Existence and decay of mild solutions to fractional dissipative quasi-geostrophic type equations on R n *

Abstract A fractional dissipative quasi-geostrophic type equation on R n is considered in this paper. First of all, we show several important properties and estimates for the linear and nonlinear parts of the integral formulation. After that, the existence and uniqueness of mild solutions are established in critical space L p c ( p c = n 2 α − 1 ), when the L p c norm of the initial data is small enough. In addition, the sharp time-decay of mild solutions in L q with q > p c is also presented. Finally, in the case of the general initial data, we provide a decay result in L p and derive the optimal decay rate in L q , where p ⩾ 2 , p c ⩽ q < ∞ . It is worth emphasizing that when n = 2, we recover the rates obtained by Niche and Schonbek (Niche and Schonbek M E 2007 Commun. Math. Phys. 276 93–115).