Spatially almost periodic complex‐valued solutions of the Boussinesq equations

Abstract

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 false(u0,θ0false)∈FM0,σfalse(R3,C3false)×FM0false(R3false). Moreover, (u,θ) is also a spatially almost periodic complex‐valued mild solution. Assume further that false(u0,θ0false)∈FM0,σ∞false(R3,C3false)×FM0∞false(R3false) (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∈double-struckN.