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Abstracts |
References and etc. |
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Navier-Stokes equations (Lectured by Luis Caffarelli)
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百万美元问题: http://www.claymath.org/millennium-problems/navier%E2%80%93stokes-equation http://www.claymath.org/sites/default/files/navierstokes.pdf |
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1. Leray-Hopf $u\in L^\infty(0,T;L^2(\bbR^3))\cap L^2(0,T;H^1(\bbR^3))$. See [Leray, Jean. Sur le mouvement d‘un liquide visqueux emplissant l‘espace. (French) Acta Math. 63 (1934), no. 1, 193--248]. 2. $\om=\n\times u\in L^\infty(0,T;L^1(\bbR^3))$. See for example [Qian, Zhongmin. An estimate for the vorticity of the Navier-Stokes equation. C. R. Math. Acad. Sci. Paris 347 (2009), no. 1-2, 89--92]. |
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在 [da Veiga, Hugo Beirao. "Open problems concerning the Holder continuity of the direction of vorticity for the Navier-Stokes equations." arXiv preprint arXiv:1604.08083 (2016)] 中, Hugo Beirao 说明了如果涡度在 $(x,t), (y,t)$ 处的涡度 $\om(x,t), \om(y,t)$ 的夹角的正弦 $\leq C|x-y|^\be$, $\be\in [1/2,1]$, 那么解是光滑的. 但是这个 $1/2$ 却不可以降低一点点. 也就是如果 $\sin\angle (\om(x,t),\om(y,t))\leq C|x-y|^\be$, $0<\be<1/2$, 那么 Leray-Hopf 弱解的正则性一点都不能抬高...更不要说是强解了. 真是奇怪. 试了下, 对任何 $r>1$, 要 $\om\in L^\infty(L^r)$, 都要 $\be\geq 1/2$. 对涡度做 $L^p$ 估计根本没用啊. |
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Vorticity directions 1: self-improving property of the vorticity |
在 [Li, Siran. "On Vortex Alignment and Boundedness of $ L^ q $ Norm of Vorticity." arXiv preprint arXiv:1712.00551 (2017)] 中, 作者证明了 |
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Geometric regularity criterion for NSE: the cross product of velocity and vorticity 1: $u\times \om$ |
在 [Chae, Dongho. On the regularity conditions of
suitable weak solutions of the 3D Navier-Stokes equations. J. Math. Fluid
Mech. 12 (2010), no. 2, 171--180] 中, 作者证明了如果 |
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在 [Lee, Jihoon. Notes on the geometric
regularity criterion of 3D Navier-Stokes system. J. Math. Phys. 53 (2012),
no. 7, 073103, 6 pp] 中, 作者证明了如果 |
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在 [Chae, Dongho; Lee, Jihoon. On the geometric
regularity conditions for the 3D Navier-Stokes equations. Nonlinear Anal. 151
(2017), 265--273] 中, 作者证明了如果 |
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Geometric regularity criterion for NSE: the cross product of velocity and vorticity 4: $u\cdot \om$ |
在 [Berselli, Luigi C.; Córdoba, Diego. On the
regularity of the solutions to the 3D Navier-Stokes equations: a remark on
the role of the helicity. C. R. Math. Acad. Sci. Paris 347 (2009), no. 11-12,
613--618] 中, 作者证明了如果 |
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Regularity criteria for NSE 1: $u$ |
经典的 Prodi-Serrin 型准则告诉我们: 如果 $$u\in L^p(0,T;L^q(\bbR^3)),\quad \f{2}{p}+\f{3}{q}=1,\quad 3\leq q\leq\infty,$$ 那么解光滑. |
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Regularity criteria for NSE 2: $\n u$ |
[Beir$\tilde a$o da Veiga, H. A new regularity class for the Navier-Stokes equations in ${\bf R}^n$. A Chinese summary appears in Chinese Ann. Math. Ser. A 16 (1995), no. 6, 797. Chinese Ann. Math. Ser. B 16 (1995), no. 4, 407--412] 则告诉我们: 如果 $$\n u\in L^p(0,T;L^q(\bbR^3)),\quad \f{2}{p}+\f{3}{q}=2,\quad \f{3}{2}\leq q\leq\infty,$$ 那么解光滑. |
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Regularity criteria for NSE 3: $-\lap u=\n\times \om$ |
一个开放性问题就是: 如果 $$-\lap u=\n\times \om\in L^p(0,T;L^q(\bbR^3)),\quad \f{2}{p}+\f{3}{q}=3,\quad 1\leq q\leq\infty$$ 能否推出解的光滑性. Sobolev 嵌入及 Beir$\tilde a$o da Veiga 的结果告诉我们如果 $1\leq q<3$, 则解光滑. 当 $q=3$ 时, weak Lebesgue 空间中的准则在 [Berselli, Luigi C. Some criteria concerning the vorticity and the problem of global regularity for the 3D Navier-Stokes equations. Ann. Univ. Ferrara Sez. VII Sci. Mat. 55 (2009), no. 2, 209--224] 中得到了: $\curl \om\in L^1(0,T;L^3_w(\bbR^3))$. 问题是当 $3<q<\infty$, 解也是光滑的么? 我试了下 $L^p$ 估计和对方程作 $\vLm^s u$ 试验, 都不行. |
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对轴对称 NSE, 我们改进了 [Pan, Xinghong. A regularity condition of 3d axisymmetric Navier-Stokes equations. Acta Appl. Math. 150 (2017), 103--109] 的正则性准则: $ru^r\geq -1$, 证明了如果 $ru^r\geq M$, 其中 $M>-2$ 是一个常数, 那么解光滑. 见 https://www.sciencedirect.com/science/article/pii/S0022247X18300040. 下载: 链接: https://pan.baidu.com/s/1ggBRbUz 密码: ecq4
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