Asymptotic expansion for solutions of the Navier-Stokes equations with non-potential body forces

We study the long-time behavior of spatially periodic solutions of the Navier-Stokes equations in the three-dimensional space. The body force is assumed to possess an asymptotic expansion or, resp., finite asymptotic approximation, in Sobolev-Gevrey spaces, as time tends to infinity, in terms of polynomial and decaying exponential functions of time. We establish an asymptotic expansion, or resp., finite asymptotic approximation, of the same type for the Leray-Hopf weak solutions. This extends previous results that were obtained in the case of potential forces, to the non-potential force case, where the body force may have different levels of regularity and asymptotic approximation. This expansion or approximation, in fact, reveals precisely how the structure of the force influences the asymptotic behavior of the solutions.