Mathematical Sciences: Evolutionary Semigroups with Applications to Differential Equations and Dynamical Systems Theory

本课题研究无穷维Banach空间上微分方程和动力系统的渐近行为。考虑这些方程的二分性和稳定性，这对于研究复杂系统的长期演化是至关重要的。无限维设置涵盖了在力学和物理应用中重要的方程。该方法的核心是研究和应用进化半群理论。从字面上看，这些半群吸收了在现代应用所需的水平上发展李亚普诺夫关于稳定性的经典思想所需的所有信息。具体应用包括：强连续半群的二分法和Banach空间上线性非自治抽象Cauchy问题的二分法；Banach空间上强连续线性斜积流的双曲性无限维半线性微分方程的中心流形理论Banach空间上非自治线性控制理论的稳定性理想导电流体磁流体动力学的运动学发电机算子统计物理中矩阵Ruelle算子的谱性质。

Abstract Latushkin This project is concerned with asymptotic behavior of differential equations and dynamical systems on infinite dimensional Banach spaces. Dichotomy and stability of these equations will be considered, which is crucial for the study of long-time evolution of complex systems. The infinite dimensional setting covers equations important for application in mechanics and physics. The core of the proposed approach is to study and apply the theory of evolution semigroups. These semigroups absorb, literally, all information needed to develop classical ideas of Lyapunov on stability at the level required by modern applications. Concrete applications include: dichotomy for strongly continuous semigroups and for linear nonautonomous abstract Cauchy problems on Banach spaces; the hyperbolicity of strongly continuous linear skew-product flows on Banach spaces; center manifolds theory for infinite dimensional semilinear differential equations; stability in non-autonomous linear control theory on Banach spaces; kinematic dynamo operator of magnetohydrodynamics for an ideally conducting fluid; and spectral properties of the matrix Ruelle operator of statistical physics.