The L ∞-Stokes semigroup in exterior domains

Abstract

Abstract. The Stokes semigroup is extended to an analytic semigroup in spaces of bounded functions in an exterior domain with C 3 boundary. Some of these spaces include vector fields non-decaying at the space infinity. Moreover, uniform bounds by a sup-norm of initial velocity are established in finite time for second spacial derivatives of velocity and also for gradient of pressure to the Stokes equations.

On the Stokes resolvent estimates for cylindrical domains

“…For a C 3 exterior domain, a similar estimate is proved in [AG2]. In both cases we need not assume that ∇u ∈ L 2 (Ω) ∩ L r (Ω).…”

“…For a general domain it is proved [AG1] that the semigroup is analytic in C 0,σ provided that the domain is "admissible" in the sense of [AG1]. Since it turns out that a bounded domain [AG1] and an exterior domain [AG2] are admissible (even strictly admissible), we conclude that S(t) forms an analytic semigroup in such a domain; for improvement of these results see [AGH] where only C 2 regularity is used.…”

“…Since g is tangential, ∇ ∂Ω can be replaced by ∇. Such a notion of a very weak solution for (2.8) is introduced by [AG1], [AG2]. These types of notions of a very weak solution are elaborated by [MR] to handle the case when the Neumann data equals the Dirac measure which corresponds to the Green function of the Neumann problem.…”

“…We only need to assume that u is a (very) weak solution. Moreover, we need not assume the regularity of g. If (1.3) holds for all very weak solutions u having finite left-hand side of (1.3), we say that Ω is strictly admissible [AG2]. Note that an infinite cylinder is not strictly admissible because a linear function u(x 1 , x ′ ) = x 1 solves (1.2a), (1.2b) with g = 0.…”

“…This type of result as well as [AG1], [AG2], [AGH] is concerned with regularizing effect locally-in-time. The boundedness of the Stokes semigroup S(t) for t > 0 is a different question.…”