Quasi-geostrophic equation in $\mathbb{R}^2$
Quasi-geostrophic equation in $\mathbb{R}^2$
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发表时间：
2014-11
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通讯作者：
Tomasz Dłotko;M. Kania;Chunyou Sun
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作者：
Tomasz Dłotko;M. Kania;Chunyou Sun
本文讨论了两相空间中次临界拟地转方程在$\mathbb{R}^2$中的Cauchy问题的可解性；$L^p(\mathbb{R}^2)$和$p> \frac{2}{2\alpha-1}$, $H^s(\mathbb{R}^2)$和$s>1$。当$(-\Delta)^\alpha$的指数$\alpha$趋向于$\frac{1}{2}^+$时，作为次临界方程$H^s$ -解的极限，得到了临界情况下方程的解。这样的观点在文献中似乎是新的。讨论了亚临界情况下全局吸引子的存在性。在第7节中，我们还讨论了当$\| \theta_0 \|_{L^\infty(\Omega)}$很小时，在有界域$\Omega \subset \mathbb{R}^2$上具有Dirichlet边界条件的临界问题的可解性。
Solvability of Cauchy's problem in $\mathbb{R}^2$ for subcritical quasi-geostrophic equation is discussed here in two phase spaces; $L^p(\mathbb{R}^2)$ with $p> \frac{2}{2\alpha-1}$ and $H^s(\mathbb{R}^2)$ with $s>1$. A solution to that equation in critical case is obtained next as a limit of the $H^s$-solutions to subcritical equations when the exponent $\alpha$ of $(-\Delta)^\alpha$ tends to $\frac{1}{2}^+$. Such idea seems to be new in the literature. Existence of the global attractor in subcritical case is discussed in the paper. In section 7 we also discuss solvability of the critical problem with Dirichlet boundary condition in bounded domain $\Omega \subset \mathbb{R}^2$, when $\| \theta_0 \|_{L^\infty(\Omega)}$ is small.