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Navier-Stokes 在 $\dot B^{-1}_{\infty,\infty}$ 中的小性	In [], we have proved that if $\sen{\bbu}_{\dot B^{-1}_{\infty,\infty}}$ is sufficiently small, then the solution to the Navier-Stokes flow is in fact smooth.
科研指导20180907
20180905研讨班
科研指导2018-07-18	(2.2) 是有问题的. 还是回到以前你问的问题. 理由如下: 我们有 Bony 分解 $uv=T_uv+T_vu+R(u,v)$. 在积分的时候, 有 $$\beex \bea \int uv\rd x &=\int \sum_j \lap_j u \sum_k \lap_k v\rd x\\ &=\sum_{j,k}\int \lap_j u\lap_k v\rd x\\ &=\sum_{j,k}\int u \lap_j\lap_k v\rd x\\ &=\sum_{|j-k|\leq 1} \int u\lap_j\lap_k v\rd x\\ &=\sum_{|j-k|\leq 1} \int \lap_ju\lap_k v\rd x. \eea \eeex$$ 这里, $\lap_j$ 可以转移是因为卷积的性质, 见  . 对一般的 $\dps{\int |uv|^p\rd x}$, 你试下能否得到?
轴对称 Navier-Stokes 方程组的一些新准则	在 [Zujin Zhang, , Computers Mathematics with Applications, (2018), accepted] 中, 我们给出了 [T. Gallay, V. \v Sver\'ak, Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations, Confluentes Mathematici, Volume 7 (2015), no. 2, 67--92] 下列估计的详细证明. 累了半天才看懂, 也才完整给出. Let $a,b\in [-1,2]$ such that $0\leq b-a<1$, and assume that $\gm,\be\in (1,+\infty)$ satisfy $$\bex \f{1}{\gm}=\f{1}{\be}-\f{1+a-b}{2}. \eex$$ If $r^b\om^\tt\in L^\be(\Om)$ with $\Om=\sed{(r,z)\in\bbR^2;r>0,\ z\in\bbR}$, then $r^au^r\in L^r(\Om)$, and we have the bound $$\bee\label{lem:weight_sobo:ineq} \sen{r^au^r}_{L^\gm(\Om)} \leq C\sen{r^b\om^\tt}_{L^\be(\Om)}. \eee$$ 利用上述估计给出了轴对称 Navier-Stokes 方程组关于 $\om^\tt$ 的进一步的带权正则性准则 $$\bee\label{thm1:reg} r^d\om^\tt \in L^\al(0,T;L^\be(\bbR^3)),\ \f{2}{\al}+\f{3}{\be}=2-d,\ \seddm{ 1<\be<\f{3}{-d},&-\f{3}{2}\leq d\leq -1\\ \f{3}{2-d}<\be<\f{3}{-d},&-1< d<0}. \eee$$ 进一步, 利用新构建的 Hardy 型不等式及 $\n\cdot\bbu=0$ 条件, 给出了另外三个准则 $$\bee\label{thm2:pr ur} \bea r^d\p_ru^r\in L^\al(0,T;L^\be(\bbR^3)),&\ \f{2}{\al}+\f{3}{\be}=2-d,\\ &\ \f{3}{2-d}<\be<\infty,\ \be\neq \f{2}{1-d},\ 0\leq d<2; \eea \eee$$ $$\bee\label{thm2:pz uz} \bea r^d\p_zu^z\in L^\al(0,T;L^\be(\bbR^3)),&\ \f{2}{\al}+\f{3}{\be}=2-d,\\ &\ \f{3}{2-d}<\be<\infty,\ \be\neq \f{2}{1-d},\ 0\leq d<2; \eea \eee$$ $$\bee\label{thm2:pr utt} \bea r^d\p_ru^\tt\in L^\al(0,T;L^\be(\bbR^3)),&\ \f{2}{\al}+\f{3}{\be}=2-d,\\ &\ \f{3}{2-d}<\be<\infty,\ \be\neq \f{2}{1-d},\ 0\leq d<2. \eea \eee$$ 这里我们证明了两个推广的 Hardy 型不等式: Let $1<p<\infty$ and $\dps{d\neq \f{p-2}{p}}$. Then there exists a positive constant $C$ depending only on $p,d$ such that $$\bee\label{cor:Hardy1:eq} \sen{r^{d-1}f}_{L^p}\leq C\sen{r^d \p_rf}_{L^p}, \eee$$ for each $f=f(r,z)\in W^{1,p}(\bbR^3)$ with $f(0,z)=0$. Here, $f(0,z)$ is in the trace sense. 及 Let $1<p<\infty$ and $\dps{d\neq \f{p-2}{p}}$. Then there exists a positive constant $C$ depending only on $p,d$ such that $$\bee\label{cor:Hardy2:eq} \sen{r^df}_{L^p}\leq C\sen{r^d \p_r(rf)}_{L^p}, \eee$$ for each $f=f(r,z)\in W^{1,p}(\bbR^3)$.
双边乘积型 Sobolev 不等式及在 Navier-Stokes 方程组中的应用	在 [Zujin Zhang, Chupeng Wu, , Annales Polonici Mathematici, (2018), accepted] 中, 我们证明了不等式 $$\bex \int_{\bbR^3} f^2g^2\rd x \leq 8 \sen{f}_{L^2}^\frac{1}{2} \sen{\p_if}_{L^2}^\frac{1}{2} \sen{\p_jf}_{L^2}^\frac{1}{2} \sen{\p_i\p_jf}_{L^2}^\frac{1}{2} \sen{g}_{L^2} \sen{\p_kg}_{L^2},\quad\sed{i,j,k}=\sed{1,2,3}. \eex$$ 利用上述不等式, 证明了 Navier-Stokes 方程组的一个正则性准则 $$\bee\label{thm:comb:eq} \ba{lll} &\p_ju_i\in L^4(0,T;L^2(\bbR^3)),\quad \p_i\p_ju_i\in L^8(0,T;L^2(\bbR^3)),\\ \mbox{or }&\p_iu_i\in L^4(0,T;L^2(\bbR^3)),\quad \p_j\p_iu_i\in L^8(0,T;L^2(\bbR^3)). \ea \eee$$ 又证明了 $$\bex \int_{\bbR^3} f^2g^2\rd x \leq 2\sen{\p_i\p_j(f^2)}_{L^1}\sen{g}_{L^2}\sen{\p_kg}_{L^2},\quad\sed{i,j,k}=\sed{1,2,3}. \eex$$ 利用上述不等式证明了 Navier-Stokes 方程组的两个正则性准则 $$\bee\label{thm:serrin_Hess:eq} \p_i\p_j(|\bbu|^2)\in L^2(0,T;L^1(\bbR^3))\qx{1\leq i\neq j\leq 3}, \eee$$ 及 $$\bee\label{thm:hess:eq} \p_j\p_i(u_i^2)\in L^\frac{8}{3}(0,T;L^1(\bbR^3))\qx{1\leq i\neq j\leq 3}. \eee$$
Navier-Stokes-Maxwell 方程组的一个正则性准则	在 [Zhang, Zujin, Jian Pan, and Shulin Qiu. "." Bulletin of the Malaysian Mathematical Sciences Society (2017): 1-8] 中, 我们考虑 Navier-Stokes-Maxwell 方程组 $$\bee\label{NSM} \seddm{ \bbu_t+(\bbu\cdot\n)\bbu -\lap\bbu+\n P=\bbj\times\bbB,\\ \bbE_t-\curl \bbB=-\bbj,\\ \bbB_t+\curl \bbE=\bbO,\\ \Div \bbu=\Div\bbB=0,\\ (\bbu,\bbE,\bbB)|_{t=0}=(\bbu_0,\bbE_0,\bbB_0). } \eee$$ 并证明了如果 $$\bee\label{thm:reg} \bea \bbu\in L^\f{2}{1-r}(0,T;\dot B^{-r}_{\infty,\infty}(\bbR^3))\mbox{ with } -1<r<1,\\ \mbox{ and }\n\bbB\in L^p(0,T;L^q(\bbR^3))\mbox{ with }\f{2}{p}+\f{3}{q}=2\mbox{ with }2\leq q\leq 3, \eea \eee$$ 则 $$\bex \bbu,\bbE,\bbB\in L^\infty(0,T;H^2(\bbR^3)),\quad \n\bbu,\bbj\in L^2(0,T;H^2(\bbR^3)), \eex$$ 而解可光滑延拓.
三维 magnetic Bénard 问题的一个轴对称解的存在性	在 [Zhang, Zujin, and Tong Tang. "." Applicable Analysis (2017): 1-11] 中, 我们考虑三维 magnetic Bénard 问题 $$\bee\label{MB} \sedd{\ba{ll} \p_t\bbu+(\bbu\cdot\n)\bbu -(\bbb\cdot\n)\bbb -\lap\bbu +\n p=\vtt \bbe_3,\\ \p_t \bbb+(\bbu\cdot\n)\bbb -(\bbb\cdot\n)\bbu -\lap\bbb=\bbO,\\ \p_t\vtt+(\bbu\cdot\n)\vtt-\lap\vtt=\bbu\cdot \bbe_3,\\ \n\cdot\bbu=\n\cdot\bbb=0,\\ \bbu(0)=\bbu_0,\ \bbb(0)=\bbb_0,\ \vtt(0)=\vtt_0, \ea} \eee$$ 并证明了它有行如 $$\bee \bea \bbu&=u^r(t,r,z)\bbe_r +u^\tt(t,r,z)\bbe_\tt +u^z(t,r,z)\bbe_z,\\ \bbb&=b^r(t,r,z)\bbe_r +b^\tt(t,r,z)\bbe_\tt +b^z(t,r,z)\bbe_z,\\ \vtt&=\vtt(t,r,z) \eea \eee$$ 的解. 各分量满足的方程为 $$\bee\label{MB:axis} \sedd{\ba{ll} \frac{\tilde D}{Dt}u^r -\sex{\p_r^2+\p_z^2+\frac{1}{r}\p_r -\frac{1}{r^2}}u^r +\p_r\pi =b^r\p_rb^r+b^z\p_zb^r +\frac{|u^\tt|^2}{r} -\frac{|b^\tt|^2}{r},\\ \frac{\tilde D}{Dt}u^\tt -\sex{\p_r^2+\p_z^2+\frac{1}{r}\p_r-\frac{1}{r^2}}u^\tt =b^r\p_rb^\tt +b^z\p_zb^\tt -\frac{u^ru^\tt}{r}+\frac{b^rb^\tt}{r},\\ \frac{\tilde D}{Dt}u^z -\sex{\p_r^2+\p_z^2+\frac{1}{r}\p_r}u^z+\p_z\pi =b^r\p_rb^z +b^z\p_zb^z+\vtt,\\ \frac{\tilde D}{Dt}b^r -\sex{\p_r^2+\p_z^2+\frac{1}{r}\p_r -\frac{1}{r^2}}b^r =b^r\p_ru^r +b^z\p_zu^r,\\ \frac{\tilde D}{Dt}b^\tt -\sex{\p_r^2+\p_z^2+\frac{1}{r}\p_r -\frac{1}{r^2}}b^\tt =b^r\p_ru^\tt+b^z\p_zu^\tt -\frac{u^\tt b^r}{r} +\frac{u^r b^\tt}{r},\\ \frac{\tilde D}{Dt}b^z -\sex{\p_r^2+\p_z^2+\frac{1}{r}\p_r}b^z =b^r\p_ru^z+b^z\p_zu^z,\\ \frac{\tilde D}{Dt}\vtt -\sex{\p_r^2+\p_z^2+\frac{1}{r}\p_r}\vtt =u^z,\\ \p_r(ru^r)+\p_z(ru^z)=\p_r(rb^r)+\p_z(rb^z)=0,\\ (u^r,u^\tt,u^z,b^r,b^\tt,b^z,\vtt)|_{t=0}=(u^r_0,u^\tt_0,u^z_0,b^r_0,b^\tt_0,b^z_0,\vtt_0). \ea} \eee$$
二维广义磁流体方程组在 Besov 空间中的正则性	在 [Zhang. . To appear in Rocky Mountain J. Math.https://projecteuclid.org/euclid.rmjm/1528164035] 中, 我们考虑了二维广义磁流体方程组 $$\bee\label{GMHD} \seddm{ \p_t\bbu +(\bbu\cdot\n)\bbu -(\bbb\cdot\n)\bbb +\vLm^{2\al}\bbu +\n \Pi=\bbO,\\ \p_t\bbb+(\bbu\cdot\n)\bbb -(\bbb\cdot\n)\bbu +\vLm^{2\be}\bbb=\bbO,\\ \n\cdot\bbu=\n\cdot\n\bbb=0,\\ (\bbu,\bbb)|_{t=0}=(\bbu_0,\bbb_0), } \eee$$ 证明了当 $\dps{\al,\be\geq \frac{1}{2}}$ 时, 如果 $$\bee\label{thm:al,be>=1/2:om} \om\in L^\frac{2\be}{2\be-r}(0,T;\dot B^{-r}_{\infty,\infty}(\bbR^2))\mbox{ for some } 0<r<\be \eee$$ or $$\bee\label{thm:al,be>=1/2:j} j\in L^\frac{2\be}{2\be-r}(0,T;\dot B^{-r}_{\infty,\infty}(\bbR^2)),\mbox{ for some }0<r<\be, \eee$$ 则解可以光滑延拓. 当 $\dps{\al,\be>0}$ 时, 如果 $$\bee\label{thm:al,be>0:om,j} \om,j\in L^{\max\sed{\frac{2\al}{2\al-r},\frac{2\be}{2\be-r}}} (0,T;\dot B^{-r}_{\infty,\infty}(\bbR^2))\mbox{ for some } 0<r<\min\sed{\al,\be}, \eee$$ 则解可以光滑延拓.
科研指导2018-07-07	NSE 关于 $u_3$ (或者等价的, $\n u_3$) 在 Besov 空间中的准则: Skalák, Zdeněk. Criteria for the regularity of the solutions to the Navier-Stokes equations based on the velocity gradient. Nonlinear Anal. 118 (2015), 1--21. Fang, Daoyuan; Qian, Chenyin. Regularity criterion for 3D Navier-Stokes equations in Besov spaces. Commun. Pure Appl. Anal. 13 (2014), no. 2, 585--603.
科研指导2018-07-04	弱解满足 $$\bex \bbu\in L^2(0,T;H^1) \subset L^2(0,T;L^6)\subset L^2(0,T;\dot B^{-\f{1}{2}}_{\infty,\infty}). \eex$$ 如果你的准则正确, 那么用你的结果 $s=-1/2$ 的情形不是把百万美元问题解决了么? 再自己检查检查. 如果正确, 那就恭喜你. (3.3) 为啥没有 Remainder 项?
Navier-Stokes 方程组在 Besov 框架下的最终准则	在 [Zhang, Zujin; Yang, Xian. Navier-Stokes equations with vorticity in Besov spaces of negative regular indices. J. Math. Anal. Appl. 440 (2016), no. 1, 415--419] 中, 我们说明: 如果 $$\bex \n\times \bbu\in L^\frac{2}{2-r}(0,T;\dot B^{-r}_{\infty,\infty}),\quad 0<r<2\lra \bbu\in L^\frac{2}{1-s}(0,T;\dot B^{-s}_{\infty,\infty}),\quad -1<s<1, \eex$$ 则解是光滑的.
带霍尔效应的磁流体方程组的正则性准则	在 [Zhang, Zujin. 3D Hall-MHD system with vorticity in Besov spaces. Ann. Polon. Math. 121 (2018), no. 1, 91--98] 中, 我们将 [Ye, Zhuan. Regularity criterion for the 3D Hall-magnetohydrodynamic equations involving the vorticity. Nonlinear Anal. 144 (2016), 182--193] 获得的结果 $$\bee\label{Ye} \n\om\in L^p(0,T;L^q(\bbR^3)),\quad \f{2}{p}+\f{3}{q}=1,\quad 3<q<\infty \eee$$ 推广到了齐次 Besov 空间: $$\bee\label{Zhang_this} \bea &\quad \n\om \in L^\f{2}{1-\f{3}{q}}(0,T;\dot B^{-\f{3}{q}}_{\infty,\infty}(\bbR^3)),\quad 3<q<\infty\\ &\lra \om \in L^\f{2}{1-\f{3}{q}}(0,T; \dot B^{1-\f{3}{q}}_{\infty,\infty}(\bbR^3)),\quad 3<q<\infty\\ &\lra \om \in L^\f{2}{s}(0,T;\dot B^s_{\infty,\infty}(\bbR^3)),\quad 0<s<1\\ &\lra \n u\in L^\f{2}{s}(0,T;\dot B^s_{\infty,\infty}(\bbR^3)),\quad 0<s<1\\ &\lra u\in L^\f{2}{r-1}(0,T;\dot B^r_{\infty,\infty}(\bbR^3)),\quad 1<r<2. \eea \eee$$
	百万美元问题:  :
	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].$\om=\n\times u\in L^\infty(0,T;L^1(\bbR^3))$. 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].
	在 [da Veiga, Hugo Beirao. "." 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.
	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 $q>\f{5}{3},\ \be\in \sez{\max\sed{0,\f{5}{q}-2},1}.$
	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 $$u\times\f{\om}{|\om|}\in L^p(0,T;L^q(\bbR^3)),\quad\f{2}{p}+\f{3}{q}=1,\quad 3<q\leq\infty,$$ 或 $$\om\times\f{u}{|u|}\in L^p(0,T;L^q(\bbR^3)),\quad\f{2}{p}+\f{3}{q}=2,\quad \f{3}{2}<q\leq\infty,$$ 则解光滑.
	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.
	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.
	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.
	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 .
	In [Zhang, Zujin. . 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 $(0,T)$.
	In [Zujin Zhang, Jinlu Li, Zheng-an Yao, , 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: 密码: wr2y
	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 . 链接: 密码: 2nu9
	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.
	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).
	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 $(0,T)$.
	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: 密码: rd2x
	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:  密码: eqfb
	在 [Zujin Zhang, Chupeng Wu, Zheng-an Yao, , 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: 密码: c3tq
	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.