湍流级联与分数场的一个线性随机模型
A Linear Stochastic Model of Turbulent Cascades and Fractional Fields
AI总结：
本文构造了一个由零阶微分算子控制的线性方程，在外力为统计均匀平稳随机过程时模拟湍流能量级联，并证明解在有限时刻光滑、在无穷时刻收敛到分数Gaussian场。
摘要：
湍流级联刻画了由大尺度上的随机力注入的能量向小尺度传递的过程。在流体动力学湍流中，当雷诺数很大时，流体的速度场变得不规则，并且即使流体黏性趋于零，能量耗散率仍然保持有下界。自Kolmogorov在流体动力学湍流中以及Zakharov在波湍流中的开创性工作以来，湍流级联的数学描述一直是一个非常活跃的研究课题。在这两种情形中，这些湍流级联机制都意味着若干统计量（例如功率谱密度）具有幂律行为。长期以来，人们认为这些级联与非线性相互作用有关，但最近的工作表明，级联也可以发生在由零阶微分算子控制的线性方程所支配的动力学中。本着这一思路，我们构造了一个线性方程，当外力是统计均匀且平稳的随机过程时，该方程能够模拟能量级联的唯象行为。在Fourier变量中，该方程可以看作一个线性输运方程，它在物理空间中对应于一个零阶算子。我们的结果给出了解的完整刻画：解在任意有限时刻都是光滑的，并且在忽略较小阶修正项的情况下，它在无穷时刻收敛到一个分数Gaussian场。
英文摘要：
Turbulent cascades characterize the transfer of energy injected by a random force at large scales towards the small scales. In hydrodynamic turbulence, when the Reynolds number is large, the velocity field of the fluid becomes irregular and the rate of energy dissipation remains bounded from below even if the fluid viscosity tends to zero. A mathematical description of the turbulent cascade is a very active research topic since the pioneering work of Kolmogorov in hydrodynamic turbulence and that of Zakharov in wave turbulence. In both cases, these turbulent cascade mechanisms imply power-law behaviors of several statistical quantities such as power spectral densities. For a long time, these cascades were believed to be associated with nonlinear interactions, but recent works have shown that they can also take place in a dynamics governed by a linear equation with a differential operator of degree 0. In this spirit, we construct a linear equation that mimics the phenomenology of energy cascades when the external force is a statistically homogeneous and stationary stochastic process. In the Fourier variable, this equation can be seen as a linear transport equation, which corresponds to an operator of degree 0 in physical space. Our results give a complete characterization of the solution: it is smooth at any finite time, and, up to smaller order corrections, it converges to a fractional Gaussian field at infinite time.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
