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

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].