Uniform local solvability for the Navier-Stokes equations with the Coriolis force

摘要

This is a supplementary note of the paper: Y. Giga, K. Inui, A. Mahalov and S. Matsui (2005, Methods and Applications of Analysis), where local-in-time existence and uniqueness of mild solution for the 3-dimensional Navier-Stokes equations with the Coriolis force were established with its uniform existence time in the Coriolis parameter. The crucial part of the proof is to seek an appropriate class for initial data which allows uniform boundedness in t ∈ R of the Riesz semigroup whose symbol is exp(t(iξ_j/|ξ|)) (ξ = (ξ_1,ξ_2, ξ_3), i= -1~(1/2) for j = 1,2,3. For this purpose we found a new space denoted by FM_0, Fourier preimage of finite Radon measures having no-point mass at the origin. In Appendix we give an observation on the Mikhlin theorem in the Besov-type space B_(z,1)~0 for a Banach space Z which is included in the space of temperd distributions S'.