On the Rate of Decay of the Oseen Semigroup in Exterior Domains and its Application to Navier–Stokes Equation

Abstract

We prove Lp-Lq estimates of the Oseen semigroup in n-dimensional exterior domains (n 3), which refine and improve those obtained by Kobayashi and Shibata [15]. As an application, we give a globally in time stability theory for the stationary Navier-Stokes flow whose velocity at infinity is a non-zero constant vector. We thus extend the result of Shibata [21]. In particular, we find an optimal rate of convergence of solutions of the non-stationary problem to those of the corresponding stationary problem. (2000). 35B40, 35M20, 73B30. Mathematics Subject Classification

Stability for the 3D Navier–Stokes Equations with Nonzero far Field Velocity on Exterior Domains

“…For the nonlinear equations, Enomoto and Shibata [12] proved that u(t) L p (Ω) = o(t − 1 2 + 3 2p ) for 3 ≤ p ≤ ∞. We extend their results for p < 3 and improve the estimates for p ≥ 3.…”

“…One should note that under the assumption in the above theorem one can also get ∇u(t) L 3 (Ω) = o(t − 1 2 ) (refer to [12]). By using the results Theorem 1.1, we will derive spatial-temporal convergence rates of nonstationary solutions to the stationary solutions.…”

“…In this work, we consider the stability for the full Navier-Stokes flow on the exterior to a fixed object. In this direction, there are just a few results [12,13,31]. For the stationary case, Finn worked on such a problem [15,16].…”

“…Our goal consists of two; first, we extend the temporal convergence results in [12] to p < 3 and obtain better estimates for p ≥ 3, and secondly, we obtain the weighted stability.…”

On boundary-driven time-dependent Oseen flows

“…The velocity u : Z T → R 3 and the pressure π : Z T → R are unknown. In previous articles, problem (1) - (3) was usually solved by semigroup theory based on estimates of the Oseen resolvent ( [4], [5], [8], [10]). Recently reference [2] proposed a potential theoretic approach which leads to solutions of (1) - (3) in the form of a sum of certain volume potentials plus a single-layer potential.…”