Existence and upper semicontinuity of random pullback attractors for 2D and 3D non-autonomous stochastic convective Brinkman-Forchheimer equations on whole domain

TL;DR

This paper proves the existence and upper semicontinuity of random pullback attractors for 2D and 3D stochastic convective Brinkman-Forchheimer equations with non-autonomous forcing, extending attractor theory to unbounded domains.

Contribution

It establishes the existence of unique global and random pullback attractors for non-autonomous stochastic CBF equations on unbounded domains, a novel extension in the field.

Findings

Existence of a unique global pullback attractor for deterministic CBF equations.

Existence of a unique random pullback attractor for stochastic CBF equations with multiplicative noise.

Upper semicontinuity of the random attractor as noise intensity approaches zero.

Abstract

In this work, we analyze the long time behavior of 2D as well as 3D convective Brinkman-Forchheimer (CBF) equations and its stochastic counter part with non-autonomous deterministic forcing term in $\mathbb{R}^d$ $ (d=2, 3)$: $$\frac{\partial\boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f},\quad \nabla\cdot\boldsymbol{u}=0,$$ where $r\geq1$. We prove the existence of a unique global pullback attractor for non-autonomous CBF equations, for $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$. For the same cases, we show the existence of a unique random pullback attractor for non-autonomous stochastic CBF equations with multiplicative white noise. Finally, we establish the upper semicontinuity of the random pullback attractor, that is, the random pullback attractor converges towards the global pullback attractor when the noise intensity approaches to zero. Since we do not have compact Sobolev embeddings on unbounded domains, the pullback asymptotic compactness of the solution is proved by the method of energy equations given by Ball. For the case of Navier-Stokes equations defined on $\mathbb{R}^d$, such results are not available and the presence of Darcy term $\alpha\boldsymbol{u}$ helps us to establish the above mentioned results for CBF equations.