The purpose of this paper is to prove L-P-boundedness of an operator phi(H-V), where H-V = -Delta + V(x) is the Schrodinger operator on an open set Omega of R-d (d >= 1). Moreover, we prove uniform L-p-estimates for phi(theta H-V) with respect to a parameter theta > 0. This paper will give an improvement of our previous paper (Iwabuchi et al., L-p-mapping properties for Schrodinger operators in open sets of R-d, submitted); assumptions of potential V and space dimension.

We consider the stability problem of time periodic solutions for the rotating Navier-Stokes equations. For the non-rotating case, it is known that time periodic solutions to the original Navier-Stokes equations are asymptotically stable under the smallness assumptions both on the time periodic solutions and on the initial disturbances. We shall treat the high-rotating cases, and prove the asymptotic stability of large time periodic solutions for large initial perturbations. © Springer Basel 2016.