Logarithmic evolutions on the incompressible Navier--Stokes flow

Abstract

Through the asymptotic expansion, the large-time behavior of the incompressible Navier-Stokes flow in $n$-dimensional whole space is drawn. In particular, the logarithmic evolution included in the flow velocity is the focus of attention. When the components of velocity are ordered from major to minor according to the parabolic scales, the logarithmically evolving components appear in a certain pattern. This work asserts that this pattern varies depending on the even-oddness of the space dimension. This means that the parity of the nonlinear drift is different in even and odd dimensions. The logarithmic term represents the nonlinear component of the phenomenon. No symmetry of the initial condition is required to prove this fact. In the preceding works, the expansion with the $2n$th order was already derived. The assertion is derived by reexamining these works in detail.