Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations

<p style='text-indent:20px;'>This work deals with the asymptotic behavior of the two as well as three dimensional convective Brinkman-Forchheimer (CBF) equations in an <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional torus (<inline-formula><tex-math id="M2">\begin{document}$ n = 2, 3 $\end{document}</tex-math></inline-formula>):</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \frac{\partial\boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p = \boldsymbol{f}, \ \nabla\cdot\boldsymbol{u} = 0, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ r\geq1 $\end{document}</tex-math></inline-formula>. We prove that the global attractor of the above system is singleton under small forcing intensity (<inline-formula><tex-math id="M4">\begin{document}$ r\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ r\geq 3 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M7">\begin{document}$ n = 3 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M8">\begin{document}$ 2\beta\mu\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M9">\begin{document}$ r = n = 3 $\end{document}</tex-math></inline-formula>). But if one perturbs the above system with an additive or multiplicative white noise, there is no sufficient evidence that the random attractor keeps the singleton structure. We obtain that the random attractor for 2D stochastic CBF equations forced by additive and multiplicative white noise converges towards the deterministic singleton attractor for all <inline-formula><tex-math id="M10">\begin{document}$ 1\leq r&lt;\infty $\end{document}</tex-math></inline-formula>, when the coefficient of random perturbation converges to zero (upper and lower semicontinuity). For the case of 3D stochastic CBF equations perturbed by additive and multiplicative white noise, we are able to establish that the random attractor converges towards the deterministic singleton attractor for <inline-formula><tex-math id="M11">\begin{document}$ 3\leq r&lt;\infty $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M12">\begin{document}$ 2\beta\mu\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M13">\begin{document}$ r = 3 $\end{document}</tex-math></inline-formula>), when the coefficient of random perturbation converges to zero.</p>

Semicontinuity of the Solution Mapping to a Parametric Generalized Weak Ky Fan Inequality

Under new assumptions, which do not contain any information about the solution set, the upper and lower semicontinuity of the solution mapping to a class of parametric generalized weak Ky Fan inequality are established by using a nonlinear scalarization technique. These results extend and improve the recent ones in the literature. Some examples are given to illustrate our results.

Continuity of the Solution Maps for Generalized Parametric Set-Valued Ky Fan Inequality Problems

Under new assumptions, we provide suffcient conditions for the (upper and lower) semicontinuity and continuity of the solution mappings to a class of generalized parametric set-valued Ky Fan inequality problems in linear metric space. These results extend and improve some known results in the literature (e.g., Gong, 2008; Gong and Yoa, 2008; Chen and Gong, 2010; Li and Fang, 2010). Some examples are given to illustrate our results.

Decomposition of Topologies Which Characterize the Upper and Lower Semicontinuous Limits of Functions

We present a decomposition of two topologies which characterize the upper and lower semicontinuity of the limit function to visualize their hidden and opposite roles with respect to the upper and lower semicontinuity and consequently the continuity of the limit. We show that (from the statistical point of view) there is an asymmetric role of the upper and lower decomposition of the pointwise convergence with respect to the upper and lower decomposition of the sticking convergence and the semicontinuity of the limit. This role is completely hidden if we use the whole pointwise convergence. Moreover, thanks to this mirror effect played by these decompositions, the statistical pointwise convergence of a sequence of continuous functions to a continuous function in one of the two symmetric topologies, which are the decomposition of the sticking topology, automatically ensures the convergence in the whole sticking topology.