Convex integration and phenomenologies in turbulence
Convex integration and phenomenologies in turbulence
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DOI：
10.4171/emss/34
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发表时间：
2019-01-01
影响因子：
2.3
通讯作者：
Vicol, Vlad
中科院分区：
文献类型：
--
作者：
Buckmaster, Tristan;Vicol, Vlad
本文讨论了一些关于不可压缩Euler方程和Navier-Stokes方程野弱解的最新结果。这些结果建立在De Lellis和Szekelyhidi Jr.开创性工作的基础上，他们将纳什关于C-1柔性等长嵌入的基本思想扩展到了流体动力学领域。这些技术统称为凸积分，与流体动力湍流的现象学理论有基本的相似之处[51,54,55,200]。湍流中出现的数学问题（如Onsager猜想）不仅引发了对凸积分的新兴趣，而且某些实验观察到的湍流特征（如间歇性）也为新的凸积分结构提供了信息。首先，我们给出了由De Lellis-Szekelyhidi Jr.[52,53]首先证明的欧拉方程的非保守C-x，t（0+）弱解的初等构造。其次，我们提出了Isett[108]最近对Onsager猜想的柔性面的解决方案。在这里，我们实际上遵循De Lellis-Szekelyhidi Jr.和本文作者的联合工作[21]，其中构造了正则类C-x，t（1/3-）中的欧拉方程的弱解，得到了任意能量分布。第三，我们给出了作者最近的结果[23]的简洁证明，该结果证明了C-t(0) L-x（2+）布尔和C-t(0) W-x（1,1+）正则类中存在无穷多个Navier-Stokes弱解。在本文的最后，我们提到了凸积分和流体动力湍流交叉处的一些开放问题。
In this review article we discuss a number of recent results concerning wild weak solutions of the incompressible Euler and Navier-Stokes equations. These results build on the groundbreaking works of De Lellis and Szekelyhidi Jr., who extended Nash's fundamental ideas on C-1 flexible isometric embeddings, into the realm of fluid dynamics. These techniques, which go under the umbrella name convex integration, have fundamental analogies with the phenomenological theories of hydrodynamic turbulence [51, 54, 55, 200]. Mathematical problems arising in turbulence (such as the Onsager conjecture) have not only sparked new interest in convex integration, but certain experimentally observed features of turbulent flows (such as intermittency) have also informed new convex integration constructions.First, we give an elementary construction of nonconservative C-x,t(0+) weak solutions of the Euler equations, first proven by De Lellis-Szekelyhidi Jr. [52, 53]. Second, we present Isett's [108] recent resolution of the flexible side of the Onsager conjecture. Here, we in fact follow the joint work [21] of De Lellis-Szekelyhidi Jr. and the authors of this paper, in which weak solutions of the Euler equations in the regularity class C-x,t(1/3-) are constructed, attaining any energy profile. Third, we give a concise proof of the authors' recent result [23], which proves the existence of infinitely many weak solutions of the Navier-Stokes in the regularity class C-t(0) L-x(2+) boolean AND C-t(0) W-x(1,1+).We conclude the article by mentioning a number of open problems at the intersection of convex integration and hydrodynamic turbulence.