Weighted estimates for the incompressible fluid in exterior domains

For a strong solution u(x,t) of the Navier-Stokes equations in exterior domain Ω in Rn where n=2,3, we study the time decay of ‖α|x|u(t)Lp_‖ 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.