We study steady vortex sheet solutions of the Navier–Stokes in the limit of vanishing viscosity at fixed energy flow. We refer to this as the turbulent limit. These steady flows correspond to a minimum of the Euler Hamiltonian as a functional of the tangent discontinuity of the local velocity parametrized as [Formula: see text]. This observation means that the steady flow represents the low-temperature limit of the Gibbs distribution for vortex sheet dynamics with the normal displacement [Formula: see text] of the vortex sheet as a Hamiltonian coordinate and [Formula: see text] as a conjugate momentum. An infinite number of Euler conservation laws lead to a degenerate vacuum of this system, which explains the complexity of turbulence statistics and provides the relevant degrees of freedom (random surfaces). The simplest example of a steady solution of the Navier–Stokes equation in the turbulent limit is a spherical vortex sheet whose flow outside is equivalent to a potential flow past a sphere, while the velocity is constant inside the sphere. Potential flow past other bodies provide other steady solutions. The new ingredient we add is a calculable gap in tangent velocity, leading to anomalous dissipation. This family of steady solutions provides an example of the Euler instanton advocated in our recent work, which is supposed to be responsible for the dissipation of the Navier–Stokes equation in the turbulent limit. We further conclude that one can obtain turbulent statistics from the Gibbs statistics of vortex sheets by adding Lagrange multipliers for the conserved volume inside closed surfaces, the rate of energy pumping, and energy dissipation. The effective temperature in our Gibbs distribution goes to zero as [Formula: see text] with Reynolds number [Formula: see text] in the turbulent limit. The Gibbs statistics in this limit reduces to the solvable string theory in two dimensions (so-called [Formula: see text] critical matrix model). This opens the way for nonperturbative calculations in the Vortex Sheet Turbulence, some of which we report here.

中文翻译：

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作为可解弦理论的涡片湍流

我们研究了在固定能量流下消失粘度极限下的 Navier-Stokes 稳定涡旋片解。我们将此称为湍流极限。这些稳定流对应于欧拉哈密顿量的最小值，作为局部速度的切线不连续性的函数，参数化为 [公式：参见文本]。这一观察意味着稳态流动代表涡流片动力学吉布斯分布的低温极限，涡流片的法向位移[公式：见文本]作为哈密顿坐标，[公式：见文本]作为共轭动量. 无限数量的欧拉守恒定律导致该系统的退化真空，这解释了湍流统计的复杂性并提供了相关的自由度（随机表面）。湍流极限中 Navier-Stokes 方程的稳定解的最简单示例是球形涡旋片，其外部的流动等效于通过球体的势流，而球体内的速度是恒定的。经过其他物体的潜在流动提供了其他稳定的解决方案。我们添加的新成分是切线速度的可计算间隙，导致异常耗散。这一系列稳定解提供了我们最近工作中提倡的欧拉瞬时子的一个例子，它被认为是导致 Navier-Stokes 方程在湍流极限中消散的原因。我们进一步得出结论，可以通过为闭合表面内的守恒体积、能量泵送速率和能量耗散添加拉格朗日乘数，从涡旋片的吉布斯统计中获得湍流统计。我们的吉布斯分布中的有效温度变为零，因为 [公式：见文本] 雷诺数 [公式：见文本] 在湍流极限。此极限中的吉布斯统计量简化为二维可解弦理论（所谓的[公式：见正文]临界矩阵模型）。这为 Vortex Sheet Turbulence 中的非微扰计算开辟了道路，我们在此报告了其中的一些。