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

TL;DR

This work proves global well-posedness and exponential stability for the Boussinesq system on real hyperbolic manifolds $\mathbb{H}^d(\mathbb{R})$ with $d\ge 2$ in $L^p$-phase spaces ($1<p\le d$). The authors formulate the problem via a vectorial matrix semigroup generated by the coupled operators $L$ and $\widetilde{L}$, establish $L^p-L^q$ dispersive and smoothing estimates, and use a fixed-point argument to obtain bounded mild solutions for the semilinear system, followed by Grönwall-type estimates to deduce exponential decay. They further derive a Serrin-type criterion to guarantee the existence of $T$-periodic mild solutions under time-periodic forcing, initialization, and gravitational fields in a small-data regime. The appendix extends the analysis to generalized gravitational fields and the three-dimensional case, showing well-posedness in weak-$L^p$ spaces and providing detailed integral estimates essential for the main results.

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.

Theorems & Definitions (19)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Remark 3.4
• Theorem 3.5
• proof
• Lemma 3.6