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Optimal weighted estimates of the flows in exterior domains. (English) Zbl 1193.35122

From the summary: We prove some decay properties of global solutions for the Navier-Stokes equaitons in an exterior domain \(\Omega\subset\mathbb R^n\), \(n=2,3\).
When a domain has a boundary, the pressure term co problems 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. C. He and Z.-P. Xin [Methods Appl. Anal. 7, No. 3, 443–458 (2000; Zbl 1007.76012)] first introduced this method and then H.-O. Bae and B. J. Jin [J. Funct. Anal. 240, No. 2, 508–529 (2006; Zbl 1115.35095), Bull. Korean Math. Soc. 44, No. 3, 547–567 (2007; Zbl 1146.35070)] modified the method to obtain better decay rates. Also, H.-O. Bae and J. Roh [J. Math. Anal. Appl. 355, No. 2, 846–854 (2009; [Zbl 1168.35407)] improved Bae-Jin’s results. Unfortunately, their results were not optimal, because there exists an unpleasant positive small \(\delta\) in their rates.
In this paper, we obtain optimal rate without \(\delta\).

MSC:

35Q30 Navier-Stokes equations
76D07 Stokes and related (Oseen, etc.) flows
76D05 Navier-Stokes equations for incompressible viscous fluids
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
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References:

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