On the Stokes-type resolvent problem associated with time-periodic flow around a rotating obstacle

Abstract

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.

On the Oseen-type resolvent problem associated with time-periodic flow past a rotating body

“…Since the relevant differential operator in (1.4) has constant coefficients, a solution formula can directly be deduced in terms of a Fourier multiplier in the group setting T × R 3 , and we can derive associated a priori estimates by L q multiplier theorems, which lead to resolvent estimates for (1.1) that are uniform for all s ∈ R satisfying dist(s, ωZ \ {s}) > δ for some fixed δ > 0. This result differs from the case λ = 0, where uniform resolvent estimates for all s ∈ R are available (see [5]). This observation parallels the known results for the non-rotating case ω = 0, that is, for the resolvent problem…”

“…This severe restriction already appears in the existence theory for the linear problem (1.2) derived in [7]. In contrast, in the cases without translation (λ = 0) or without rotation (ω = 0), existence of time-periodic solutions to (1.2) can be shown without further restrictions on the time-period T ; see [5,14]. A leading question of this article is whether the condition ω = 2π…”

“…Instead, we shall handle this term by a method recently developed by Galdi and Kyed [12,13] to investigate steady-state solutions to (1.2), that is, solutions to (1.1) for s = 0. This method was successfully applied to the time-periodic problem (1.2) for λ = 0 in [5] and for λ > 0 in [7]. As mentioned above, while the linear theory in [5] holds for all T , ω > 0, in [7] the assumption 2π T = ω was imposed.…”

“…This method was successfully applied to the time-periodic problem (1.2) for λ = 0 in [5] and for λ > 0 in [7]. As mentioned above, while the linear theory in [5] holds for all T , ω > 0, in [7] the assumption 2π T = ω was imposed. This restriction makes it possible to absorb the term ω(e 1 ∧u − e 1 ∧x • ∇u) into the time derivative ∂ t u by a suitable transformation when Ω = R 3 , and to reduce the problem to the case ω = 0.…”

“…Note that if ω = 2π T , then the associated transformation is not an isomorphism between T -periodic functions. To circumvent this problem, we proceed as in [5], and adapt this method to first derive well-posedness of the resolvent problem (1.1) by a reduction to the auxiliary problem…”