科研过程

PDE 研讨班 (http://math.funbbs.me/forumdisplay.php?fid=9)

 

一些工作的介绍

 

Abstracts

References and etc.

Navier-Stokes equations (Lectured by Luis Caffarelli)

 

百万美元问题: http://www.claymath.org/millennium-problems/navier%E2%80%93stokes-equation 

Official Problem Description:

http://www.claymath.org/sites/default/files/navierstokes.pdf

视频

NSE: known a priori estimate

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].

开问题: 涡度的方向与解的正则性

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.

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
$$\serdm{|\sin \angle(\om(x,t),\om(y,t))|\leq C|x-y|^\be\\ \om\in L^q(\bbR^3\times (0,T))}\ra \om \in L^\infty(0,T;L^q(\bbR^3)),$$
其中 $q>\f{5}{3},\ \be\in \sez{\max\sed{0,\f{5}{q}-2},1}.$

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
$$u\times\f{\om}{|\om|}\in L^p(0,T;L^q(\bbR^3)),\quad\f{2}{p}+\f{3}{q}=1,\quad 3<q\leq\infty,$$

$$\om\times\f{u}{|u|}\in L^p(0,T;L^q(\bbR^3)),\quad\f{2}{p}+\f{3}{q}=2,\quad \f{3}{2}<q\leq\infty,$$
则解光滑.

Geometric regularity criterion for NSE: the cross product of velocity and vorticity 2: $u\times \om\cdot \n\times \om$

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
$$\f{u}{|u|}\times \f{\om}{|\om|}\cdot \f{\n\times \om}{|\n\times \om|}$$
充分小, 则解光滑.

Geometric regularity criterion for NSE: the cross product of velocity and vorticity 3: $u\times \f{\om}{|\om|}\cdot \f{\vLm^\be u}{|\vLm^\be u|}$

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
$$u\times \f{\om}{|\om|}\cdot \f{\vLm^\be u}{|\vLm^\be u|}\in L^p(0,T;L^q(\bbR^3)),\quad \f{2}{p}+\f{3}{q}=1,\quad 3<q\leq\infty,\quad 1\leq \be\leq 2,$$
则解光滑.

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
$$|u(x+y,t)\cdot \om(x,t)|\leq c_1|y||u(x+y,t)||\om(x,t),\ |y|\leq \del,$$
则解光滑.

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.

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.

Regularity criteria for NSE 3: $-\lap u=\n\times \om$

一个开问题就是: 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.

Regularity criteria for NSE 4: $\p_3u$

In [Zhang, Zujin. An improved regularity criterion for the Navier–Stokes equations in terms of one directional derivative of the velocity field. Bull. Math. Sci. 8 (2018), no. 1, 33--47] we have improved the results in Kukavica and Ziane (J Math Phys 48:065203, 2007) and Cao (Discrete Contin Dyn Syst 26:1141–1151, 2010) simultaneously. The result reads: the condition
$$\bee\label{me}\p_3\bbu\in L^p(0,T;L^q(\bbR^3)),\quad \frac{2}{p}+\frac{3}{q}=2,\quad \frac{3\sqrt{37}}{4}-3\leq q\leq 3\eee$$
could ensure the regularity of the solution.

see https://link.springer.com/article/10.1007/s13373-016-0098-x.

Regularity criteria for NSE 5: $u_3,\om_3$

In [Zhang, Zujin. Serrin-type regularity criterion for the Navier-Stokes equations involving one velocity and one vorticity component. Czechoslovak Math. J. 68 (2018), no. 1, 219--225], we give an affirmative answer to an open problem in [Penel, Patrick; Pokorn\'y, Milan. Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity. Appl. Math. 49 (2004), no. 5, 483--493], that is, whether or not we could obtain a regularity criterion involving only $u_3$ and $\om_3=\p_1u_2-\p_2u_1$. Our result reveals that if $$\bee\label{this} \bea u_3\in L^p(0,T;L^q(\bbR^3));&\quad \omega_3\in L^r(0,T;L^s(\bbR^3)),\\ \frac{2}{p}+\frac{3}{q}=1,\ 3<q\leq\infty;&\quad \frac{2}{r}+\frac{3}{s}=2,\quad \frac{3}{2}< s\leq \infty, \eea \eee$$ then the solution is smooth on $(0,T)$.

Regularity criteria for NSE 6: $u_3,\p_3u_1,\p_3u_2$

In [Zujin Zhang, Jinlu Li, Zheng-an Yao, A remark on the global regularity criterion for the 3D Navier-Stokes equations based on end-point Prodi-Serrin conditions, Applied Mathematics Letters, 83 (2018), 182—187], we take full advantage of the regularity of the vertical velocity component, and show that $$\bee\label{u_3,p_3u_1,p_3u_2} u_3\in L^\infty(0,T;L^3(\bbR^3))\mbox{ and }\p_3\bbu_h\in L^\be(0,T;L^\al(\bbR^3)),\quad \f{2}{\be}+\f{3}{\al}=2,\quad 2\leq \al\leq \infty, \eee$$ could ensure the smoothness of the solution. This improves the result $$\bee\label{Qian16} u_3\in L^\infty(0,T;L^3(\bbR^3))\mbox{ and }\p_3\bbu_h\in L^\be(0,T;L^\al(\bbR^3)),\quad \f{2}{\be}+\f{3}{\al}=2,\quad 2\leq \al\leq 3, \eee$$ in [Qian, Chenyin. A remark on the global regularity for the 3D Navier-Stokes equations. Appl. Math. Lett. 57 (2016), 126--131] significantly.

 

链接: https://pan.baidu.com/s/1OUvQqQexmloZOFPxcyTiSQ 密码: wr2y

轴对称 Navier-Stokes 方程组的一个点态正则性准则

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.

链接: https://pan.baidu.com/s/1pMIyx31 密码: 2nu9

轴对称 Navier-Stokes 方程组的点态正则性准则 I

在 [Lei, Zhen; Zhang, Qi. Criticality of the axially symmetric Navier-Stokes equations. Pacific J. Math. 289 (2017), no. 1, 169--187] 中, 作者证明了如果 $$\bex \sup_{t\geq 0} |ru^\tt(t,r,z)|\leq C_*|\ln r|^{-2},\quad r\leq \del_0\in\sex{0,\f{1}{2}},\quad C_*<\infty, \eex$$ 则解光滑. 

轴对称 Navier-Stokes 方程组的点态正则性准则 II

在 [Wei, Dongyi. Regularity criterion to the axially symmetric Navier-Stokes equations. J. Math. Anal. Appl. 435 (2016), no. 1, 402--413] 中, 作者证明了如果 $$\bex \sup_{t\geq 0} |ru^\tt(t,r,z)|\leq |\ln r|^{-\f{3}{2}},\quad r\leq \del_0\in\sex{0,\f{1}{2}}, \eex$$ 则解光滑. 那么一个开问题就是能否让指标 $-3/2$ 提高 (最终目标是 $0$, 那样轴对称 NSE 的整体解就解决了). 

A fine property of the convective terms of axisymmetric MHD system, and a regularity criterion in terms of $\om^\tt$

In [Zhang, Zujin; Yao, Zheng-an. 3D axisymmetric MHD system with regularity in the swirl component of the vorticity. Comput. Math. Appl. 73 (2017), no. 12, 2573--2580], we have obtained the following fine property of the convective terms of axisymmetric MHD system Let $u,v,w$ be smooth axisymmetric $\bbR^3$-valued functions. Then $$\bee\label{lem:me:equal} \bea &\quad\sum_{i,j,k=1}^3\p_ku_j\cdot \p_jv_i\cdot \p_kw_i\\ &=\frac{u^r}{r}\cdot \frac{v^r}{r}\cdot \frac{w^r}{r} +\frac{u^r}{r}\cdot \frac{v^\tt}{r}\cdot \frac{w^\tt}{r}\\ &\quad+\frac{u^\tt}{r}\cdot \p_rv^r\cdot \frac{w^\tt}{r} -\frac{u^\tt}{r}\cdot \p_rv^\tt\cdot \frac{w^r}{r}\\ &\quad+ \p_ru^\tt\cdot \frac{v^r}{r}\cdot\p_rw^\tt +\p_zu^\tt \cdot \frac{v^r}{r}\cdot \p_zw^\tt -\p_ru^\tt\cdot \frac{v^\tt}{r}\cdot \p_rw^r -\p_zu^\tt\cdot \frac{v^\tt}{r}\cdot \p_zw^r \\ &\quad +\p_ru^r\cdot \p_rv^r\cdot \p_rw^r +\p_ru^r\cdot \p_rv^\tt\cdot \p_rw^\tt +\p_ru^r\cdot \p_rv^z\cdot \p_rw^z\\ &\quad +\p_ru^z\cdot \p_zv^r\cdot \p_rw^r +\p_ru^z\cdot \p_zv^\tt\cdot \p_rw^\tt +\p_ru^z\cdot \p_zv^z\cdot \p_rw^z\\ &\quad +\p_zu^r\cdot \p_rv^r\cdot \p_zw^r +\p_zu^r\cdot \p_rv^\tt\cdot \p_zw^\tt +\p_zu^r\cdot \p_rv^z\cdot \p_zw^z\\ &\quad +\p_zu^z\cdot \p_zv^r\cdot \p_zw^r +\p_zu^z\cdot \p_zv^\tt\cdot \p_zw^\tt +\p_zu^z\cdot \p_zv^z\cdot \p_zw^z. \eea \eee$$ With this above fine property, we could be able to find a regularity criterion in terms of $\om^\tt$ and $j^\tt$. Moreover, using the governing equations of $j^\tt$: $$\bee\label{j_tt} \bea &\p_t j^\tt +u^r\p_rj^\tt+u^z\p_zj^\tt -\sex{\p_r^2+\p_z^2+\frac{1}{r}\p_r-\frac{1}{r^2}}j^\tt\\ &=b^r\p_r\om^\tt +b^z\p_z\om^\tt +(\p_ru^r-\p_zu^z)(\p_zb^r+\p_rb^z) -(\p_zu^r+\p_ru^z) (\p_rb^r-\p_zb^z), \eea \eee$$ we could show that if $$\bee\label{thm:me:om^tt} \om^\tt\in L^p(0,T;L^q(\bbR^3)),\quad\frac{2}{p} +\frac{3}{q}=2,\quad 2\leq q\leq 3, \eee$$ then the solution is smooth on $(0,T)$. 

QGE 在齐次 Besov 空间中的准则

在 [Zhang, Zujin. On the blow-up criterion for the quasi-geostrophic equations in homogeneous Besov spaces. Comput. Math. Appl. 75 (2018), no. 3, 1038--1043] 中, 我们将 Dong-Pavlovic 在非齐次 Besov 空间中的准则推广到齐次 Besov 空间, 证明了如下爆破准则: $$\beex \int_0^{T^*} \sen{\tt(\tau)}_{\dot B^s_{\infty,\infty}}^\frac{\gm}{\gm+s-1}\rd \tau=\infty,\quad \forall\ 1-\frac{\gm}{2}<s<1, \eeex$$ 其中 $T^*$ 是强解的极大存在时间. 

链接: https://pan.baidu.com/s/1eSWDDdC 密码: rd2x

液晶流在齐次 Besov 空间中的正则性准则

在 [Zhang, Zujin. Regularity criteria for the three dimensional Ericksen–Leslie system in homogeneous Besov spaces. Comput. Math. Appl. 75 (2018), no. 3, 1060--1065] 中, 我们讨论了 $$\bee\label{EL:Simple} \seddm{ \p_t\bbu   +(\bbu\cdot\n)\bbu     -\lap\bbu+\n P     =-\n\cdot[\n\bbd \odot\n\bbd],\\ \p_t\bbd+(\bbu\cdot\n)\bbd   =\lap \bbd     -\bbf(\bbd),\\ \Div\bbu=0,\\ (\bbu,\bbd)|_{t=0}=(\bbu_0,\bbd_0), } \eee$$ 说明如果 $$\bee\label{thm:EL:Simple:reg} \bbu\in L^\frac{2}{1+r}(0,T;\dot B^r_{\infty,\infty}(\bbR^3)),\quad 0<r<1, \eee$$ 则解光滑. 也讨论了 $$\bee\label{EL:d=1}   \seddm{   \p_t\bbu     +(\bbu\cdot\n)\bbu     -\lap \bbu     +\n P=-\n\cdot (\n\bbd\odot\n\bbd),\\   \p_t\bbd+(\bbu\cdot\n)\bbd     =\lap\bbd+|\n\bbd|^2\bbd,\\   \Div\bbu=0,\quad |\bbd|=1,\\   (\bbu,\bbd_0)|_{t=0}=(\bbu_0,\bbd_0).   }   \eee$$ 说明如果 $$\bee\label{thm:EL:Simple:d=1:reg}   \bbu\in L^\frac{2}{1+r}(0,T;\dot B^r_{\infty,\infty}(\bbR^3)),\quad   \n\bbd\in L^\frac{2}{1+s}(0,T;\dot B^s_{\infty,\infty}(\bbR^3)),\quad -1<r,s<1,   \eee$$   则解光滑. 最后讨论了一般的 Ericksen-Leslie 系统 $$\bee\label{EL}   \seddm{   \p_t\bbu     +(\bbu\cdot\n)\bbu     -\lap\bbu     +\n P       =-\Div \sez{(\n \bbd)^t \cfrac{\p W(\bbd,\n\bbd)}{\p (\n\bbd)}},\\   \p_t\bbd     +(\bbu\cdot\n)\bbd       =\bbh-(\bbd\cdot \bbh)\bbd,\\   \Div\bbu=0,\quad |\bbd|=1,\\   (\bbu,\bbd)|_{t=0}=(\bbu_0,\bbd_0),   }   \eee$$ 说明如果 $$\bee\label{thm:EL:reg}   \bbu\in L^\frac{2}{1+r}(0,T;\dot B^r_{\infty,\infty}(\bbR^3)),\quad   \n\bbd\in L^\frac{2}{1+s}(0,T;\dot B^s_{\infty,\infty}(\bbR^3)),\quad -1<r,s<1,   \eee$$   则解光滑. 

链接: https://pan.baidu.com/s/1raiKJeO 密码: eqfb

带阻尼的磁流体方程组的整体适定性

在 [Zujin Zhang, Chupeng Wu, Zheng-an Yao, Remarks on global regularity for the 3D MHD system with damping, Applied Mathematics and Computation, 333 (2018), 1—7] 中, 我们考虑带阻尼的磁流体方程组 $$\bee\label{MHD_damping} \sedd{\ba{ll} \p_t\bbu+(\bbu\cdot\n)\bbu -(\bbb\cdot\n)\bbb -\lap\bbu +|\bbu|^{\al-1}\bbu +\n\pi=\bf{0},\\ \p_t\bbb+(\bbu\cdot\n)\bbb -(\bbb\cdot\n)\bbu -\lap\bbb +|\bbb|^{\beta-1}\bbb =\bf{0},\\ \n\cdot\bbu=\n\cdot\bbb=0,\\ \bbu|_{t=0}=\bbu_0,\quad \bbb|_{t=0}=\bbb_0, \ea} \eee$$ 并证明了如果 $$\bee\label{thm:1} 3\leq \al\leq \f{27}{8},\quad \be\geq 4; \eee$$ $$\bee\label{thm:2} \f{27}{8}<\al\leq\f{7}{2},\quad \be\geq \f{7}{2\al-5}; \eee$$ $$\bee\label{thm:3} \f{7}{2}<\al<4,\quad \be\geq \f{5\al+7}{2\al}; \eee$$ $$\bee\label{thm:4} \al\geq 4,\quad \be\geq 1. \eee$$ 那么 \eqref{MHD_damping} 有一个唯一的整体强解. 主要想法有两个: 一是阻尼越强, 整体适定性应该更好做; 二是速度场如果足够好, 那么磁场可不要阻尼. 

链接: https://pan.baidu.com/s/1iaKn0_4ABafC55yuofZSKw 密码: c3tq

$\be$-QGE 的弱强唯一性

在 [Zhao, Jihong; Liu, Qiao. Weak-strong uniqueness criterion for the $\beta$-generalized surface quasi-geostrophic equation. Monatsh. Math. 172 (2013), no. 3-4, 431--440] 中, 作者考虑 $$\bee\label{be qge} \seddm{ \p_t\tt+(\bbu\cdot\n)\tt+\nu \vLm^\al \tt=0,\\ \tt|_{t=0}=\tt_0, } \eee$$ 其中 $$\bex \bbu=(u_1,u_2)=\vLm^{1-\be} \calR^\perp \tt =\vLm^{1-\be}(-\calR_2\tt,\calR_1\tt). \eex$$ 证明了如果 $$\bee\label{ws:Zhao-Liu} \n\tt\in L^p(0,T;L^q(\bbR^3)),\quad \f{\al}{p}+\f{2}{q} =\al+\be-1,\quad \f{2}{\al+\be-1}<q<\infty, \eee$$ 则有弱强唯一性. 想将其推广到 Besov 空间, 发现不行. 最起码 $q>2/\gm$ 的时候不行. 不知道有啥办法没有. 困难在于所估计的两个函数的正则性不一样. 嗨. 再等等. http://math.funbbs.me/viewthread.php?tid=69&extra=page%3D1

 

posted @ 2018-01-28 08:09 张祖锦 阅读(...) 评论(...) 编辑 收藏