Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in \(\mathbb{R}^n\)
type: Item
instance of: scholarly article
title: Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in \(\mathbb{R}^n\)
zbMATH Open document ID: 0960.35081
author: Marcel Oliver
author: E. S. Titi
published in: Journal of Functional Analysis
publication date: 2001-05-10T00:00:00Z
review text: Some estimations for the decay of higher-order derivatives of solutions to the unsteady unforced Navier-Stokes (N-S) equations in \(\mathbb{R}^n\) are obtained. The main result is an extension to \(\mathbb{R}^n\) of [\textit{C. Foias} and \textit{R. Temam}, J. Funct. Anal. 87, No. 2, 359-369 (1989; Zbl 0702.35203)], where periodic boundary conditions were considered. Some assumptions are used, concerning the algebraic decay of N-S solutions, but only for \(n\leq 3\). The main point is to prove that the solutions enter a subclass of real analytic functions in an arbitrary short time. For a large class of initial data, the N-S solution stays close to a solution of the heat equation in the \(L^2\) norm. By using this fact, lower bounds concerning the decay of derivatives can be obtained.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1443261
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: comparison with heat equation
reviewed by: Gelu Paşa
MaRDI profile type: Publication
full work available at URL: jfan.1999.3550
OpenAlex ID: W1971630350
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DOI: 10.1006/JFAN.1999.3550
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description: scientific article; zbMATH DE number 1443261