Stability of purely convective steady-states of fractional Boussinesq equations in an exterior domain

TL;DR

This work analyzes the stability of a purely convective steady state for fractional Boussinesq equations in the exterior of a sphere. By formulating the problem with a fractional Stokes operator $A^{\kappa}$ and employing an analytic semigroup framework for the linearized operator $L$, it proves the global existence of weak solutions in $L^2$ for $\frac{3}{4}<\kappa\le1$ under small buoyancy $0<\alpha<2^{-\kappa}$ and initial data in $L^2_\sigma(\Omega)^3\times L^2(\Omega)$. The paper establishes an $L^2$ decay to zero and shows an algebraic decay propagation: if the linearized system decays as $t^{-\beta}$, the nonlinear solution inherits the same rate, with a bootstrap argument demonstrating decay rates up to any $\beta<\tfrac{1}{2}$ as long as $\kappa\in(\tfrac{3}{4},1]$. The approach combines energy inequalities, semigroup estimates for the fractional operator, and compactness methods to handle the exterior-domain setting, extending stability analysis to fractional dissipations in an unbounded geometry.

Abstract

A thermal convection flow in the three-dimensional unbounded fluid domain exterior to a sphere is considered. The viscosity force is determined by a fractional power of the Stokes operator. A purely conductive steady state arises due to the fluid heated from the sphere. A weak solution of the fluid motion problem is obtained and global stability of the steady-state solution in $L^2$ is provided.

Theorems & Definitions (18)

• Definition 1.1
• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof