We consider a formulation for the Hopf functional differential equation which governs statistical solutions of the Navier-Stokes equations. By introducing an exponential operator with a functional derivative, we recast the Hopf equation as an integro-differential functional equation by the Duhamel principle. On this basis we introduce a successive approximation to the Hopf equation. As an illustration we take the Burgers equation and carry out the approximations to the leading order. Scale invariance of the statistical Navier-Stokes equations in d dimensions is formulated and contrasted with that of the deterministic Navier-Stokes equations. For the statistical Navier-Stokes equations, critical scale invariance is achieved for the characteristic functional of the dth derivative of the vector potential in d dimensions. The deterministic equations corresponding to this choice of the dependent variable acquire the linear Fokker-Planck operator under dynamic scaling. In three dimensions it is the vorticity gradient that behaves like a fundamental solution (more precisely, source-type solution) of deterministic Navier-Stokes equations in the long-time limit. Physical applications of these ideas include study of a self-similar decaying profile of fluid flows. Moreover, we reveal typical physical properties in the late-stage evolution by combining statistical scale invariance and the source-type solution. This yields an asymptotic form of the Hopf functional in the long-time limit, improving the well-known Hopf-Titt solution. In particular, we present analyses for the Burgers equations to illustrate the main ideas and indicate a similar analysis for the Navier-Stokes equations.

中文翻译：

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关于湍流的霍普夫方程的研究：杜哈默原理和动力学定标。

我们考虑了Hopf泛函微分方程的公式，该方程控制着Navier-Stokes方程的统计解。通过引入具有函数导数的指数算子，我们通过Duhamel原理将Hopf方程重铸为积分微分方程。在此基础上，我们对Hopf方程进行了逐次逼近。作为说明，我们采用Burgers方程，并对前导阶进行近似。公式化了d维统计Navier-Stokes方程的尺度不变性，并将其与确定性Navier-Stokes方程的尺度不变性进行了对比。对于统计Navier-Stokes方程，对于d维矢量势的dth导数的特征函数，实现了临界尺度不变性。对应于因变量选择的确定性方程式在动态缩放下获取线性Fokker-Planck算子。在三个维度上，涡度梯度的行为类似于长期限制下确定性Navier-Stokes方程的基本解（更精确地说，是源类型解）。这些想法的物理应用包括对流体自衰减曲线的研究。此外，通过结合统计尺度不变性和源类型解，我们揭示了后期演化的典型物理性质。在长时间限制下，这会产生Hopf函数的渐近形式，从而改善了众所周知的Hopf-Titt解决方案。特别是，