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Asymptotic Behavior of the Navier-Stokes Equations with Nonzero Far-Field Velocity

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  • Jaiok Roh

Abstract

Concerning the nonstationary Navier-Stokes flow with a nonzero constant velocity at infinity, the temporal stability has been studied by Heywood (1970, 1972) and Masuda (1975) in space and by Shibata (1999) and Enomoto-Shibata (2005) in spaces for . However, their results did not include enough information to find the spatial decay. So, Bae-Roh (2010) improved Enomoto-Shibata's 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 decay estimates obtained by Roh (2011). Bae-Roh (2010) proved the temporal rate becomes slower by if a weighted function is for . In this paper, we prove that the temporal decay becomes slower by where if a weighted function is . For the proof, we deduce an integral representation of the solution and then establish the temporal decay estimates of weighted -norm of solutions. This method was first initiated by He and Xin (2000) and developed by Bae and Jin (2006, 2007, 2008).

Suggested Citation

  • Jaiok Roh, 2011. "Asymptotic Behavior of the Navier-Stokes Equations with Nonzero Far-Field Velocity," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-15, November.
    • Handle: RePEc:hin:jnlaaa:369571
    DOI: 10.1155/2011/369571
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