学术报告报告主题: The Keller-Segel approximation in vasculogenesis
报 告 人: 寿凌云 讲师
报告时间： 2024年 11月13日（星期三）上午10:00-11:00
报告地点： 37号楼3A01
邀 请 人: 武乐云 教授
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数学学院
2024年 11月11日
报告摘要：
In vitro experiments on blood vessels show that endothelial cells spontaneously organize into a coherent vascular network. This process can be modeled by by a quasi-linear hyperbolic-parabolic chemotaxis system, which consists of the relaxed compressible Euler equations coupled with a parabolic equation governing the chemoattractant concentration. The purpose of this work is to justify the validity of the singular limit from this hyperbolic-parabolic system to the Keller–Segel equations as the relaxation parameter approaches zero. First, we prove the global existence of a unique classical solution to the Cauchy problem and establish uniform regularity estimates with respect to the relaxation parameter in a critical regularity setting assuming that the initial data is a small perturbation of the constant equilibrium. Furthermore, we establish global-in-time error estimates between solutions to the hyperbolic-parabolic system and the Keller-Segel equations for ill-prepared data, with an explicit convergence rate. Additionally, we study the optimal time convergence rates of the solution to equilibrium, uniformly in the relaxation parameter.
Due to the presence of chemotaxis, the analysis of the hyperbolic-parabolic system falls outside the scope of the classical Shizuta-Kawashima theorem, and it is thus a challenge to study its dynamics. We characterize the influence of the relaxation parameter on the dissipative structure by introducing a sharp threshold between low and high frequencies. In different frequency regimes, the use of new effective unknowns and the Lyapunov functional inspired by hypocoercivity leads to uniform a priori estimates.
报告人介绍：
寿凌云，目前任职于南京师范大学，主要从事偏微分方程的研究，研究方向为Fourier分析及其应用。主要从事流体力学方程的长时间行为、双曲松弛系统奇异极限等方面的研究。相关结果发表在Adv. Math., Nonlinearity, SIMA J. Math. Anal., J. Differential Equations等国际知名期刊上。