| Abstract: | We are concerned with a family of dissipative active scalar equation on  . By using similar methods from the previous paper of Y. Giga et al. (see Introduction below), we construct a unique real, spatially almost periodic mild solution θ of 1.1 satisfying 1.11 . In this paper, we consider some countable sum‐closed frequency sets (see Remark 1.1 ). We show that the property of the solution is rather different from Chae et al 1 and obtain that  with some initial data θ0 for all t≥0,  and 0≤α≤ω, where ω is a fixed constant. Furthermore, arranging the elements of a countable sum‐closed frequency set Fδ as in Remark 1.3 , we have for any 0≤α≤ω that  belongs to  , where Fδ is defined in 1.4 or 1.5 . |
