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

Abstract

In this work, we consider the two and three-dimensional stochastic convective Brinkman-Forchheimer (2D and 3D SCBF) equations driven by irregular additive white noise $$\mathrm{d}\mathbf{u}-[\mu \Delta \mathbf{u}-(\mathbf{u}\cdot \nabla )\mathbf{u}-\alpha \mathbf{u}-\beta |\mathbf{u}{|}^{r-1}\mathbf{u}-\nabla p]\mathrm{d}t=\mathbf{f}\mathrm{d}t+\mathrm{dW},\nabla \cdot \mathbf{u}=0,$$ for $r\in [1,\infty ),$ $\mu ,\alpha ,\beta >0$ in unbounded domains (like Poincaré domains) $\mathcal{O}\subset {\mathbb{R}}^d$ ($d=2,3$) where $\mathrm{W}(\cdot )$ is a Hilbert space valued Wiener process on some given filtered probability space, and discuss the asymptotic behavior of its solution. For $d=2$ with $r\in [1,\infty )$ and $d=3$ with $r\in [3,\infty )$ (for $d=r=3$ with $2\beta \mu \ge 1$), we first prove the existence and uniqueness of a weak solution (in the analytic sense) satisfying the energy equality for SCBF equations driven by an irregular additive white noise in Poincaré domains by using a Faedo-Galerkin approximation technique. Since the energy equality for SCBF equations is not immediate, we construct a sequence which converges in Lebesgue and Sobolev spaces simultaneously and it helps us to demonstrate the energy equality. Then, we establish the existence of random attractors for the stochastic flow generated by the SCBF equations. One of the technical difficulties connected with the irregular white noise is overcome with the help of the corresponding Cameron-Martin space (or Reproducing Kernel Hilbert space). Finally, we address the existence of a unique invariant measure for 2D and 3D SCBF equations defined on Poincaré domains (bounded or unbounded). Moreover, we provide a remark on the extension of the above mentioned results to general unbounded domains also.

References in corpus (2)

• Stochastic convective Brinkman-Forchheimer equations on general Unbounded Domains
• Asymptotic analysis of the 2D convective Brinkman-Forchheimer equations in unbounded domains: Global attractors and upper semicontinuity

Cited by in corpus (3)

• Weak pullback mean random attractors for the stochastic convective Brinkman-Forchheimer equations and locally monotone stochastic partial differential equations
• Asymptotic analysis of the 2D convective Brinkman-Forchheimer equations in unbounded domains: Global attractors and upper semicontinuity
• Long term behavior of 2D and 3D non-autonomous random convective Brinkman-Forchheimer equations driven by colored noise