Advances in Differential Equations

On the boundedness and decay of moments of solutions to the Navier-Stokes equations

Maria E. Schonbek and Tomas P. Schonbek

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In this paper we consider the existence and decay of moments of the solutions to the Navier-Stokes equations in the whole space ${{\bf R}^n}$; $n \geq 2$ for existence, $2 \leq n \leq 5$ for decay. The decay obtained is algebraic, of order \[ \int_{{{\bf R}^n}} |x|^k |u|^2\,dx \leq C(t+1)^{-2\mu(1-k/n)} \] for $0\leq k\leq n$, for solutions $u$ of appropriate data for which the $L^2$ norm decays at a rate of order $\mu$. That is for solutions that satisfy $||u(t)||_2 \leq C(t+1) ^{-\mu}$. Where $\mu >1/2$. Such solutions are easy to obtain as for example it suffices for $n=3$ that the data $u_0$ is in $L^2 \cap L^1$.

Article information

Adv. Differential Equations, Volume 5, Number 7-9 (2000), 861-898.

First available in Project Euclid: 27 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]
Secondary: 35A07 35A35: Theoretical approximation to solutions {For numerical analysis, see 65Mxx, 65Nxx} 35B41: Attractors 76D03: Existence, uniqueness, and regularity theory [See also 35Q30] 76D05: Navier-Stokes equations [See also 35Q30]


Schonbek, Maria E.; Schonbek, Tomas P. On the boundedness and decay of moments of solutions to the Navier-Stokes equations. Adv. Differential Equations 5 (2000), no. 7-9, 861--898.

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