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

摘要

It is proved that there exists a local-in-time solution u is an element of C([0,T),bmo(Rd)d)$$uin C([0,T),bmo({mathbb {R}}<^>d)<^>d)$$end{document} 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 u0 is a local BMO function. Our method for proving this is a combination and refinement of the work by Grujic and Kukavica (J Funct Anal 152(2):447-466, 1998. ), Guberovic (Discrete Contin Dyn Syst 27(1):231-236, 2010. ), and Kozono et al. (Kyushu J Math 57(2):303-324, 2003. ). One challenging step is the estimation of the heat and Stokes semigroups from BMO-type spaces to L infinity a result that is itself of independent interest. We also apply the idea to the analyticity of vorticity with the assistance of Calderon-Zygmund theory.