Spatially almost periodic complex‐valued solutions of the Boussinesq equations.

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]