Random attractors and invariant measures for stochastic convective Brinkman-Forchheimer equations on 2D and 3D unbounded domains

Abstract

In this work, we consider the 2D and 3D SCBF equations driven by irregular additive white noise$ \mathrm{d}\boldsymbol{u}-[\mu \Delta\boldsymbol{u}-(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}-\alpha\boldsymbol{u}-\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}-\nabla p]\mathrm{d}t = \boldsymbol{f}\mathrm{d}t+\mathrm{d}\mathrm{W},\ \nabla\cdot\boldsymbol{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 a 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\geq 1 $), we first prove the existence and uniqueness of a weak solution (in the analytic sense) satisfying the energy equality for SCBF equations by using a Faedo-Galerkin approximation technique. Then, we establish the existence of random attractors for the stochastic flow generated by the SCBF equations. One of the technical difficulties associated with irregular white noise is overcome with the help of the corresponding Reproducing Kernel Hilbert space. Furthermore, we observe that the regularity of irregular white noise needed to obtain random attractors for the SCBF equations for $ d = 2 $ with $ r\in[1,3] $ and $ d = r = 3 $ with $ 2\beta\mu\geq1 $, is the same as that in the case of 2D Navier-Stokes equations, whereas for the cases $ d = 2,3 $ and $ r\in(3,\infty) $, we require more spatial regularity on the noise. Finally, we address the existence of a unique invariant measure for 2D and 3D SCBF equations defined on Poincaré domains (bounded or unbounded).

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