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Subject: Evolution equations ×

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    Maximal Regularity for Non-autonomous Equations with Measurable Dependence on Time
    (Dordrecht [u.a.] : Springer Science + Business Media B.V, 2016) Gallarati, Chiara; Veraar, Mark
    In this paper we study maximal L p-regularity for evolution equations with time-dependent operators A. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the L p-boundedness of a class of vector-valued singular integrals which does not rely on Hörmander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of m-th order elliptic operators A with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an L p(L q)-theory for such equations for p,q∈(1,∞). In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.
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    Periodic Lp estimates by R-boundedness: Applications to the Navier--Stokes equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Eiter, Thomas; Kyed, Mads; Shibata, Yoshihiro
    General evolution equations in Banach spaces are investigated. Based on an operator-valued version of de Leeuw's transference principle, time-periodic Lp estimates of maximal regularity type are established from R-bounds of the family of solution operators (R-solvers) to the corresponding resolvent problems. With this method, existence of time-periodic solutions to the Navier--Stokes equations is shown for two configurations: in a periodically moving bounded domain and in an exterior domain, subject to prescribed time-periodic forcing and boundary data.
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    Sobolev-Morrey spaces associated with evolution equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Griepentrog, Jens A.
    In this text we introduce new classes of Sobolev-Morrey spaces being adequate for the regularity theory of second order parabolic boundary value problems on Lipschitz domains of space dimension n ≥ 3 with nonsmooth coefficients and mixed boundary conditions. We prove embedding and trace theorems as well as invariance properties of these spaces with respect to localization, Lipschitz transformation, and reflection. In the second part [11] of our presentation we show that the class of second order parabolic systems with diagonal principal part generates isomorphisms between the above mentioned Sobolev-Morrey spaces of solutions and right hand sides.