Existence of time-periodic solutions to the Navier–Stokes equations around a moving body

Abstract

We demonstrate the existence of time-periodic motions of an incompressible Navier-Stokes fluid subject to a time-periodic body force, occupying the region exterior to a body that performs a periodic rigid motion of same period.

Time-Periodic Solutions to the Navier-Stokes Equations

“…In other words, one then obtains a time-periodic Navier-Stokes problem in a frame of reference that is given by the motion of the body, and which is not necessarily an inertial frame. This special type of time-periodic Navier-Stokes problem was investigated for the first time by Galdi and Silvestre in [25]. Galdi and Silvestre assumed a 1 Introduction prescribed motion of the body given by a time-periodic translational velocity ξ(t) and time-periodic angular velocity ω(t), which yields…”

“…Using a combination of an "invading domain" technique and the ProuseYudovich method, Galdi and Silvestre showed existence of a weak time-periodic solution to (1.7). In addition, existence of a strong solution is established in [25] under the assumption that the data ξ(t), ω(t), and f are sufficiently "small". In a further investigation of this problem, the same authors showed a few years later in [27] a similar result for the unconstrained motion of a body under the influence of a time-periodic body force, thereby extending a famous result of Weinberger [58,59] to the time-periodic case.…”

“…These methods can therefore not easily be adapted to the case u ∞ = 0. Only the approach of Galdi and Silvestre from [25] can be modified to show existence of a time-periodic solution in the case u ∞ = 0. Their approach yields existence of a weak solution in the space L 2 0, T ; D 1,2 0 (R 3 ) 3 , where T denotes the time period of the data f .…”

“…Thus, (6.4) is an appropriate weak formulation. Definition 6.1.1 is equivalent to the definition of a weak time-periodic solution introduced in [25].…”

“…We can estimate the last term on the right-hand side in (6.22) by 25) where f l denotes the l'th Fourier coefficient of f . Collecting (6.22)-(6.25) and recalling (6.14), we can now deduce…”