Stability for the Boussinesq system on real hyperbolic Manifolds and application

Abstract

In this paper we study the global existence and stability of mild solution for the Boussinesq system on the real hyperbolic manifold $\mathbb{H}^d(\mathbb{R})$ ($d \geqslant 2$). We will consider a couple of Ebin-Marsden's Laplace and Laplace-Beltrami operators associated with the corresponding linear system which provides a vectorial heat semigoup. First, we prove the existence and the uniqueness of the bounded mild solution for the linear system by using certain dispersive and smoothing estimates of the vectorial heat semigroup. Next, using the fixed point arguments, we can pass from the linear system to the semilinear system to establish the existence of the bounded mild solution. We will prove the exponential stability of such solution by using the cone inequality. Finally, we give an application of stability to the existence of periodic mild solution for the Boussinesq system.Comment: 23 pages. arXiv admin note: substantial text overlap with arXiv:2209.0780