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

• 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 $\delta \in [0,\delta_1]$.

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

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