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- ItemPeriodic 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, YoshihiroGeneral 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.
- ItemPressure reconstruction for weak solutions of the two-phase incompressible Navier--Stokes equations with surface tension(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Abels, Helmut; Daube, Johannes; Kraus, ChristianeFor the two-phase incompressible Navier--Stokes equations with surface tension, we derive an appropriate weak formulation incorporating a variational formulation using divergence-free test functions. We prove a consistency result to justify our definition and, under reasonable regularity assumptions, we reconstruct the pressure function from the weak formulation.