Global regularity of weak solutions to the generalized Leray equations and its applications
Global regularity of weak solutions to the generalized Leray equations and its applications
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广义Leray方程弱解的全局正则性及其应用
DOI：
10.1090/tran/8455
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发表时间：
2019-09
影响因子：
1.3
通讯作者：
Zheng Xiaoxin
中科院分区：
文献类型：
--
作者：
Lai Baishun;Miao Changxing;Zheng Xiaoxin
We investigate the 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 similar solutions to the generalized Naiver-Stokes equations in $\mathbb R^3$. Firstly, by making use of the vanishing viscosity and developing the non-local effect of the fractional diffusion operator, we give the unifrom estimate for weak solution $V$ in the weighted Hilbert space $H^\alpha_{\omega}(\R^3)$. Via the differences characterization of Besov spaces and the bootatrap argument, we improve the regularity for weak solution from $H^\alpha_{\omega}(\R^3)$ to $H_{\omega}^{1+\alpha}(\R^3)$. This regularity result, together linear theory for the non-local stokes system, deduces the pointwise estimates of $V$ which makes us to obtain the natural pointwise property of the self-similar solution constructed in [12]. In particular, we obtain the optimal decay estimate to the self-similar solution of the classical Naiver-Stokes equations by means of the special structure of Oseen tensor under the low regularity condition on initial data.
This answers the question proposed by Tsai in \cite{Tsai}.
We investigate the 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 similar solutions to the generalized Naiver-Stokes equations in $\mathbb R^3$. Firstly, by making use of the vanishing viscosity and developing the non-local effect of the fractional diffusion operator, we give the unifrom estimate for weak solution $V$ in the weighted Hilbert space $H^\alpha_{\omega}(\R^3)$. Via the differences characterization of Besov spaces and the bootatrap argument, we improve the regularity for weak solution from $H^\alpha_{\omega}(\R^3)$ to $H_{\omega}^{1+\alpha}(\R^3)$. This regularity result, together linear theory for the non-local stokes system, deduces the pointwise estimates of $V$ which makes us to obtain the natural pointwise property of the self-similar solution constructed in [12]. In particular, we obtain the optimal decay estimate to the self-similar solution of the classical Naiver-Stokes equations by means of the special structure of Oseen tensor under the low regularity condition on initial data.
This answers the question proposed by Tsai in \cite{Tsai}.
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影响因子：
1.6
作者：
Jonathan C. Mattingly;Y. Sinai
通讯作者：
Jonathan C. Mattingly;Y. Sinai
影响因子：
1.4
作者：
C. Miao;Baoquan Yuan;Bo Zhang
通讯作者：
C. Miao;Baoquan Yuan;Bo Zhang
影响因子：
0.8
作者：
S. Dubois
通讯作者：
S. Dubois
影响因子：
2.4
作者：
Colombo, Maria;De Lellis, Camillo;De Rosa, Luigi
通讯作者：
De Rosa, Luigi
DOI：
10.2140/apde.2016.9.1811
发表时间：
2014-09
期刊：
arXiv: Analysis of PDEs
影响因子：
--
作者：
M. Korobkov;Tai-Peng Tsai
通讯作者：
M. Korobkov;Tai-Peng Tsai