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Periodic Lp estimates by R-boundedness: Applications to the Navier-Stokes equations

General evolution equations in Banach spaces are investigated. Based on an operator-valued version of de Leeuw's transference principle, time-periodic $L^p$ estimates of maximal regularity type are established from $\mathscr{R}$-bounds of the family of solution operators ($\mathscr{R}$-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.