## Advances in Differential Equations

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

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

Maria E. Schonbek and Tomas P. Schonbek

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 27 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1356651290

**Mathematical Reviews number (MathSciNet)**

MR1776344

**Zentralblatt MATH identifier**

1027.35095

**Subjects**

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]

#### Citation

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. https://projecteuclid.org/euclid.ade/1356651290