PFDE with nonautonomous past
Simon Brendle Rainer Nagel
Discrete & Continuous Dynamical Systems - A 2002, 8(4): 953-966 doi: 10.3934/dcds.2002.8.953
We study Cauchy problems associated to partial differential equations with infinite delay where the history function is modified by an evolution family. Using sophisticated tools from semigroup theory such as evolution semigroups, extrapolation spaces, or the critical spectrum, we prove well-posedness and characterize the asymptotic behavior of the solution semigroup by an operator-valued characteristic equation.
keywords: nonautonomous Cauchy problem evolution semigroups stability. pde with delay
Arithmetic progressions -- an operator theoretic view
Tanja Eisner Rainer Nagel
Discrete & Continuous Dynamical Systems - S 2013, 6(3): 657-667 doi: 10.3934/dcdss.2013.6.657
Motivated by the recent Green--Tao theorem on arithmetic progressions in the primes, we discuss some of the basic operator theoretic techniques used in its proof. In particular, we obtain a complete proof of Szemerédi's theorem for arithmetic progressions of length $3$ (Roth's theorem) and the Furstenberg--Sárközy theorem.
keywords: Roth's theorem Jacobs-Glicksberg-deLeeuw decomposition. multiple ergodic theorems Arithmetic progressions operator theoretic methods
Vertex control of flows in networks
Klaus-Jochen Engel Marjeta Kramar Fijavž Rainer Nagel Eszter Sikolya
Networks & Heterogeneous Media 2008, 3(4): 709-722 doi: 10.3934/nhm.2008.3.709
We study a transport equation in a network and control it in a single vertex. We describe all possible reachable states and prove a criterion of Kalman type for those vertices in which the problem is maximally controllable. The results are then applied to concrete networks to show the complexity of the problem.
keywords: operator semigroups boundary control transport equation networks
Floquet representations and asymptotic behavior of periodic evolution families
Fatih Bayazit Ulrich Groh Rainer Nagel
Discrete & Continuous Dynamical Systems - A 2013, 33(11&12): 4795-4810 doi: 10.3934/dcds.2013.33.4795
We use semigroup techniques to describe the asymptotic behavior of contractive, periodic evolution families on Hilbert spaces. In particular, we show that such evolution families converge almost weakly to a Floquet representation with discrete spectrum.
keywords: evolution semigroups Nonautonomous abstract Cauchy problems asymptotic behavior periodic evolution families Floquet theory.

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