Hong Kong Polytechnic University).

Short bio

赵昆，美国杜兰大学数学系教授，中国科学技术大学本科，美国乔治亚理工学院博士，美国俄亥俄州立大学生物研究所博士后，曾任美国爱荷华大学数学系访问助理教授。从事生物数学、流体力学、数学物理领域中非线性偏微分方程的定性和定量分析研究。在Arch. Ration .Mech. Anal., Indiana Univ.Math.,J.,SIAM J.Math.Anal.,Math.Models Methods Appl.Sci.、Physical D、J.Differential Equations., J. Geom. Mech.、J.Math.Biol.、European J.Appl.Math.、Nonlinearlity等国际知名期刊上发表SCI论文50多篇。现担任杂志《Annals of Applied Mathematics》编委，主持完成多项美国自然科学基金项目。