We study the well-posedness of the primitive equations for the ocean and atmosphere on two particular domains : a bounded domain $Ω_1 := (-1, 1)^3$ with periodic boundary conditions and the strip $Ω_2 := \mathbb{R}^2 \times (-1, 1)$ with a periodic boundary condition for the vertical coordinate. An existence theorem for global solutions on a suitable Besov space is derived. Then, in a second step, we rigorously justify the passage to the limit from the rescaled anisotropic Navier-Stokes equations to these primitive equations in the same functional framework as that found for the solutions of the primitive equations.

Results concerning the existence and spectral stability/instability of multiple periodic standing wave solutions for a cubic nonlinear Schrödinger system will be shown in this manuscript.Our approach considers periodic perturbations that have the same period of the standing wave solution.To obtain the quantity and multiplicity of non-positive eigenvalues for the corresponding linearized operator, we use the comparison theorem and tools of Floquet theory.The results are then obtained by applying the spectral stability theory via Krein signature as established in [20] and [21].