Strong solutions of fractional Boussinesq equations in an exterior domain
Zhi-Min Chen, Qiuyue Zhang
TL;DR
The paper analyzes the fractional Boussinesq equations in the exterior domain around a heated sphere, addressing local stability of the purely conductive steady state and the existence of small strong solutions in a Hilbert space. It adopts a spectral decomposition of a self-adjoint linear operator $\mathcal{L}$ and introduces fractional powers $\mathcal{L}^\kappa$ for $\tfrac{3}{4}<\kappa\le1$, deriving analytic semigroup estimates and nonlinear bounds to formulate a Banach fixed-point problem. The main result is the existence and uniqueness of a global-in-time strong solution $U=(u,T)$ for small initial data and buoyancy parameter $\alpha>0$, with decay properties $\|\mathcal{A}^{5/4-\kappa}U(t)\|_2 \to 0$ and $t^{1-3/(4\kappa)}\|\nabla U(t)\|_2 \to 0$ as $t\to\infty$. This demonstrates local stability of the conductive steady state under small perturbations and extends known $\kappa=1$ results to the fractional range $\tfrac{3}{4}<\kappa\le1$.
Abstract
A thermal convection fluid motion in the three-dimensional domain exterior to a sphere is considered. A purely conductive steady state arises due to the fluid heated from the sphere. A fractional equation system is introduced by using spectral presentation. The existence of small strong solutions in a Hilbert space is obtained. The strong solution existence implies the local stability of the steady state, which attracts asymptotically the flows evolving initially from the vector fields close to the steady state.