Remarks on upper and lower bounds of solutions to the Navier-Stokes equations in R2

In the study of long time asymptotic behaviors of the solutions of the two-dimensional Navier-Stokes equations in the whole space, some optimal upper and lower bounds for the decay of higher order derivatives of the solutions are derived through a simple approach without assuming any decay properties of the solutions. This method is mainly based on some auxiliary decay estimates of the solutions and a generalized Gronwall type argument.