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

TL;DR

This paper studies the long-term behavior of 2D and 3D convective Brinkman-Forchheimer equations, showing that their attractors are singletons under certain conditions and analyzing how stochastic perturbations affect convergence to these attractors.

Contribution

It proves the convergence of random attractors to deterministic singleton attractors for stochastic CBF equations in 2D and 3D under specific parameter regimes.

Findings

Deterministic attractors are singletons for small forcing.

Random attractors converge to deterministic attractors as noise diminishes.

Convergence results depend on the dimension and the nonlinearity parameter r.

Abstract

This work deals with the asymptotic behavior of the two as well as three dimensional convective Brinkman-Forchheimer (CBF) equations in periodic domains: $$\frac{\partial\boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f},\ \nabla\cdot\boldsymbol{u}=0,$$ where $r\geq1$. We prove that the global attractor of the above system is a singleton under small forcing intensity ($r\geq 1$ for $n=2$ and $r\geq 3$ for $n=3$ with $2\beta\mu\geq 1$ for $r=n=3$). After perturbing the above system with additive or multiplicative white noise, the random attractor does not have a singleton structure. But we obtain that the random attractor for 2D stochastic CBF equations with additive and multiplicative white noise converges towards the deterministic singleton attractor for $1\leq r\leq 2$ and $1\leq r<\infty$, respectively, when the coefficient of random perturbation converges to zero (upper and lower semicontinuity). Interestingly in the case of 3D stochastic CBF equations perturbed by multiplicative white noise, we are able to establish that the random attractor converges towards the deterministic singleton attractor for $3\leq r\leq 5$ ($2\beta\mu\geq 1$ for $r=3$), when the coefficient of random perturbation converges to zero.