The asymptotic analysis of solutions of the two dimensional convective Brinkman-Forchheimer (CBF) equations

$$ \partial_{t}\boldsymbol{u}-\mu {\Delta}\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u} +\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f}, \ \nabla\cdot\boldsymbol{u}=0, $$

for r ∈ [1,3], in unbounded domains Ω is carried out in this work. If the forcing term f is in the space \(\mathbb {L}^{2}({\Omega })\), 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 \(\mathbb {L}^{2}\)-norm but also in the \(\mathbb {H}^{1}\)-norm. The enstrophy equation as well as the asymptotic compactness of the semigroup associated with the 2D CBF equations is exploited in the proofs.

中文翻译：

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bound 1 $ \ mathbb {H} ^ {1} $-无界域中二维对流Brinkman-Forchheimer方程的紧凑全局吸引子

二维对流Brinkman-Forchheimer（CBF）方程解的渐近分析

$$ \ partial_ {t} \ boldsymbol {u}-\ mu {\ Delta} \ boldsymbol {u} +（\ boldsymbol {u} \ cdot \ nabla）\ boldsymbol {u} + \ alpha \ boldsymbol {u} + \ beta | \ boldsymbol {u} | ^ {r-1} \ boldsymbol {u} + \ nabla p = \ boldsymbol {f}，\ \ nabla \ cdot \ boldsymbol {u} = 0，$$

为[R ∈[1,3]，在无界区域Ω在此工作中进行。如果强迫项f在空间\（\ mathbb {L} ^ {2}（{\ Omega}）\）中，则表明2D CBF方程的全局吸引子。在Poincaré域和一般无界域中定义的值不仅在\（\ mathbb {L} ^ {2} \）-范数中而且在\（\ mathbb {H} ^ {1} \）-范数中都是紧凑的。在证明中利用了与2 CBF方程相关的半群的熵方程和渐近紧性。