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

抽象拉图什金 本项目主要研究无穷维Banach空间上微分方程和动力系统的渐近性态。 这些方程的二分法和稳定性将被考虑，这对于研究复杂系统的长期演化至关重要。 无限维的设置涵盖了在力学和物理学中应用的重要方程。 该方法的核心是发展半群理论的研究和应用。 这些半群吸收，从字面上看，所有需要发展的经典思想的稳定性的现代应用所需的水平上的李雅普诺夫的信息。 具体应用包括：Banach空间上强连续半群和线性非自治抽象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力学的运动发电机算子;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.