Local-in-time Solvability and Space Analyticity for the Navier-Stokes Equations with BMO-type Initial Data

Abstract

It is proved that there exists a local-in-time solution $u\in C([0,T),bmo({\mathbb{R}}^d{)}^d)$ of the Navier-Stokes equations such that every $u(t)$ has an analytic extension on a complex domain whose size only depends on $t$ (and increases with $t$) and the external force $f$, assuming only that the initial velocity $u_0$ is a local $BMO$ function. Our method for proving is a combination and refinement of the work by Grujić and Kukavica [13], Guberović [15] and Kozono et al. [18]. One challenging step is the estimation of the heat and Stokes semigroups from $BMO$-type spaces to $L^{\infty }$; a result itself of independent interest. We also apply the idea to the analyticity of vorticity with the assistance of Calderón-Zygmund theory.

31 pages