We construct an invariant measure \(\mu \) for the Surface Quasi-Geostrophic (SQG) equation and show that almost all functions in the support of \(\mu \) are initial conditions of global, unique solutions of SQG that depend continuously on the initial data. In addition, we show that the support of \(\mu \) is infinite dimensional, meaning that it is not locally a subset of any compact set with finite Hausdorff dimension. Also, there are global solutions that have arbitrarily large initial condition. The measure a \(\mu \) is obtained via fluctuation–dissipation method, that is, as a limit of invariant measures for stochastic SQG with a carefully chosen dissipation and random forcing.

我们为表面准地转方程（SQG）方程构造了一个不变测度\（\ mu \），并证明了支持\（\ mu \）的几乎所有函数都是SQG全局唯一解的连续性的初始条件。在初始数据上。另外，我们证明\（\ mu \）的支持是无限维的，这意味着它不是任何具有有限Hausdorff维数的紧集的局部子集。同样，有些全局解决方案具有任意大的初始条件。通过波动-耗散方法获得度量\（\ mu \），即作为随机SQG的不变度量的极限，并具有经过仔细选择的耗散和随机强迫。