Spatial pointwise behavior of time-periodic Navier-Stokes flow induced by oscillation of a moving obstacle

TL;DR

This paper proves the existence and spatial decay of unique time-periodic solutions to the 3D Navier-Stokes equations around a moving obstacle, under certain conditions on the obstacle's motion, extending previous results on flow behavior at infinity.

Contribution

It establishes the existence of a unique time-periodic Navier-Stokes flow with specific spatial decay properties in 3D exterior domains, under conditions on the obstacle's motion.

Findings

Existence of a unique time-periodic flow in weak-$L^3$ space.

Pointwise decay of the flow at rate $|x|^{-1}$.

Conditions on obstacle motion ensuring decay and uniqueness.

Abstract

We study the spatial decay of time-periodic Navier-Stokes flow at the rate $|x|^{-1}$ with/without wake structure in 3D exterior domains when a rigid body moves periodically in time. In this regime the existence of time-periodic solutions was established first in the 2006 paper by Galdi and Silvestre, however, with little information about spatial behavior at infinity so that uniqueness of solutions was not available. This latter issue has been addressed by Galdi, who has recently succeeded in construction of a unique time-periodic solution with spatial behavior mentioned above if translational and angular velocities of the body fulfill, besides smallness and regularity, either of the following assumptions: (i) translation or rotation is absent; (ii) both velocities are parallel to the same constant vector. This paper shows the existence of a unique time-periodic Navier-stokes flow in the small with values in the weak-$L^3$ space and then deduces the desired pointwise decay of the solution under some condition on the rigid motion of the body, that covers the cases (i), (ii) mentioned above.