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

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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.