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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 $f\in M\sb r\sp n(M)\cap\circover M\sb 1\sp n(M)$, $r>1$, the Navier- Stokes equation on $M$ with the initial data $f$ has a unique solution $$u\in C([0,T], M\sb r\sp n(M))\cap C\sp \infty((0,T]\times M), \qquad t\sp{2/n}u\in C([0,T]\times M)$$ for some $T>0$, where $M\sb q\sp p(M)$ is the Morrey space on $M$ which is the natural analogue of $M\sb q\sp p(\bbfR\sp n)$ consisting of functions such that $$R\sp{-n}\int\sb{B\sb R}\vert f(x)\vert\sp q dx\leq CR\sp{-nq/p}$$ for any ball $B\sb R$ of radius $R\leq 1$ and $\circover M\sb q\sp p(\bbfR\sp n)$ is the set of all $f\in M\sb q\sp p(\bbfR\sp n)$ for which the left side of the above inequality is $o(R\sp{-nq/p})$ as $R\to 0$. This theorem is an extension of the results of {\it T. Kato} [Math. Z. 187, 471-480 (1984; Zbl 0537.35065)], and {\it Y. Giga} and {\it 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 $\bbfR\sp n$.

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
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