AI 中文总结
研究具有粘性且无热扩散的二维Boussinesq方程的弱σ吸引子\(\mathcal{A}\)结构，证明其在温度分量投影是满射，且对任意\(r>0\)，\(\mathcal{A}_r\)在速度分量投影含无限维希尔伯特椭球。
AI 中文摘要
本文继续研究具有粘性且无热扩散的二维Boussinesq方程的弱σ吸引子\(\mathcal{A}\)的结构。具体而言，证明了吸引子\(\mathcal{A}\)在温度分量上的投影是满射，即\(P_{\theta}\mathcal{A}=L^2\)；对于任意\(r>0\)，\(\mathcal{A}_r\)在速度分量上的投影包含一个无限维希尔伯特椭球，这意味着\(P_u\mathcal{A}_r\)立即是无限维的。
英文摘要
In this paper, we continue investigate the structure of the weak $σ$-attractor $\mathcal{A}$ for the 2D Boussinesq equations with viscosity and without heat diffusion. Specifically, we show that the projection of the attractor $\mathcal{A}$ onto the temperature component is surjective, i.e., $P_θ\mathcal{A}=L^2$; and, for any $r>0$, the projection of $\mathcal{A}_r$ onto the velocity component contains a infinite-dimensional Hilbert ellipsoid, which implies $P_u\mathcal{A}_r$ is infinite dimensional immediately.
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