Long-time asymptotic behavior of the generalized two-dimensional quasi-geostrophic equation

In this paper, we are concerned with the long-time asymptotic behavior of the generalized two-dimensional quasi-geostrophic equation. More precisely, we obtain the sharp time-decay of the smooth solution of the generalized quasi-geostrophic equation with the sub-critical dissipation and the critical dissipation for the general initial data, and the sup-critical dissipation for the small initial data in the scale invariant Besov space. In addition, the sharp time-decay of any weak solution of the generalized quasi-geostrophic equation for the general initial data is also derived. Moreover, we also derive the decay estimates for the difference between the full solution and the solution to the corresponding linear part. (c) 2022 Elsevier Inc. All rights reserved.