Decay estimates of gradient of a generalized Oseen evolution operator arising from time-dependent rigid motions in exterior domains

Abstract

Let us consider the motion of a viscous incompressible fluid past a rotating rigid body in 3D, where the translational and angular velocities of the body are prescribed but time-dependent. In a reference frame attached to the body, we have the Navier-Stokes system with the drift and (one half of the) Coriolis terms in a fixed exterior domain. The existence of the evolution operator $T(t,s)$ in the space $L^q$ generated by the linearized non-autonomous system was proved by Hansel and Rhandi [26] and the large time behavior of $T(t,s)f$ in $L^r$ for $(t-s)\to \infty$ was then developed by the present author [33] when $f$ is taken from $L^q$ with $q\le r$. The contribution of the present paper concerns such $L^q$-$L^r$ decay estimates of $\nabla T(t,s)$ with optimal rates, which must be useful for the study of stability/attainability of the Navier-Stokes flow in several physically relevant situations. Our main theorem completely recovers the $L^q$-$L^r$ estimates for the autonomous case (Stokes and Oseen semigroups, those semigroups with rotating effect) in 3D exterior domains, which were established by [37], [42], [39], [36] and [44].

arXiv comment45pages

References in corpus (2)

• On the Navier-Stokes equations with rotating effect and prescribed outflow velocity
• Navier-Stokes flow past a rigid body: attainability of steady solutions as limits of unsteady weak solutions, starting and landing cases

Cited by in corpus (5)

• Navier-Stokes Flow past a Rigid Body that Moves by Time-Periodic Motion
• Stability of time-dependent motions for fluid-rigid ball interaction
• Regularity properties of a generalized Oseen evolution operator in exterior domains, with applications to the Navier-Stokes initial value problem
• Spatial pointwise behavior of time-periodic Navier-Stokes flow induced by oscillation of a moving obstacle
• Large time decay of the Oseen flow in exterior domains subject to the Navier slip-with-friction boundary condition