# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] DOI: 10.1007/BF01212349 · Zbl 0573.76029 [2] DOI: 10.1016/0362-546X(85)90039-2 · Zbl 0621.76027 [3] Bony J., Ann. Sci. Ecole Norm. Sup. 14 pp 209– (1981) [4] DOI: 10.1080/03605308808820568 · Zbl 0659.35115 [5] DOI: 10.1002/cpa.3160350604 · Zbl 0509.35067 [6] J. Y. Chemin Remarques sur l’existence globale pour le systeme de Navier-Stokes incompressible Preprint [7] Cotter G., 1 303, in: C. R. Acad. Sci. pp 105– (1986) [8] DOI: 10.1002/cpa.3160400304 · Zbl 0850.76730 [9] DOI: 10.1080/03605308408820341 · Zbl 0548.76005 [10] DOI: 10.2307/1970699 · Zbl 0211.57401 [11] DOI: 10.1007/BF00281533 · Zbl 0254.35097 [12] J. Y.Chemin Remarques sur l’existence globale pour le systeme de Navier-Stokes incompressible Preprint p. Federbush Navier and Stokes meet the wavelet Preprint [13] DOI: 10.1007/BF02392215 · Zbl 0257.46078 [14] DOI: 10.1007/BF00276188 · Zbl 0126.42301 [15] DOI: 10.1080/03605308908820621 · Zbl 0681.35072 [16] DOI: 10.1007/BF00281355 · Zbl 0666.76052 [17] Hörmander L., Lectures Notes (1986) [18] DOI: 10.1070/SM1991v069n02ABEH002116 · Zbl 0724.35088 [19] DOI: 10.1007/BF01174182 · Zbl 0545.35073 [20] J. Y. Chemin Remarques sur l’existence globale pour le systeme de Navier-Stokes incompressible Preprint p. Federbush Navier and Stokes meet the wavelet Preprint T. kato Strong solutions of the Navier-Stokes equation in Morrey spaces Preprint [21] Majda A., Appl. Math. Sci. [22] Meyer Y., Rend. del Circolo mat. di palermo pp 1– (1981) [23] DOI: 10.1080/03605307708820054 · Zbl 0397.35071 [24] DOI: 10.1007/BF02414340 · Zbl 0149.09102 [25] DOI: 10.1016/0022-1236(69)90022-6 · Zbl 0175.42602 [26] Serrin J., Encly. of Physics 8 pp 125– (1959) [27] Stein E., Graduate Lecture Notes (1972) [28] Taylor, M. 1981. ”Pseudodifferential Operators”. Princeton Univ. Press. · Zbl 0453.47026 [29] DOI: 10.1007/978-1-4612-0431-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.