On the Dynamics of Nonlinear Systems in Applied Sciences

多相流模型的分析，例如控制流体-颗粒相互作用和细胞演化的模型，与工程、物理和生物医学科学中的几个实际应用有关。本课题主要研究非线性模型的稳定性、唯一性和规律性以及数值逼近问题。特别是，这些项目包括分析涉及移动域或自由边界的系统和随机强迫的系统。本项目包括本科生、研究生和博士后参与研究。本研究项目侧重于物理和生物科学中出现的非线性系统的建模和数学分析，并在两个相互关联的方向上解决主题：(a)运动域内的流体动力学模型和自由边界问题，以及(b)可压缩流体和多相流模型的随机扰动。这些非线性模型的数学分析需要创新的思想来构建合适的方案来逼近它们的控制系统，以及根据应用中出现的模型的特殊特征开发新的分析技术来证明适定性和收敛结果。该项目的目标是开发一种变分框架，能够以分析和计算的方式处理大类多相流。所研究的非线性系统包括由Navier-Stokes和Euler系统控制的可压缩流体模型，欧拉动力学流体模型，固定或移动域中的混合型双曲-椭圆系统以及自由边界模型。

The analysis of multiphase flow models, such as models governing fluid-particle interaction and evolution of cells, is relevant to several practical applications in engineering and in physical and biomedical sciences. This project focuses on issues of stability, uniqueness and regularity, and numerical approximations for nonlinear models. In particular, the projects include analyzing systems involving moving domains or free boundaries and systems with stochastic forcing. The project includes undergraduates, graduate students, and post-doctoral associates in the research. This research project focuses on the modeling and mathematical analysis of nonlinear systems arising in physical and biological science and addresses themes in two interconnected directions: (a) hydrodynamic models within moving domains and free boundary problems, and (b) random perturbations of models of compressible fluids and multiphase flows. The mathematical analysis of these nonlinear models requires innovative ideas for the construction of suitable schemes for the approximation of their governing systems, as well as the development of new analytical techniques for the proof of well-posedness and convergence results in light of special features of the models that arise in applications. The goal of the project is the development of a variational framework able to treat a large class of multi-phase flows both analytically and computationally. The nonlinear systems under investigation include models of compressible fluids governed by the Navier-Stokes and Euler systems, Euler-kinetic fluid models, mixed-type hyperbolic-elliptic systems within fixed or moving domains, and free-boundary models.