Long-Time Behavior for the Two-Dimensional Motion of a Disk in a Viscous Fluid

Abstract

In this article, we study the long-time behavior of solutions of the two-dimensional fluidrigid disk problem. The motion of the fluid is modeled by the two-dimensional Navier-Stokes equations, and the disk moves under the influence of the forces exerted by the viscous fluid. We first derive L p -L q decay estimates for the linearized equations and compute the first term in the asymptotic expansion of the solutions of the linearized equations. We then apply these computations to derive time-decay estimates for the solutions to the full Navier-Stokes fluid-rigid disk system.

Small Moving Rigid Body into a Viscous Incompressible Fluid

“…Proof. For ε = 1, the statements of the theorem were proved in [12], see therein Theorem 1.1, Corollaries 3.10 and 3.11.…”

“…For the fluid-solid problem, such a Cauchy theory is not yet established. As the optimal decay estimates for the Stokes semigroup are now known in L p − L q [12], we guess that it would be possible to extend it by interpolation to Marcinkiewicz spaces, and then to prove a well-posedness result for the full fluid-solid system. Such an extension would require more work and could be interesting, but the main goal of this article is to stay in the standard framework for the fluid solid problem and to treat the same question as [10,33].…”

“…In dimension two, the global optimal L p − L q decay estimates for the Stokes semigroup were obtained by Ervedoza, Hillairet and Lacave in [12] when the solid is a disk in the whole plane. We guess that their analysis can be adapted in the exterior of a ball in dimension three, but it would require a considerable work, decomposing the Stokes equations on spherical coordinate system instead to polar decomposition, exhibiting the "good unknown" (see [12,Proposition 2.3] for the two dimensional case), and adapting the elliptic lemmas.…”

A Small Solid Body with Large Density in a Planar Fluid is Negligible

“…On the other hand, the hypothesis that Ω ε is a disk seems to be essential in the result of [11]. Indeed, a key ingredient are the estimates of [3] and the proof of that result relies heavily on the fact that Ω ε is a disk because it uses explicit formulae valid only for the case of a disk. Moreover, it is also hard to see how the smallness condition (10) could be removed in their argument.…”

On the Asymptotic Stability of Steady Flows with Nonzero Flux in Two-Dimensional Exterior Domains

“…For vanishing forcing u * = 0 or in R 2 , Borchers & Miyakawa (1992) established the asymptotic stability of trivial solutionū = 0 under L 2 -perturbations. Even under more general hypotheses, the solution is known to be asymptotic to the Oseen vortex (Gallay & Wayne, 2002, 2005Iftimie et al, 2011;Gallay & Maekawa, 2013;Ervedoza et al, 2014;Maekawa, 2015). The aim of this paper is to stud the stability of the steady solution for a nonzero forcing u * .…”