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

TL;DR

This paper proves the global existence, uniqueness, and exponential stability of mild solutions for Boussinesq systems on real hyperbolic manifolds, extending understanding of these systems in curved geometric settings.

Contribution

It introduces a novel analysis of Boussinesq systems on hyperbolic manifolds using vectorial matrix semigroups and fixed point methods, establishing stability and periodic solutions.

Findings

Existence and uniqueness of mild solutions for linear and semilinear Boussinesq systems.

Exponential decay and stability of solutions.

Application to periodic mild solutions on hyperbolic manifolds.

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 $d \geqslant 2$ and $1<p\leq d$. 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.