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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
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