June  2010, 3(2): 361-370. doi: 10.3934/dcdss.2010.3.361

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

1. 

Institute of Hydrodynamics, Pod Patankou 30/5, 166 12 Prague 6, Czech Republic

Received  February 2009 Revised  August 2009 Published  April 2010

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 $\delta \in [0,\delta_1]$.

Citation: Zdeněk Skalák. On the asymptotic decay of higher-order norms of the solutions to the Navier-Stokes equations in R3. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 361-370. doi: 10.3934/dcdss.2010.3.361
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