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.