Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations

Abstract

<p style='text-indent:20px;'>Stability problem on perturbations near the hydrostatic balance is one of the important issues for Boussinesq equations. This paper focuses on the asymptotic stability and large-time behavior problem of perturbations of the 2D fractional Boussinesq equations with only fractional velocity dissipation or fractional thermal diffusivity. Since the linear portion of the Boussinesq equations plays a crucial role in the stability properties, we firstly study the linearized fractional Boussinesq equations with only fractional velocity dissipation or fractional thermal diffusivity and complete the following work: 1) assessing the stability and obtaining the precise large-time asymptotic behavior for solutions to the linearized system satisfied the perturbation; 2) understanding the spectral property of the linearization; 3) showing the <inline-formula><tex-math id="M1">\begin{document}$ H^2 $\end{document}</tex-math></inline-formula>-stability for the linearized system, and prove that the <inline-formula><tex-math id="M2">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm of <inline-formula><tex-math id="M3">\begin{document}$ \nabla{u} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \Delta{u} $\end{document}</tex-math></inline-formula> (or <inline-formula><tex-math id="M5">\begin{document}$ \nabla\theta $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ \Delta\theta $\end{document}</tex-math></inline-formula>), the <inline-formula><tex-math id="M7">\begin{document}$ L^\varrho $\end{document}</tex-math></inline-formula>-norm <inline-formula><tex-math id="M8">\begin{document}$ (2<\varrho<\infty) $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M9">\begin{document}$ u $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ \nabla{u} $\end{document}</tex-math></inline-formula> (or <inline-formula><tex-math id="M11">\begin{document}$ \theta $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ \nabla\theta $\end{document}</tex-math></inline-formula>) are all approaching to zero as <inline-formula><tex-math id="M13">\begin{document}$ t\rightarrow\infty $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M14">\begin{document}$ \alpha = 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M15">\begin{document}$ \eta = 0 $\end{document}</tex-math></inline-formula> (or <inline-formula><tex-math id="M16">\begin{document}$ \nu = 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M17">\begin{document}$ \beta = 1 $\end{document}</tex-math></inline-formula>). Secondly, we obtain the <inline-formula><tex-math id="M18">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula>-stability for the full nonlinear system and prove the <inline-formula><tex-math id="M19">\begin{document}$ L^\varrho $\end{document}</tex-math></inline-formula>-norm <inline-formula><tex-math id="M20">\begin{document}$ (2<\varrho<\infty) $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M21">\begin{document}$ \theta $\end{document}</tex-math></inline-formula> and the <inline-formula><tex-math id="M22">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm of <inline-formula><tex-math id="M23">\begin{document}$ \nabla\theta $\end{document}</tex-math></inline-formula> approaching to zero as <inline-formula><tex-math id="M24">\begin{document}$ t\rightarrow\infty $\end{document}</tex-math></inline-formula>.</p>