Abstract
We consider the large-time behavior such as the existence of attractors for the 3D autonomous and nonautonomous Brinkman-Forchheimer equations. By means of the decomposition method we overcome the difficulties for the existence of absorbing sets and asymptotical compactness of the semigroup generated by a global solution to prove the attractors for the autonomous Brinkman-Forchheimer equation. Under suitable assumptions on the external force $\sigma(t,x)$ and initial data $u_{\tau}(x)$ , we prove the existence of a uniform attractor for a 3D nonautonomous Brinkman-Forchheimer equation. Moreover, we apply the theory of weak continuity and weak convergence method to establish the asymptotical compactness of the processes.
Highlights
1 Introduction The large-time behavior of global solutions and the associated infinite dimensional dynamical systems have become the essential aspects in the field of nonlinear evolutionary equations such as the existence of global attractors, inertial manifold, uniform attractors, pullback attractors, and their fractal dimensions for the autonomous and nonautonomous systems with the unique solution have attracted attention of many mathematicians since the s
Choosing a nonautonomous fixed external force σ (t, x) ∈ L b(R, H), the global solution u(t, x) generates a process {Uσ (τ, t)} (τ ∈ R, t > τ, σ ∈ ), which is continuous with respect to uτ, where σ is a symbol that belongs to the symbol space = H(σ ) = [{σ (s + h)|h ∈ R}]L loc(R,H)with [·]E meaning the closure in the topology of E
VV, Vishik, MI: Attractors for Equations of Mathematical Physics
Summary
The large-time behavior of global solutions and the associated infinite dimensional dynamical systems have become the essential aspects in the field of nonlinear evolutionary equations such as the existence of global attractors, inertial manifold, uniform attractors, pullback attractors, and their fractal dimensions for the autonomous and nonautonomous systems with the unique solution have attracted attention of many mathematicians since the s. We consider the large time behavior (i.e., the existence of global and uniform attractors for the autonomous and nonautonomous Brinkman-Forchheimer equations by decomposition method and weak continuous method to establish asymptotical compactness for the semigroups and processes, respectively) for the D BrinkmanForchheimer equation that governs the motion of fluid in a saturated porous medium:. Choosing a nonautonomous fixed external force σ (t, x) ∈ L b(R, H), the global solution u(t, x) generates a process {Uσ (τ , t)} (τ ∈ R, t > τ , σ ∈ ), which is continuous with respect to uτ , where σ is a symbol that belongs to the symbol space = H(σ ) = [{σ (s + h)|h ∈ R}]L loc(R,H)with [·]E meaning the closure in the topology of E.
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