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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 \(f\in M_ r^ n(M)\cap\mathring M_ 1^ n(M)\), \(r>1\), the Navier- Stokes equation on \(M\) with the initial data \(f\) has a unique solution \[ u\in C([0,T], M_ r^ n(M))\cap C^ \infty((0,T]\times M), \qquad t^{2/n}u\in C([0,T]\times M) \] for some \(T>0\), where \(M_ q^ p(M)\) is the Morrey space on \(M\) which is the natural analogue of \(M_ q^ p(\mathbb{R}^ n)\) consisting of functions such that \[ R^{-n}\int_{B_ R}| f(x)|^ q dx\leq CR^{-nq/p} \] for any ball \(B_ R\) of radius \(R\leq 1\) and \(\mathring M_ q^ p(\mathbb{R}^ n)\) is the set of all \(f\in M_ q^ p(\mathbb{R}^ n)\) for which the left side of the above inequality is \(o(R^{-nq/p})\) as \(R\to 0\). 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 \(\mathbb{R}^ n\).

MSC:
35Q30 Navier-Stokes equations
58D25 Equations in function spaces; evolution equations
46N20 Applications of functional analysis to differential and integral equations
34G20 Nonlinear differential equations in abstract spaces
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