Search

We extend several ideas developed in E. Dinaburg, Ya. G. Sinai and D. Li's papers (see Lemma , Remark  , Lemma , Remark  , Remark) and construct a unique mild solution (u,θ) of the system satisfying with spatially almost periodic initial data (u0,θ0)∈FM0,σ(R3,C3)×FM0(R3). Moreover, (u,θ) is also a spatially almost periodic complex‐valued mild solution. Assume further that (u0,θ0)∈FM0,σ∞(R3,C3)×FM0∞(R3) (see Definition). Then, we overcome the difficulty in the work of Y. Giga et al. (see Introduction below) and show that the above mild solution also satisfies for all S∈N. [ABSTRACT FROM AUTHOR]

We are concerned with a family of dissipative active scalar equation on R 2. 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 satisfying . In this paper, we consider some countable sum‐closed frequency sets (see Remark ). We show that the property of the solution is rather different from Chae et al and obtain that ( − Δ ) α 2 θ ( · , t ) ∉ L p ( R 2 ) with some initial data θ0 for all t≥0, 0 < p < ∞ 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 , we have for any 0≤α≤ω that ( − Δ ) α 2 θ ( x , t ) belongs to B U C ( R 2 × ( 0 , ∞ ) ), where Fδ is defined in or . [ABSTRACT FROM AUTHOR]