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Liapunov functions and monotonicity in the Navier-Stokes equation. (English) Zbl 0727.35107

Functional-analytic methods for partial differential equations, Proc. Conf. Symp., Tokyo/Jap. 1989, Lect. Notes Math. 1450, 53-63 (1990).
[For the entire collection see Zbl 0707.00017.]
It is well known, that the Cauchy problem for the Navier-Stokes equations \[ \partial u/\partial t-\nu \Delta u+(u\nabla)u+\nabla p=0,\quad div u=0\text{ on } {\mathbb{R}}^ n \] has a global strong solution, provided \(\| u_ 0\|_ n\nu^{-1}\) is small, see the author [Math. Z. 187, 471-480 (1984; Zbl 0537.35065)]. The author shows, that one can obtain more information, namely the existence of several types of Lyapunov functions. Special types are the norms \(\| u\|_{s,p}=\| (I-A)^{s/2}u\|_ p\) (for \(p\geq 2\), if \(s>0)\), which decrease monotonically in time provided \(\| u_ 0\|_ n\nu^{-1}\) is small enough.

MSC:

35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs