The Littlewood-Paley decomposition for periodic functions and applications to the Boussinesq equations

The Littlewood-Paley decomposition for functions defined on the whole space R-d and related Besov space techniques have become indispensable tools in the study of many partial differential equations (PDEs) with R-d as the spatial domain. This paper intends to develop parallel tools for the periodic domain T-d. Taking advantage of the boundedness and convergence theory on the square-cutoff Fourier partial sum, we define the Littlewood-Paley decomposition for periodic functions via the square cutoff. We remark that the Littlewood-Paley projections defined via the circular cutoff in T-d with d > 1 in the literature do not behave well on the Lebesgue space L-q except for q = 2. We develop a complete set of tools associated with this decomposition, which would be very useful in the study of PDEs defined on T-d. As an application of the tools developed here, we study the periodic weak solutions of the d-dimensional Boussinesq equations with the fractional dissipation (-Delta)(alpha)u and without thermal diffusion. We obtain two main results. The first assesses the global existence of L-2-weak solutions for any alpha > 0 and the existence and uniqueness of the L-2-weak solutions when alpha >= 1/2 + d/4 for d >= 2. The second establishes the zero thermal diffusion limit with an explicit convergence rate.