Optimal weighted estimates of the flows in exterior domains

In this paper, we prove some decay properties of global solutions for the Navier-Stokes equations in an exterior domain Ω⊂Rⁿ, n=2,3.When a domain has a boundary, the pressure term is troublesome since we do not have enough information on the pressure near the boundary. To overcome this difficulty, by multiplying a special form of test functions, we obtain an integral equation. He-Xin (2000) 12] first introduced this method and then Bae-Jin (2006, 2007) 1] and 13] modified their method to obtain better decay rates. Also, Bae-Roh (2009) 11] improved Bae-Jin’s results. Unfortunately, their results were not optimal, because there exists an unpleasant positive small δ in their rates.In this paper, we obtain the following optimal rate without δ,