Weighted estimates for the incompressible fluid in exterior domains

Abstract

For a strong solution u ( x , t ) of the Navier–Stokes equations in exterior domain Ω in R n where n = 2 , 3 , we study the time decay of ‖ | x | α u ( t ) ‖ L p for α < n . 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 [H.-O. Bae, B.J. Jin, Temporal and spatial decay rates of Navier–Stokes solutions in exterior domains, Bull. Korean Math. Soc. 44 (3) (2007) 547–567; H.-O. Bae, B.J. Jin, Asymptotic behavior for the Navier–Stokes solutions in 2D exterior domains, J. Funct. Anal. 240 (2006) 508–529] and we will extend Bae and Jin's results by modifying their methods.