Abstract:We investigate a regularity for weak solutions of the following generalized Leray equations \begin{equation*} (-\Delta )^{\alpha }V- \frac {2\alpha -1}{2\alpha }V+V\cdot \nabla V-\frac {1}{2\alpha }x\cdot \nabla V+\nabla P=0, \end{equation*} which arises from the study of self-similar solutions to the generalized Navier-Stokes equations in $\mathbb R^3$. Firstly, by making use of the vanishing viscosity and developing non-local effects of the fractional diffusion operator, we prove uniform estimates for weak solutions $V$ in the weighted Hilbert space $H^\alpha _{\omega }(\mathbb {R}^3)$. Via the differences characterization of Besov spaces and the bootstrap argument, we improve the regularity for weak solution from $H^\alpha _{\omega }(\mathbb {R}^3)$ to $H_{\omega }^{1+\alpha }(\mathbb {R}^3)$. This regularity result, together with linear theory for the non-local Stokes system, leads to pointwise estimates of $V$ which allow us to obtain a natural pointwise property of the self-similar solution constructed by Lai, Miao, and Zheng [Adv. Math. 352 (2019), pp. 981–1043]. In particular, we obtain an optimal decay estimate of the self-similar solution to the classical Navier-Stokes equations by means of the special structure of Oseen tensor. This answers the question proposed by Tsai Comm. Math. Phys., 328 (2014), pp. 29–44.

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中文翻译：

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广义Leray方程弱解的全局规律及其应用

摘要：我们研究了以下广义 Leray 方程的弱解的规律性 \begin{equation*} (-\Delta )^{\alpha }V- \frac {2\alpha -1}{2\alpha }V+V \cdot \nabla V-\frac {1}{2\alpha }x\cdot \nabla V+\nabla P=0, \end{equation*} 源于对广义纳维-斯托克斯的自相似解的研究$\mathbb R^3$ 中的方程。首先，通过利用消失的粘度和发展分数扩散算子的非局部效应，我们证明了加权希尔伯特空间 $H^\alpha _{\omega }(\mathbb { R}^3)$。通过 Besov 空间的差异表征和 bootstrap 论证，我们将弱解的正则性从 $H^\alpha _{\omega }(\mathbb {R}^3)$ 提高到 $H_{\omega }^{1 +\alpha }(\mathbb {R}^3)$。这种规律性的结果，与非局部斯托克斯系统的线性理论一起，导致 $V$ 的逐点估计，这使我们能够获得由 Lai、Miao 和 Zheng 构建的自相似解的自然逐点属性 [Adv. 数学。352 (2019)，第 981-1043 页]。特别是，我们通过 Oseen 张量的特殊结构获得了经典 Navier-Stokes 方程自相似解的最优衰减估计。这回答了蔡英文提出的问题。数学。Phys., 328 (2014)，第 29-44 页。我们通过Oseen张量的特殊结构获得了经典Navier-Stokes方程自相似解的最优衰减估计。这回答了蔡英文提出的问题。数学。Phys., 328 (2014)，第 29-44 页。我们通过Oseen张量的特殊结构获得了经典Navier-Stokes方程自相似解的最优衰减估计。这回答了蔡英文提出的问题。数学。Phys., 328 (2014)，第 29-44 页。

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