论文标题
不可压缩的流体中的间歇性和较低的尺寸耗散
Intermittency and lower dimensional dissipation in incompressible fluids
论文作者
论文摘要
在不可压缩的流体的背景下，湍流的奇异结构无法实现空间填充的观察被称为``间歇性''，并且具有强大的实验基础。因此，正如Landau首先指出的那样，真正的动荡流不满足K41理论中同质性和自相似性的中心假设，而K41结构功能指数的预测$ζ_P= \ frac {p} {p} {3} {3} {3} $可能是不合理的。在这项工作中，我们证明，在无关紧要的情况下，从适当意义上讲是较低维度的能量耗散意味着在$ p> 3 $的每个$ p-$ th订单结构功能中都偏离K41预测。通过利用Lagrangian型Minkowski尺寸，它非常让人联想到泰勒的冷冻湍流假设，我们对$ζ_P$ $ζ_P$的最强上限与70年代后期Frisch，Sulem和Nelkin提出的$β-$模型相吻合，并在70年代后期添加了一些严格的分析基础。更普遍地，我们探讨了维数假设对耗散支持的关系与$ p- $ th订单绝对结构功能的限制之间的关系。这种方法与当前关于间歇性的数学作品不同，其重点是几何而不是纯粹的分析假设。证明是基于著名的cancertin-e-titi参数的新局部变体，该变体的使用是使用三阶换向器估计值，压力的特殊双重规则性以及沿矢量场流动的动机。
In the context of incompressible fluids, the observation that turbulent singular structures fail to be space filling is known as ``intermittency'' and it has strong experimental foundations. Consequently, as first pointed out by Landau, real turbulent flows do not satisfy the central assumptions of homogeneity and self-similarity in the K41 theory, and the K41 prediction of structure function exponents $ζ_p=\frac{p}{3}$ might be inaccurate. In this work we prove that, in the inviscid case, energy dissipation that is lower-dimensional in an appropriate sense implies deviations from the K41 prediction in every $p-$th order structure function for $p>3$. By exploiting a Lagrangian-type Minkowski dimension that is very reminiscent of the Taylor's frozen turbulence hypothesis, our strongest upper bound on $ζ_p$ coincides with the $β-$model proposed by Frisch, Sulem and Nelkin in the late 70s, adding some rigorous analytical foundations to the model. More generally we explore the relationship between dimensionality assumptions on the dissipation support and restrictions on the $p-$th order absolute structure functions. This approach differs from the current mathematical works on intermittency by its focus on geometrical rather than purely analytical assumptions. The proof is based on a new local variant of the celebrated Constantin-E-Titi argument that features the use of a third order commutator estimate, the special double regularity of the pressure, and mollification along the flow of a vector field.
