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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 $$ \partial u/\partial t-\nu \Delta u+(u\nabla)u+\nabla p=0,\quad div u=0\text{ on } {\bbfR}\sp n $$ has a global strong solution, provided $\Vert u\sb 0\Vert\sb n\nu\sp{-1}$ 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 $\Vert u\Vert\sb{s,p}=\Vert (I-A)\sp{s/2}u\Vert\sb p$ (for $p\ge 2$, if $s>0)$, which decrease monotonically in time provided $\Vert u\sb 0\Vert\sb n\nu\sp{-1}$ is small enough.

35Q30Stokes and Navier-Stokes equations
35B40Asymptotic behavior of solutions of PDE