Uniform pointwise asymptotics of solutions to quasi-geostrophic equation

Abstract

We provide two-sided pointwise estimates and uniform asymptotics of the solutions to the subcritical quasi-geostrophic equation with initial data in L 2/(α−1) (R 2 ). Furthermore, we give upper bound of similar type for any derivative of the solutions. Initial data in L p (R 2 ), p > 2/(α − 1), are also discussed.where (P t ) t 0 is the semi-group generated by ∆ α/2 . Although all of the aforementioned results provide precise bounds for L p norms of the solutions, they do not say much about pointwise behaviour of these solutions. In particular, there are not known any lower bounds. In fact, this is rather common problem in the theory of nonlinear 2010 Mathematics Subject Classification. 35B40; 35K55; 35S10.