Algebraic $L\sp 2$ decay for Navier-Stokes flows in exterior domains. II

The Navier-Stokes equations in the weak- $L^n$ space with time-dependent external force

“…Standard L p -L q estimates are proved by Kato [13], Ukai [36], Iwashita [12], Giga and Sohr [11] and Borchers and Miyakawa [3], [4]. We start with a fact which corresponds to [4,Theorem 4.4]. In view of (2.2), it suffices to show the estimate…”

On a larger class of stable solutions to the Navier-Stokes equations in exterior domains

“…Let us first recall some previous results on the Stokes operator A r in L r σ given by Giga-Sohr [12] and Borchers-Miyakawa [2]. Proposition 2.1 Theorem 3.1]) Let π 2 < ω < π.…”

“…See Giga [10] and Giga-Sohr [12]. Moreover, if we impose restrictions on r, then the following sharper estimates hold: [12,Theorem 6.5], Borchers-Miyakawa [2,) (1) Let 1 < r < n and 0 ≤ α < n/2r. Then we have an embedding D(A α r ) ⊂ L q for 1/q = 1/r − 2α/n with the estimate…”

“…If w ≡ 0, we have L r = A r , and in this case our problem coincides with the initial boundary value problem for the usual nonstationary Navier-Stokes equations. Many efforts had been made to obtain the L p −L r estimates for the Stokes semigroup e −tAr in unbounded domains ( [2], [12] [15], [17], [37]). To treat L r as such perturbation of A r as the L p,∞ − L r -estimate remains to be obtained, Kozono-Ogawa [19] and Borchers-Miyakawa [3] required smallness of ∇w L n 2 and of C w , respectively.…”

On estimates in Hardy spaces for the Stokes flow in a half space

“…For example, a regularity-decay estimate (0. 2) ∇u(t) p ≤ Ct −1/2 u 0 p for t > 0 holds for all 1 ≤ p ≤ ∞ with C independent of t and u 0 , where f (t) p := Ω |f (t, x)| p dx 1/p and ∇ denotes the gradient in the space variables. If…”