Asymptotic behavior of the Navier-Stokes equations with nonzero far-field velocity
Summary: Concerning the nonstationary Navier-Stokes flow with a nonzero constant velocity at infinity, the temporal stability has been studied by \textit{J.G. Heywood} [Arch. Ration. Mech. Anal. 37, 48--60 (1970; Zbl 0194.41402), Acta Math. 129, 11--34 (1972; Zbl 0237.35074)] and \textit{K. Masuda} [J. Math. Soc. Japan 27, 294--327 (1975; Zbl 0303.76011)] in \(L^2\) space and by \textit{Y. Shibata} [Q. Appl. Math. 57, No. 1, 117--155 (1999; Zbl 1157.35451)] and \textit{Y. Enomoto} and \textit{Y. Shibata} [J. Math. Fluid Mech. 7, No. 3, 339--367 (2005; Zbl 1094.35097)] in \(L^p\) spaces for \(p \geq 3\). However, their results did not include enough information to find the spatial decay. So, \textit{H.-O. Bae} and \textit{J. Roh} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 5, 1350--1363 (2010; Zbl 1193.35122)] improved Enomoto and Shibata's [loc. cit.] results in some sense and estimated the spatial decay even though their results are limited. In this paper, we will prove temporal decay with a weighted function by using \(L^r\)-\(L^p\) decay estimates obtained by \textit{J. Roh} [Abstr. Appl. Anal. 2011, Article ID 856896, 15 p. (2011; Zbl 1230.35088)]. Bae and Roh [loc. cit.] proved the temporal rate becomes slower by \((1 + \sigma)/2\) if a weighted function is \(|x|^\sigma\) for \(0 < \sigma < 1/2\). In this paper, we prove that the temporal decay becomes slower by \(\sigma\), where \(0 < \sigma < 3/2\) if a weighted function is \(|x|^\sigma\). For the proof, we deduce an integral representation of the solution and then establish the temporal decay estimates of weighted \(L^p\)-norm of solutions. This method was first initiated by \textit{C. He} and \textit{Z.-P. Xin} [Methods Appl. Anal. 7, No.3, 443--458 (2000; Zbl 1007.76012)] and developed by \textit{H.-O. Bae} and \textit{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); J. Math. Anal. Appl. 355, No. 2, 846-854 (2009; Zbl 1168.35407)].
scientific article; zbMATH DE number 5997790
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