Journal of Evolution Equations ( IF 1.4 ) Pub Date : 2024-03-15 , DOI: 10.1007/s00028-024-00947-w Wilberclay G. Melo
Our interest in this research is to prove the decay, as time tends to infinity, of a unique global solution for the supercritical case of the modified quasi-gesotrophic equation (MQG)
$$\begin{aligned} \theta _t \;\!+\, (-\Delta )^{\alpha }\,\theta \,+\, u_{\theta } \cdot \nabla \theta \;=\; 0, \quad \hbox {with } u_{\theta }\;=\;(\partial _2(-\Delta )^{\frac{\gamma -2}{2}}\theta , -\partial _1(-\Delta )^{\frac{\gamma -2}{2}}\theta ), \end{aligned}$$in the non-homogenous Sobolev space \(H^{1+\gamma -2\alpha }(\mathbb {R}^2)\), where \(\alpha \in (0,\frac{1}{2})\) and \(\gamma \in (1,2\alpha +1)\). To this end, we need consider that the initial data for this equation are small. More precisely, we assume that \(\Vert \theta _0\Vert _{H^{1+\gamma -2\alpha }}\) is small enough in order to obtain a unique \(\theta \in C([0,\infty );H^{1+\gamma -2\alpha }(\mathbb {R}^2))\) that solves (MQG) and satisfies the following limit:
$$\begin{aligned} \lim _{t\rightarrow \infty } \Vert \theta (t)\Vert _{H^{1+\gamma -2\alpha }}=0. \end{aligned}$$中文翻译：
修正的超临界耗散准地转方程唯一全局解的存在及其无穷大衰减
我们对这项研究的兴趣是证明随着时间趋于无穷大，修正准地转方程 (MQG) 超临界情况的唯一全局解的衰减
$$\begin{对齐} \theta _t \;\!+\, (-\Delta )^{\alpha }\,\theta \,+\, u_{\theta } \cdot \nabla \theta \;= \; 0, \quad \hbox {与 } u_{\theta }\;=\;(\partial _2(-\Delta )^{\frac{\gamma -2}{2}}\theta , -\partial _1( -\Delta )^{\frac{\gamma -2}{2}}\theta ), \end{对齐}$$在非齐次 Sobolev 空间\(H^{1+\gamma -2\alpha }(\mathbb {R}^2)\)中，其中\(\alpha \in (0,\frac{1}{2 })\)和\(\gamma \in (1,2\alpha +1)\)。为此，我们需要考虑到该方程的初始数据很小。更准确地说，我们假设\(\Vert \theta _0\Vert _{H^{1+\gamma -2\alpha }}\)足够小，以便在 C([ 0,\infty );H^{1+\gamma -2\alpha }(\mathbb {R}^2))\)求解 (MQG) 并满足以下极限：
$$\begin{对齐} \lim _{t\rightarrow \infty } \Vert \theta (t)\Vert _{H^{1+\gamma -2\alpha }}=0。\end{对齐}$$
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