| We study the incompressible, isotropic Navier-Stokes system and other semi-linear parabolic equations for which the initial data belong to Banach spaces modeled on Besov spaces. Specifically, we consider the class, introduced by Hideo Kozono and Masao Yamazaki, of Besov spaces based on Morrey spaces, which we call KY spaces. We first produce a wavelet decomposition and obtain sharper embeddings. 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 KY 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 supercritical regularity. We exploit pseudo-differential calculus to extend the analysis to compact, smooth, boundary-less, Riemannian manifolds. KY spaces are defined by means of partitions of unity and coordinate patches, and intrinsically in terms of functions of the Laplace operator. We also study some abstract functional equations with non-linearities satisfying a log-Lipschitz condition. We construct examples for which existence and uniqueness hold. |
