论文标题
Onsager对子网格规模的猜想$α$ - 湍流模型
Onsager's Conjecture for Subgrid Scale $α$-Models of Turbulence
论文作者
论文摘要
Onsager的猜想的前半部分指出，如果$ u（\ cdot，t）\ in C^{0，θ}（\ Mathbb {t}^3）$带有$θ> \ frac {1} {3} {3} $。在本文中，我们证明了Onsager对几个子网格量表的猜想的类似物$α$ - 湍流模型。特别是，我们找到了确保能量般数量的解决方案所需的hölder规律性（$ h^1（\ mathbb {t}^3）$或$ l^2（\ Mathbb {t}^3）$ narms of to un我们为Leray-$α$模型，Euler- $α$方程（也称为Inviscid Camassa-Holm方程或拉格朗日平均欧拉方程），修改后的leray- $α$型号，clark-$α$模型，最终建立了这种结果。从某种意义上说，所有这些模型都是Euler方程的正规化。并正式收敛到Euler方程为正则长度尺度$α\ rightarrow 0^+$。对于这些模型的规律性（也是用BESOV和Sobolev空间而言），发现不同的Hölder指数，小于$ 1/3 $，可保证保存相应的能量样数量。与Euler方程相比，这是预期的，这是由于更平滑的非线性。这些结果与（Gwiazda等，2018; Bardos等，2019）的通用系统的$ 1/3 $ ONSAGER指数形成了鲜明对比。
The first half of Onsager's conjecture states that the Euler equations of an ideal incompressible fluid conserve energy if $u (\cdot ,t) \in C^{0, θ} (\mathbb{T}^3)$ with $θ> \frac{1}{3}$. In this paper, we prove an analogue of Onsager's conjecture for several subgrid scale $α$-models of turbulence. In particular we find the required Hölder regularity of the solutions that ensures the conservation of energy-like quantities (either the $H^1 (\mathbb{T}^3)$ or $L^2 (\mathbb{T}^3)$ norms) for these models. We establish such results for the Leray-$α$ model, the Euler-$α$ equations (also known as the inviscid Camassa-Holm equations or Lagrangian averaged Euler equations), the modified Leray-$α$ model, the Clark-$α$ model and finally the magnetohydrodynamic Leray-$α$ model. In a sense, all these models are inviscid regularisations of the Euler equations; and formally converge to the Euler equations as the regularisation length scale $α\rightarrow 0^+$. Different Hölder exponents, smaller than $1/3$, are found for the regularity of solutions of these models (they are also formulated in terms of Besov and Sobolev spaces) that guarantee the conservation of the corresponding energy-like quantity. This is expected due to the smoother nonlinearity compared to the Euler equations. These results form a contrast to the universality of the $1/3$ Onsager exponent found for general systems of conservation laws by (Gwiazda et al., 2018; Bardos et al., 2019).
