Multivalued Random Dynamics of Convective Brinkman–Forchheimer Equations Driven by Nonlinear Colored Noise on Unbounded Domains

ABSTRACT We investigate the multivalued random dynamics of the nonautonomous‐autonomous convective Brinkman–Forchheimer equations driven by nonlinear colored noise on unbounded domains. The presence of non‐Lipschitz continuous diffusion terms leads to the non‐uniqueness of the solutions, and hence we use the theory of multivalued random dynamical systems (MRDSs) to study the large‐time behavior of solutions of the equations. We first establish the existence of MRDSs of the equations in the nature energy space, and then prove the existence and uniqueness of tempered pullback random attractors of the MRDSs. The measurability of MRDSs is justified by showing the weak upper semi‐continuity of the solutions, which leads to the measurability of random attractors. The main difficulty for establishing the pullback asymptotic compactness of the solution operators is the non‐compactness of Sobolev embeddings on unbounded domains. We overcome this difficulty and establish the pullback asymptotic compactness for the nonautonomous‐autonomous and random Brinkman–Forchheimer equations by using Ball's idea of energy equations originally developed for proving the existence of global attractors of deterministic and autonomous Navier–Stokes flows and damped wave equations.