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⊂RnΩ⊂Rn, n=2,3n=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 δδ in their rates.In this paper, we obtain the following optimal rate without δδ, ‖|x|αu(t)‖Lp≤Ct−n2(1r−1p)+α2, for the initial condition u0 with ∣x∣αu0∈Lr and for any α