Abstract
AbstractWe aim at the stability of time-dependent motions, such as time-periodic ones, of a rigid body in a viscous fluid filling the exterior to it in 3D. The fluid motion obeys the incompressible Navier–Stokes system, whereas the motion of the body is governed by the balance for linear and angular momentum. Both motions are affected by each other at the boundary. Assuming that the rigid body is a ball, we adopt a monolithic approach to deduce $$L^q$$
L
q
–$$L^r$$
L
r
decay estimates of solutions to a non-autonomous linearized system. We then apply those estimates to the full nonlinear initial value problem to find temporal decay properties of the disturbance. Although the shape of the body is not allowed to be arbitrary, the present contribution is the first attempt at analysis of the large time behavior of solutions around nontrivial basic states, that can be time-dependent, for the fluid–structure interaction problem and provides us with a stability theorem which is indeed new even for steady motions under the self-propelling condition or with wake structure.
Funder
Japan Society for the Promotion of Science London
Publisher
Springer Science and Business Media LLC
Reference52 articles.
1. Bogovskiĭ, M.E.: Solution of the first boundary value problem for the equation of continuity of an incompressible medium. Sov. Math. Dokl. 20, 1094–1098 (1979)
2. Borchers, W., Sohr, H.: On the equations $$ \text{ rot } v=g$$ and $$ \text{ div } u=f$$ with zero boundary conditions. Hokkaido Math. J. 19, 67–87 (1990)
3. Brezis, H.: Remarks on the preceding paper by M. Ben-Artzi, “Global solutions of two-dimensional Navier–Stokes and Euler equations”. Arch. Ration. Mech. Anal. 128, 359–360 (1994)
4. Chen, Z.-M.: Solutions of the stationary and nonstationary Navier–Stokes equations in exterior domains. Pacific J. Math. 159, 227–240 (1993)
5. Cumsille, P., Takahashi, T.: Wellposedness for the system modelling the motion of a rigid body of arbitrary form in an incompressble viscous fluid. Czech. Math. J. 58, 961–992 (2008)