We provide sufficient conditions for mathematically rigorous proofs of the third order universal laws capturing the energy flux to large scales and enstrophy flux to small scales for statistically stationary, forced-dissipated 2d Navier–Stokes equations in the large-box limit. These laws should be regarded as 2d turbulence analogues of the 4/5 law in 3d turbulence, predicting a constant flux of energy and enstrophy (respectively) through the two inertial ranges in the dual cascade of 2d turbulence. Conditions implying only one of the two cascades are also obtained, as well as compactness criteria which show that the provided sufficient conditions are not far from being necessary. The specific goal of the work is to provide the weakest characterizations of the “0-th laws” of 2d turbulence in order to make mathematically rigorous predictions consistent with experimental evidence.

中文翻译：

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随机二维 Navier-Stokes 方程中双级联通量定律的充分条件

我们为三阶通用定律的数学上严格的证明提供了充分的条件，这些定律捕获了大范围限制中统计平稳、强制耗散的 2d Navier-Stokes 方程的大尺度能量通量和小尺度熵通量。这些定律应被视为 3d 湍流中 4/5 定律的 2d 湍流类似物，通过 2d 湍流的双级联中的两个惯性范围预测能量和熵的恒定通量（分别）。还获得了暗示仅两个级联之一的条件，以及表明所提供的充分条件并非必需的紧凑性标准。