论文标题
断断续续的主持人定理
An Intermittent Onsager Theorem
论文作者
论文摘要
对于任何规律性指数$β<\ frac 12 $，我们构建了$ c^0_t（H^β\ cap l^{\ frac {1}} {（1-2β）}}}} $ c^0_t（H^β\ cap l^{\ cap l^{\ cap l^{\ cap l^{\ cap l^{\ cap l^{\ cap l^{（1-2β）}}}}}}}）的3D不可压缩欧拉方程的非保守弱解。通过插值，此类解决方案属于$ c^0_tb^{s} _ {3，\ infty} $，$ s $接近$ \ frac 13 $ a a $β$ a $β$ the the the the $ \ frac 12 $。因此，此结果为基于$ l^3 $的Onsager猜想的灵活方面提供了新的证明。同样重要的是，我们的解决方案的间歇性质与动荡流的性质相匹配，湍流的基于$ l^2 $的规律性指数超过$ \ frac 13 $。因此，我们的结果并不暗示，也不是由Isett的工作暗示[Onsager的猜想的证明，数学年鉴，188（3）：871，2018]，他给出了基于Hölder的Onsager猜想的证明。我们的证明是基于作者以前与Buckmaster等人的联合合作的基础。 （3D Euler方程的间歇性凸集成：（AMS-217），普林斯顿大学出版社，2023年），其中为3D不可压缩的Euler方程开发了间歇性凸集成方案。我们采用了具有高阶雷诺应力的方案，该方案是通过最佳相对间歇性的间歇性管流的组合放置来纠正的。
For any regularity exponent $β<\frac 12$, we construct non-conservative weak solutions to the 3D incompressible Euler equations in the class $C^0_t (H^β \cap L^{\frac{1}{(1-2β)}})$. By interpolation, such solutions belong to $C^0_tB^{s}_{3,\infty}$ for $s$ approaching $\frac 13$ as $β$ approaches $\frac 12$. Hence this result provides a new proof of the flexible side of the $L^3$-based Onsager conjecture. Of equal importance is that the intermittent nature of our solutions matches that of turbulent flows, which are observed to possess an $L^2$-based regularity index exceeding $\frac 13$. Thus our result does not imply, and is not implied by, the work of Isett [A proof of Onsager's conjecture, Annals of Mathematics, 188(3):871, 2018], who gave a proof of the Hölder-based Onsager conjecture. Our proof builds on the authors' previous joint work with Buckmaster et al. (Intermittent convex integration for the 3D Euler equations: (AMS-217), Princeton University Press, 2023), in which an intermittent convex integration scheme is developed for the 3D incompressible Euler equations. We employ a scheme with higher-order Reynolds stresses, which are corrected via a combinatorial placement of intermittent pipe flows of optimal relative intermittency.
