Optimal weighted estimates of the flows in exterior domains Original Research Article

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) and 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, View the MathML source‖|x|αu(t)‖Lp≤Ct−n2(1r−1p)+α2, Turn MathJax on for the initial condition View the MathML sourceu0 with View the MathML source∣x∣αu0∈Lr and for any α