On periodic solution for the Boussinesq system on real hyperbolic Manifolds

In this work we study the existence and uniqueness of the periodic mild solutions of the Boussinesq system on the real hyperbolic manifold $\mathbb{H}^d(\mathbb{R})$ ($d \geqslant 2$). We will consider Ebin-Marsden's Laplace operator associated with the corresponding linear system. Our method is based on the dispertive and smoothing estimates of the semigroup generated by Ebin-Marsden's Laplace operator. First, we prove the existence and the uniqueness of the bounded periodic mild solution for the linear system. Next, using the fixed point arguments, we can pass from the linear system to the semilinear system to establish the existence of the periodic mild solution. Finally, we prove the unconditional uniqueness of large periodic mild solutions for the Boussinesq system on the framework of hyperbolic spaces.Comment: 23 page