论文标题
部分可观测时空混沌系统的无模型预测
Anomalous dissipation and lack of selection in the Obukhov-Corrsin theory of scalar turbulence
论文作者
论文摘要
标量湍流的Obukhov-Corrsin理论[OBU49，COR51]在湍流方面对被动量表对流进行定量预测，并且可以被视为Kolmogorov的Kolmogorov K41 K41完全发展的湍流理论的被动标量的类似物[KOL41]。从1949 - 1951年开始对Obukhov和Corrsin的缩放分析确定了对流扩散方程的关键规律性阈值，并预测了超临界状态中消失的扩散率的限制。在本文中，我们通过构建速度字段和初始基准来提供对该预测的完全严格的数学验证，以使对流扩散方程的独特有限解决方案在任何固定的超临界obukhov-corrsin规律性方面都具有均匀的扩散性，同时也表现出了抗动物的消散。我们的方法依赖于对溶液的空间尺度与布朗运动的尺度之间的相互作用进行精细的定量分析，而布朗运动的尺度代表了随机拉格朗日环境中的扩散。这为异常耗散提供了直接的Lagrangian方法，这是为了详细了解解决方案行为的基础。利用这种方法，我们还表明，对于$ c^α$的速度场（对于时空的$ c^α$）（对于任意$ 0 \ leqleqα<1 $ $）既不会消失的扩散率，也不是卷积的正则化，为对流方程的有限溶液提供了选择标准。这是由选择欧拉方程解决方案解决方案的基本开放问题的动机，因为Navier-Stokes方程的解决方案的消失粘度极限，并在被动对流的情况下提供了完全负面的答案。
The Obukhov-Corrsin theory of scalar turbulence [Obu49, Cor51] advances quantitative predictions on passive-scalar advection in a turbulent regime and can be regarded as the analogue for passive scalars of Kolmogorov's K41 theory of fully developed turbulence [Kol41]. The scaling analysis of Obukhov and Corrsin from 1949-1951 identifies a critical regularity threshold for the advection-diffusion equation and predicts anomalous dissipation in the limit of vanishing diffusivity in the supercritical regime. In this paper we provide a fully rigorous mathematical validation of this prediction by constructing a velocity field and an initial datum such that the unique bounded solution of the advection-diffusion equation is bounded uniformly-in-diffusivity within any fixed supercritical Obukhov-Corrsin regularity regime while also exhibiting anomalous dissipation. Our approach relies on a fine quantitative analysis of the interaction between the spatial scale of the solution and the scale of the Brownian motion which represents diffusion in a stochastic Lagrangian setting. This provides a direct Lagrangian approach to anomalous dissipation which is fundamental in order to get detailed insight on the behavior of the solution. Exploiting further this approach, we also show that for a velocity field in $C^α$ of space and time (for an arbitrary $0 \leq α< 1$) neither vanishing diffusivity nor regularization by convolution provide a selection criterion for bounded solutions of the advection equation. This is motivated by the fundamental open problem of the selection of solutions of the Euler equations as vanishing-viscosity limit of solutions of the Navier-Stokes equations and provides a complete negative answer in the case of passive advection.
