输运微结构和耗散欧拉流
Transporting microstructure and dissipative Euler flows
摘要：
最近，第二和第三作者开发了一个迭代方案，用于获得Hölder空间中三维不可压缩Euler方程的粗糙解（arXiv：1202.1751和arXiv：1205.3626（2012））。其动机来自Onsager的猜想。该结构涉及到无穷多个尺度上的弱相互作用扰动Beltrami流的叠加。一个障碍，以更好的规律性产生的错误，在一个快速的周期性流的线性传输的速度慢场。在最近的一篇论文中，P. Isett（arXiv：1211.4065）改进了我们的方法，引入了一些关于如何处理这种障碍的新想法，从而获得了更好的Hölder指数-尽管低于Onsager的指数。本文对Isett的最终结果给出了一个较短的证明，它更好地坚持了原方案，并引入了一些新的方法。更精确地说，我们证明了对于任何正的$ε$，存在3D不可压缩欧拉方程的周期解，该方程耗散了总动能，并且属于Hölder类$C^{\frac{1}{5}-ε}$。
英文摘要：
Recently the second and third author developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in Hölder spaces (arXiv:1202.1751 and arXiv:1205.3626 (2012)). The motivation comes from Onsager's conjecture. The construction involves a superposition of weakly interacting perturbed Beltrami flows on infinitely many scales. An obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field. In a recent paper P. Isett (arXiv:1211.4065) has improved upon our methods, introducing some novel ideas on how to deal with this obstruction, thereby reaching a better Hölder exponent - albeit below the one conjectured by Onsager. In this paper we give a shorter proof of Isett's final result, adhering more to the original scheme and introducing some new devices. More precisely we show that for any positive $ε$ there exist periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy and belong to the Hölder class $C^{\frac{1}{5}-ε}$.
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