The ℍ1documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$mathbb {H}^{1}$end{document}-Compact Global Attractor for Two-Dimensional Convective Brinkman-Forchheimer Equations in Unbounded Domains

摘要

The asymptotic analysis of solutions of the two dimensional convective Brinkman-Forchheimer (CBF) equations ∂tu−μΔu+(u⋅∇)u+αu+β|u|r−1u+∇p=f,∇⋅u=0,documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ partial_{t}boldsymbol{u}-mu {Delta}boldsymbol{u}+(boldsymbol{u}cdotnabla)boldsymbol{u} +alphaboldsymbol{u}+beta|boldsymbol{u}|^{r-1}boldsymbol{u}+nabla p=boldsymbol{f}, nablacdotboldsymbol{u}=0, $$end{document} for r ∈ [1,3], in unbounded domains Ω is carried out in this work. If the forcing term f is in the space L2(Ω)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$mathbb {L}^{2}({Omega })$end{document}, then we show that the global attractor for 2D CBF equations. defined in Poincaré domains and general unbounded domains is compact not only in the L2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$mathbb {L}^{2}$end{document}-norm but also in the ℍ1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$mathbb {H}^{1}$end{document}-norm. The enstrophy equation as well as the asymptotic compactness of the semigroup associated with the 2D CBF equations is exploited in the proofs.