Space-time asymptotics of the two dimensional Navier–Stokes flow in the whole plane

Abstract

We consider the space-time behavior of the two dimensional Navier–Stokes flow. Introducing some qualitative structure of initial data, we succeed to derive the first order asymptotic expansion of the Navier–Stokes flow without moment condition on initial data in L1(R2)∩Lσ2(R2). Moreover, we characterize the necessary and sufficient condition for the rapid energy decay ‖u(t)‖2=o(t−1) as t→∞ motivated by Miyakawa–Schonbek [21]. By weighted estimated in Hardy spaces, we discuss the possibility of the second order asymptotic expansion of the Navier–Stokes flow assuming the first order moment condition on initial data. Moreover, observing that the Navier–Stokes flow u(t) lies in the Hardy space H1(R2) for t>0, we consider the asymptotic expansions in terms of Hardy-norm. Finally we consider the rapid time decay ‖u(t)‖2=o(t−32) as t→∞ with cyclic symmetry introduced by Brandolese [2].