Asymptotic behavior for the Navier–Stokes equations in 2D exterior domains

Abstract

We show that the L p spatial-temporal decay rates of solutions of incompressible flow in an 2D exterior domain. When a domain has a boundary, pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary. To overcome the difficulty, we adopt the ideas in He, Xin [C. He, Z. Xin, Weighted estimates for nonstationary Navier-Stokes equations in exterior domain, Methods Appl. Anal. 7 (3) (2000) 443-458], and our previous results [H.-O. Bae, B.J. Jin, Asymptotic behavior of Stokes solutions in 2D exterior domains, J. Math. Fluid Mech., in press; H.-O. Bae, B.J. Jin, Temporal and spatial decay rates of Navier-Stokes solutions in exterior domains, submitted for publication].For the spatial decay rate estimate, we first extend temporal decay rate result of the Navier-Stokes solutions for general L p space when the initial velocity is in L 2 σ ∩ L r , 1 < r q < ∞ (1 < r < q = ∞).

Large time behavior for the incompressible Navier–Stokes flows in 2D exterior domains

“…In addition, let α > 0, we consider the decay rates of |x| α u(t) L q ( ) with 1 < q < ∞, α 2 + 1 q < 1; |x| α ∇u(t) L q ( ) with 1 ≤ q < ∞, α 2 + 1 q < 3 2 ; and |x| α ∇ 2 u(t) L q ( ) with 1 ≤ q < 4, α 2 + 1 q < 2, which improve Theorem 1.2 in [5]. Here our aim is to characterize the relations of α and q, which is the main reason allowing for more general weights compared to [5].…”

“…which improves Theorem 1.2 in [5]: Let 2 ≤ q < ∞ and 0 < α ≤ 1. Then for large t, and any small δ > 0,…”

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

“…For our purpose we modify significant part of the idea in [5,6] which was used to obtain weighted estimate for the Navier-Stokes flow with zero constant velocity at infinity.…”

“…Spatial and temporal behaviors of incompressible flows with zero velocity at infinity have been studied well for various domains: temporal decay estimates by Kato [25], Fujigaki and Miyakawa [17], Miyakawa and Schonbek [27], Miyakawa [26], Wiegner [33], Schonbek [29], Galdi and Maremonti [19], Iwashita [22], Kozono [23], Kozono and Ogawa [24], Borchers and Miyakawa [9], Bae and Choe [3], etc; spatial ones by He and Xin [20], Takahashi [32], Farwig and Sohr [14], Brandolese and Vigneron [11], Bae and Jin [4][5][6][7], Bae [1,2], etc.…”

Navier–Stokes equations on exterior domains: review for the stability for the incompressible flow on exterior domains

“…Bae-Jin [3] improved the temporal decay for generalized space L p with u 0 ∈ L 2 σ ∩ L r and obtained…”