Decay rates for the incompressible Navier–Stokes flows in 3D exterior domains

Abstract

The exterior nonstationary problem is studied for the 3D Navier–Stokes equations, for which the associated total net force to the boundary may not vanish. The time-decay properties of the strong solution including the first and second derivatives are shown in Lq and weighted spaces. In particular, the relation of (weighted) L1-summability for smooth solutions is discussed in details between the time decay and the total net force exerted by the fluid to the body. The conclusions in this article improve and extend results in: [H. Bae, B. Jin, Asymptotic behavior for the Navier–Stokes equations in 2D exterior domains, J. Funct. Anal. 240 (2006) 508–529; H. Bae, B. Jin, Temporal and spatial decay rates of Navier–Stokes solutions in exterior domains, Bull. Korean Math. Soc. 44 (2007) 547–567] and in: [C. He, T. Miyakawa, On L1-summability and asymptotic profiles for smooth solutions to Navier–Stokes equations in a 3D exterior domain, Math. Z. 245 (2003) 387–417; C. He, T. Miyakawa, On weighted-norm estimates for nonstationary incompressible Navier–Stokes flows in a 3D exterior domain, J. Differential Equations 246 (2009) 2355–2386], respectively.