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Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations. (English) Zbl 0771.35047

Let M be a compact manifold of dimension n. The main result is that for fM r n (M)M 1 n (M), r>1, the Navier- Stokes equation on M with the initial data f has a unique solution

uC([0,T],M r n (M))C ((0,T]×M),t 2/n uC([0,T]×M)

for some T>0, where M q p (M) is the Morrey space on M which is the natural analogue of M q p ( n ) consisting of functions such that

R -n B R |f(x)| q dxCR -nq/p

for any ball B R of radius R1 and M q p ( n ) is the set of all fM q p ( n ) for which the left side of the above inequality is o(R -nq/p ) as R0. This theorem is an extension of the results of T. Kato [Math. Z. 187, 471-480 (1984; Zbl 0537.35065)], and Y. Giga and T. Miyakawa [Commun. Partial Differ. Equations 14, No. 5, 577-618 (1989; Zbl 0681.35072)]. It was announced that T. Kato recently obtained results similar to those of this paper, especially in the context of viscous flow one n .


MSC:
35Q30Stokes and Navier-Stokes equations
58D25Differential equations and evolution equations on spaces of mappings
46N20Applications of functional analysis to differential and integral equations
34G20Nonlinear ODE in abstract spaces