Asymptotic behavior of stochastic g-Navier-Stokes equations on a sequence of expanding domains

Abstract

The limiting dynamics of stochastic 2D nonautonomous g-Navier-Stokes equations defined on a sequence of expanding domains are investigated, where the limiting domain is unbounded. By generalizing the energy-equation method, we show that the sequence of expanding cocycles is weakly equicontinuous and strongly equiasymptotically compact, which lead to both existence and upper semicontinuity of the null-expansion of the corresponding random attractor when the bounded domain approaches to the unbounded domain.

$\mathbb{H}^1$-Random attractors for 2D stochastic convective Brinkman-Forchheimer equations in unbounded domains

“…In the literature, the concept of upper semicontinuity of random attractors with respect to domain was introduced in [41] and the authors proved upper semicontinuity of random attractors with respect to domain for stochastic FitzHugh-Nagumo equations (SFNE). Later, using this concept, upper semicontinuity of random attractors with respect to domain for g-Navier-Stokes equations (g-NSE) was proved in [42]. While using the concept introduced in [41] to the models SFNE and g-NSE, the authors have not taken care of the boundary conditions, while proving the strong convergence of the solutions on the sequence of bounded domains to the solution on the unbounded domain, which is a crucial step in the theory.…”

“…While using the concept introduced in [41] to the models SFNE and g-NSE, the authors have not taken care of the boundary conditions, while proving the strong convergence of the solutions on the sequence of bounded domains to the solution on the unbounded domain, which is a crucial step in the theory. Due to this reason, it appears to us that the method developed in the works [41,42], etc have serious flaws, and we have not used the theory established in [41]. But, one can prove the upper semicontinuity of random attractors with respect to domain for SFNE and g-NSE using the same method which we apply for 2D SCBF equation in Section 6.…”

Weak pullback mean random attractors for the stochastic convective Brinkman-Forchheimer equations and locally monotone stochastic partial differential equations

“…The theory of pathwise pullback random attractors was first introduced in [6,15], and thereafter several authors used this theory and proved the existence of random attractors for several SPDEs, for e.g. [2,3,4,5,7,8,9,10,16,19,29,38,42,43,44] etc and the references therein. The existence of pathwise pullback random attractors for the two and three dimensional SCBF equations in bounded, periodic and unbounded domains with additive noise is established in [23,24,25,26], etc.…”

Random attractors for 2D and 3D stochastic convective Brinkman-Forchheimer equations in some unbounded domains

“…The concept of an asymptotically compact cocycle was introduced in [16] and authors have established the existence of attractors for the non-autonomous 2D Navier-Stokes equations. Later, this concept has been utilized to prove the existence of random attractors for several SPDEs like 1D stochastic lattice differential equation [5], stochastic Navier-Stokes equations on the 2D unit sphere [4], stochastic g-Navier-Stokes equations [27,36,39], stochastic nonautonomous Kuramoto-Sivashinsky equations [38], stochastic heat equations in materials with memory on thin domains [48], stochastic reaction-diffusion equations [6,47,51], 3D stochastic Benjamin-Bona-Mahony equations [51], etc and the references therein.…”