On the Oseen equation in the three dimensional exterior domains

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

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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.

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“…The main tool in [21] was the L p -L q estimate of the Oseen semigroup corresponding to (1.3), which was obtained by Kobayashi and Shibata [15]. Successively, Galdi, Heywood and Shibata [10] used the results of [15] to solve the so called Finn's starting problem.…”

“…Successively, Galdi, Heywood and Shibata [10] used the results of [15] to solve the so called Finn's starting problem. All the above results [21,10] were obtained under some smallness assumption.…”

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.…”

Variational Properties of a Generic Model Equation in Exterior 3D Domains

“…A common approach to study the asymptotic properties of the solutions to the Dirichlet problem of the classical steady Oseen flow (the case aðxÞ 1 0) is to use convolutions with Oseen fundamental tensor and its first and second gradients for the velocity (or with the fundamental solution of Laplace equation for the pressure), see e.g. [3,10,11,8]. So, the situation is completely di¤erent in the study of our model equation.…”