Well-posedness and exponential stability for Boussinesq systems on real hyperbolic Manifolds and application

Abstract

We investigate the global existence and exponential decay of mild solutions for the Boussinesq systems in $L^p$-phase spaces on the framework of real hyperbolic manifold ${\mathbb{H}}^d(\mathbb{R})$, where $d⩾2$ and $1<p\le d$. We consider a couple of Ebin-Marsden's Laplace and Laplace-Beltrami operators associated with the corresponding linear system which provides a vectorial matrix semigoup. First, we show the existence and the uniqueness of the bounded mild solution for the linear system by using dispersive and smoothing estimates of the vectorial matrix 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 solutions. By using Gronwall's inequality, we establish the exponential stability of such solutions. Finally, we give an application of stability to the existence of periodic mild solutions for the Boussinesq systems.

25 pages, Accepted for publication in Asymptotic Analysis

References in corpus (4)

• Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces
• Asymptotically almost periodic solutions to parabolic equations on the real hyperbolic manifold
• On asymptotically almost periodic solutions to the Navier-Stokes equations on hyperbolic manifolds
• Periodic solutions for Boussinesq systems in weak-Morrey spaces