We consider the asymptotic behavior of the surface quasi-geostrophic equation, subject to a small external force. Under suitable assumptions on the forcing, we first construct the steady states and we provide a number of useful a posteriori estimates for them. Importantly, to do so, we only impose minimal cancellation conditions on the forcing function. Our main result is that all \(L^1\cap L^\infty \) localized initial data produces global solutions of the forced SQG, which converge to the steady states in \(L^p({\mathbf {R}}^2), 1<p\le 2\) as time goes to infinity. This establishes that the steady states serve as one point attracting set. Moreover, by employing the method of scaling variables, we compute the sharp relaxation rates, by requiring slightly more localized initial data.

我们考虑了在较小外力作用下表面拟地转方程的渐近行为。在关于强迫的适当假设下，我们首先构造稳态，然后为它们提供许多有用的后验估计。重要的是，为此，我们仅对强制函数施加最小的取消条件。我们的主要结果是，所有\（L ^ 1 \ cap L ^ \ infty \）本地化的初始数据都产生了强制SQG的全局解，这些全局解收敛到\（L ^ p（{\ mathbf {R}}中的稳态^ 2），1 <p \ le 2 \）随着时间到无穷远。这就确定了稳态是一个点吸引集合。此外，通过采用缩放变量的方法，我们通过要求稍微本地化的初始数据来计算急剧的松弛率。