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Start date: 2020 × End date: 2024 × Start date: 2020 × DDC: 510 × Type: Report × Subject: time-periodic solutions × WGL Institute: WIAS ×

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Now showing 1 - 5 of 5
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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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    On the spatially asymptotic structure of time-periodic solutions to the Navier--Stokes equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Eiter, Thomas
    The asymptotic behavior of weak time-periodic solutions to the Navier--Stokes equations with a drift term in the three-dimensional whole space is investigated. The velocity field is decomposed into a time-independent and a remaining part, and separate asymptotic expansions are derived for both parts and their gradients. One observes that the behavior at spatial infinity is determined by the corresponding Oseen fundamental solutions.
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    Spatial decay of the vorticity field of time-periodic viscous flow past a body
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Eiter, Thomas; Galdi, Giovanni P.
    We study the asymptotic spatial behavior of the vorticity field associated to a time-periodic Navier-Stokes flow past a body in the class of weak solutions satisfying a Serrin-like condition. We show that outside the wake region the vorticity field decays pointwise at an exponential rate, uniformly in time. Moreover, decomposing it into its time-average over a period and a so-called purely periodic part, we prove that inside the wake region, the time-average has the same algebraic decay as that known for the associated steady-state problem, whereas the purely periodic part decays even faster, uniformly in time. This implies, in particular, that ``sufficiently far'' from the body, the time-periodic vorticity field behaves like the vorticity field of the corresponding steady-state problem.
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    On the Oseen-type resolvent problem associated with time-periodic flow past a rotating body
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Eiter, Thomas
    Consider the time-periodic flow of an incompressible viscous fluid past a body performing a rigid motion with non-zero translational and rotational velocity. We introduce a framework of homogeneous Sobolev spaces that renders the resolvent problem of the associated linear problem well posed on the whole imaginary axis. In contrast to the cases without translation or rotation, the resolvent estimates are merely uniform under additional restrictions, and the existence of time-periodic solutions depends on the ratio of the rotational velocity of the body motion to the angular velocity associated with the time period. Provided that this ratio is a rational number, time-periodic solutions to both the linear and, under suitable smallness conditions, the nonlinear problem can be established. If this ratio is irrational, a counterexample shows that in a special case there is no uniform resolvent estimate and solutions to the time-periodic linear problem do not exist.
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    On the Stokes-type resolvent problem associated with time-periodic flow around a rotating obstacle
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Eiter, Thomas
    Consider the resolvent problem associated with the linearized viscous flow around a rotating body. Within a setting of classical Sobolev spaces, this problem is not well posed on the whole imaginary axis. Therefore, a framework of homogeneous Sobolev spaces is introduced where existence of a unique solution can be guaranteed for every purely imaginary resolvent parameter. For this purpose, the problem is reduced to an auxiliary problem, which is studied by means of Fourier analytic tools in a group setting. In the end, uniform resolvent estimates can be derived, which lead to the existence of solutions to the associated time-periodic linear problem.