Weighted estimate of Stokes semigroup in unbounded domains

Abstract

We show LP -Lq estimates with the (x) 8 type weight of Stokes semigroup in exterior domains and perturbed half-spaces. Moreover as application of these estimates, we obtain weighted estimates for global solution to the N avier-Stokes equations with small data in these domains. §1. IntroductionLet n c ~n(n 2: 2) be an exterior domain or a perturbed halfspace with smooth boundary. We consider the following Navier-Stokes equations in these unbounded domains n c ~n:

Anisotropically weighted Lq$L^q$‐Lr$L^r$ estimates of the Oseen semigroup in exterior domains, with applications to the Navier–Stokes flow past a rigid body

“…Remark By the similar argument to the proof of Proposition 5.3, Kobayashi and Kubo [42] also derived the homogeneous estimate (107), but it is not clear whether we can obtain the homogeneous one if $$\beta >0$$ or if $$a>0$$. For the relation between (105) with $$\beta =0$$ and (107), we can recover (107) by taking the limit $$a\rightarrow 0$$.…”

“…We already know from [19] that $$-A$$ generates an analytic C 0 ‐semigroup (Stokes semigroup) in weighted $$L^q$$ space whenever the weight belongs to $$\mathcal {A}_q(D)$$ and the Stokes semigroup is bounded in $$L^q_{{(1+|x|)}^{\alpha q}}(D)$$, see [19, Theorem 1.5]. Its $$L^q$$‐$$L^r$$ smoothing action near the initial time was derived by Kobayashi and Kubo [42, Theorem 1], see also [41]. We state in the following theorem that $$-A_a$$ generates an analytic C 0 ‐semigroup in $$L^{q}_{\rho ,\sigma }(D)$$ possessing the $$L^q$$‐$$L^r$$ smoothing action near the initial time.…”

“…Remark It is not known whether we can obtain homogeneous $$L^q$$‐$$L^r$$ estimates if $$\beta >0$$ or if $$a>0$$, while Kobayashi and Kubo [42] derived $$$$\begin{align*} \Vert {(1+|x|)}^\alpha e^{-tA}P_Df\Vert _{r,D}\le Ct^{-\frac{3}{2}{\left(\frac{1}{q}-\frac{1}{r}\right)}} \Vert {(1+|x|)}^\alpha f\Vert _{q,D} \end{align*}$$$$for $$t>0,f\in L^q_{{(1+|x|)}^{\alpha q}}(D)$$ provided $$1<q\le r<\infty$$ and (16) for $$t>0,f\in L^q_{{(1+|x|)}^{\alpha q}}(D)$$ provided …”

“…The proof of Theorem 1.1 consists of two steps: One is the decay estimate near the boundary of D ; the other is the decay estimate near infinity. This procedure is employed by Iwashita [38], Kobayashi and Kubo [42] for the Stokes semigroup, and by Kobayashi and Shibata [43], Enomoto and Shibata [16, 17], and Hishida [37] for the Oseen semigroup. In those papers, they derived the decay rate $$t^{-3/2q}$$ for given $$f\in L^q(D)$$ in the first step by carrying out a cut‐off procedure based on the $$L^q$$‐$$L^r$$ estimates of $$S_a(t)$$ and on the decay estimate near the boundary of D for initial velocity with compact support, called the local energy decay estimate, see Proposition 6.1 in Section 6.…”

The Green tensor of the nonstationary Stokes system in the half space

“…For mild solutions of (NS) in the half-space, the unique local and global existence in L q (R n + ) were established by Weissler [48] for 3 ≤ n < q < ∞, by Ukai [47] for 2 ≤ n ≤ q < ∞, and by Kozono [29] for 2 ≤ n = q. Canaone-Planchon-Schonbek [3] established unique existence of solutions in L ∞ L 3 with initial data in the homogeneous Besov space Ḃ3/q−1 q,∞ (R 3 + ). For mild solutions in weighted L q spaces, we refer the reader to [25,26].…”