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Besov-Morrey spaces: Function space theory and applications to non-linear PDE

Author: Anna L. Mazzucato
Journal: Trans. Amer. Math. Soc. 355 (2003), 1297-1364
MSC (2000): Primary 35S05, 42B35; Secondary 76B03, 42C40, 35K55
Published electronically: December 5, 2002
MathSciNet review: 1946395
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Abstract: This paper is devoted to the analysis of function spaces modeled on Besov spaces and their applications to non-linear partial differential equations, with emphasis on the incompressible, isotropic Navier-Stokes system and semi-linear heat equations. Specifically, we consider the class, introduced by Hideo Kozono and Masao Yamazaki, of Besov spaces based on Morrey spaces, which we call Besov-Morrey or BM spaces. We obtain equivalent representations in terms of the Weierstrass semigroup and wavelets, and various embeddings in classical spaces. We then establish pseudo-differential and para-differential estimates. Our results cover non-regular and exotic symbols. Although the heat semigroup is not strongly continuous on Morrey spaces, we show that its action defines an equivalent norm. In particular, homogeneous BM spaces belong to a larger class constructed by Grzegorz Karch to analyze scaling in parabolic equations. We compare Karch's results with those of Kozono and Yamazaki and generalize them by obtaining short-time existence and uniqueness of solutions for arbitrary data with subcritical regularity. We exploit pseudo-differential calculus to extend the analysis to compact, smooth, boundaryless, Riemannian manifolds. BM spaces are defined by means of partitions of unity and coordinate patches, and intrinsically in terms of functions of the Laplace operator.

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Additional Information

Anna L. Mazzucato
Affiliation: Department of Mathematics, Yale University, P.O. Box 208283, 10 Hillhouse Ave., New Haven, Connecticut 06520-8283

Keywords: Navier-Stokes equation, function spaces, Littlewood-Paley, pseudo-differential calculus, global analysis
Received by editor(s): September 24, 2001
Received by editor(s) in revised form: September 9, 2002
Published electronically: December 5, 2002
Additional Notes: This work was completed while the author was on leave from Yale University and visiting the Mathematical Sciences Research Institute (MSRI). Research at MSRI was supported in part by NSF grant DMS-9810361.
Article copyright: © Copyright 2002 American Mathematical Society

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