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Lower bounds of rates of decay for solutions to the Navier-Stokes equations. (English) Zbl 0739.35070
The author gives explicit lower bounds for the $$L^ 2$$-decay of solutions to the Navier-Stokes equations in two and three space dimensions $u_ t-\Delta u+(u\cdot\nabla)u+\nabla p=0,\quad \hbox{div} u=0\quad \hbox{ in }\mathbb{R}^ n\times(0,\infty),\quad u(0)=u_ 0\hbox{ in }\mathbb{R}^ n.$ She especially concentrates on the case that the Fourier transform of the initial data vanishes at the origin. Under some additional conditions on $$u_ 0$$ she is able to show that $| u(\centerdot,t) |^ 2 _{L^ 2(\mathbb{R}^ n)} \geq C(1+t)^{- \alpha(n)}$ where $$\alpha(n)=n/2+1$$. Roughly speaking, the result is proved as follows: let $$v$$ denote the solution of the heat equation with initial data $$u(\centerdot,t_ 0)$$, where the time $$t_ 0$$ is chosen in such a way as to ensure that $$v$$ decays at a slow rate. Then it is shown that $$u$$ cannot decay faster than $$v$$.

##### MSC:
 35Q30 Navier-Stokes equations 35K55 Nonlinear parabolic equations
##### Keywords:
lower bounds of rates of decay; energy decay
Full Text:
##### References:
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