Pointwise decay estimate of Navier–Stokes flows in the half space with slowly decreasing initial value. (July 2017)

• Record Type: Journal Article
• Authors: Chang, Tongkeun; Jin, Bum Ja
• Is Part Of: Nonlinear analysis. Volume 157(2017)
• Page Start: 167
• Page End: 188
• Publication Date: 2017-07
• Subjects: primary 35K61 -- secondary 76D07; Navier–Stokes -- Spatial decay -- Half sapce; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2017.03.012
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we study the spatial and temporal decay estimate of the Navier–Stokes flow in the half space corresponding to auniformly but slowly decreasing initial velocity. We show the local in time solvability of the Navier–Stokes equations with | u ( x, t ) | ≤ C 0 ( 1 + | x | + t ) − α and | ∇ u ( x, t ) | ≤ C 0 t − 1 2 ( 1 + | x | + t ) − α when ( 1 + | x | + t ) α e − t A h ∈ L ∞ ( R + n × ( 0, ∞ ) ) and t 1 2 ( 1 + | x | + t ) α e − t A h ∈ L ∞ ( R + n × ( 0, ∞ ) ), 0 < α ≤ n . Asymptotically, it holds that u ( x, t ) = e − t A h ( x ) + o t ( | x | − α ), 0 < α < n, as | x | → + ∞ . The solution exists globally in time for α ≥ 1 when the data is small enough, and it holds that u ( x, t ) = e − t A h ( x ) + o ( ( | x | + t ) − α ), 1 < α < n, as | x | + t → + ∞ .