On the asymptotic decay of higher-order normsof the solutions to the Navier-Stokes equations in R3

Zdeněk Skalák

doi:10.3934/dcdss.2010.3.361

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2010, Volume 3, Issue 2: 361-370. Doi: 10.3934/dcdss.2010.3.361

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On the asymptotic decay of higher-order norms of the solutions to the Navier-Stokes equations in R³

Zdeněk Skalák^1,

1.
Institute of Hydrodynamics, Pod Patankou 30/5, 166 12 Prague 6

Received: January 31, 2009

Revised: July 31, 2009

Published: May 2010

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Abstract

Let $A$ be the Stokes operator. We show as the main result of the paper that if $w$ is a global weak solution to the Navier-Stokes equations satisfying the strong energy inequality, $\beta \in [0,1/2]$ and $\alpha \in [\beta,\infty)$, then there exist $t_0 \ge 1$, $C_1>1$ and $\delta_1 \in (0,1)$ such that

$ ||A^\alpha w(t)|| \le C_1 ||A^\beta w(t+\delta)|| $

for every $t \ge t_0$ and every $\delta \in [0,\delta_1]$.

Keywords:

Navier-Stokes equations,
large-time decay.

Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D05.

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On the asymptotic decay of higher-order norms of the solutions to the Navier-Stokes equations in R³