Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0771.35047
Taylor, Michael E.
Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations. (English)
[J] Commun. Partial Differ. Equations 17, No.9-10, 1407-1456 (1992). ISSN 0360-5302; ISSN 1532-4133/e

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 $\bbfR\sp n$.
[H.Tanabe (Toyonaka)]

MSC 2000:: *35Q30 Stokes and Navier-Stokes equations
58D25 Differential equations in function spaces
46N20 Appl. of functional analysis to differential and integral equations
34G20 Nonlinear ODE in abstract spaces

Keywords: Morrey space; Riemannian manifold; Zygmund space; pseudodifferential operator; existence; uniqueness

Citations: Zbl 0537.35065; Zbl 0581.35072; Zbl 0681.35072

Cited in: Zbl 0951.35005 Zbl 0904.58012

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]