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

Kush Kinra; Manil T. Mohan

doi:10.3934/eect.2021061

Article Contents

2022, Volume 11, Issue 5: 1701-1744. Doi: 10.3934/eect.2021061

This issue Previous Article Stability estimate for a partial data inverse problem for the convection-diffusion equation Next Article Two simple criterion to obtain exact controllability and stabilization of a linear family of dispersive PDE's on a periodic domain

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

Kush Kinra and
Manil T. Mohan^,

Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, INDIA

^*Corresponding author: Manil T. Mohan
^*Corresponding author: Manil T. Mohan

Received: May 2021

Revised: November 2021

Early access: December 2021

Published: October 2022

The second author is supported by DST INSPIRE Faculty Award (IFA17-MA110).

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Abstract

This work deals with the asymptotic behavior of the two as well as three dimensional convective Brinkman-Forchheimer (CBF) equations in an $ n $-dimensional torus ($ n = 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}, \ \nabla\cdot\boldsymbol{u} = 0, $

where $ r\geq1 $. We prove that the global attractor of the above system is singleton under small forcing intensity ($ r\geq 1 $ for $ n = 2 $ and $ r\geq 3 $ for $ n = 3 $ with $ 2\beta\mu\geq 1 $ for $ r = n = 3 $). 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 $ 1\leq r<\infty $, 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 $ 3\leq r<\infty $ ($ 2\beta\mu\geq 1 $ for $ r = 3 $), when the coefficient of random perturbation converges to zero.

Keywords:

Deterministic and stochastic convective Brinkman-Forchheimer equations,
small forcing intensity,
singleton attractor,
upper semicontinuity,
lower semicontinuity.

Mathematics Subject Classification: Primary 35B41, 35Q35; Secondary 37L55, 37N10, 35R60.

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