We consider the forced surface quasi-geostrophic equation with supercritical dissipation. We show that linear instability for steady state solutions leads to their nonlinear instability. When the dissipation is given by a fractional Laplacian, the nonlinear instability is expressed in terms of the scaling invariant norm, while we establish stronger instability claims in the setting of logarithmically supercritical dissipation. A key tool in treating the logarithmically supercritical setting is a global well-posedness result for the forced equation, which we prove by adapting and extending recent work related to nonlinear maximum principles. We believe that our proof of global well-posedness is of independent interest, to our knowledge giving the first large-data supercritical result with sharp regularity assumptions on the forcing term.

我们考虑具有超临界耗散的强迫表面准地转方程。我们表明稳态解的线性不稳定性会导致它们的非线性不稳定性。当耗散由分数拉普拉斯算子给出时，非线性不稳定性用标度不变范数表示，而我们在对数超临界耗散的设置中建立更强的不稳定性声明。处理对数超临界设置的一个关键工具是强制方程的全局适定结果，我们通过调整和扩展与非线性最大值原理相关的最新工作来证明这一点。我们相信我们对全局适定性的证明是独立的，据我们所知，给出了第一个大数据超临界结果，在强迫项上有严格的规律性假设。