The main goal of this paper is to obtain sufficient conditions that allow us to rigorously derive local versions of the 4/5 and 4/3 laws of hydrodynamic turbulence, by which we mean versions of these laws that hold in bounded domains. This is done in the context of stationary martingale solutions of the Navier–Stokes equations driven by an Ornstein–Uhlenbeck process. Specifically, we show that under an assumption of ‘on average’ precompactness in L ³, the local structure functions are expressed up to first order in the length scale as nonlinear fluxes, in the vanishing viscosity limit and within an appropriate range of scales. If in addition one assumes local energy equality, this is equivalent to expressing the structure functions in terms of the local dissipation. Our precompactness assumption is also shown to produce stationary martingale solutions of the Euler equations with the same type of forcing in the vanishing viscosity limit.

本文的主要目标是获得足够的条件，使我们能够严格推导出 4/5 和 4/3 流体动力学湍流定律的局部版本，我们指的是这些定律在有界域中的版本。这是在由 Ornstein-Uhlenbeck 过程驱动的 Navier-Stokes 方程的平稳鞅解的上下文中完成的。具体来说，我们表明，在L ³中“平均”预紧性的假设下，局部结构函数在长度尺度上以非线性通量、粘度消失极限和适当的尺度范围内的一阶表示。如果另外假设局部能量相等，这相当于用局部耗散来表示结构函数。我们的预紧性假设也被证明可以产生 Euler 方程的固定鞅解，在消失的粘度极限中具有相同类型的强迫。