Let be a compact manifold of dimension . The main result is that for , , the Navier- Stokes equation on with the initial data has a unique solution
for some , where is the Morrey space on which is the natural analogue of consisting of functions such that
for any ball of radius and is the set of all for which the left side of the above inequality is as . 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 .