A remark for spatial analyticity around straining flows

Abstract

Time-local existence of unique smooth solutions to the Navier-Stokes equations in the whole space with linearly growing initial data has been established, via smoothing properties of Ornstein-Uhlenbeck semigroup. It has also been shown that the solution is real-analytic in spatial variables around rotating flows. This note is devoted to prove the spatial analyticity for cases of straining flows and shear flows. It is estimated the size of radius of convergence of Taylor series, due to estimates for higher order derivatives and Cauchy-Hadamard theorem.