# 科研过程

 Abstracts References and etc. 我的介绍 能活着就是我的目标, 如果死了当然我也不知道了就是. 研究生培养 PDE研讨班 Bedrossian, Jacob; Masmoudi, Nader. Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations. Publ. Math. Inst. Hautes \'Etudes Sci. 122 (2015), 195--300.   Bae, Hantaek; Biswas, Animikh; Tadmor, Eitan. Analyticity and decay estimates of the Navier-Stokes equations in critical Besov spaces. Arch. Ration. Mech. Anal. 205 (2012), no. 3, 963--991.   Bae, Hantaek. Analyticity of the inhomogeneous incompressible Navier–Stokes equations. Appl. Math. Lett. 83 (2018), 200--206.   Cheskidov, A.; Shvydkoy, R. The regularity of weak solutions of the 3D Navier-Stokes equations in $B^{-1}_{\infty,\infty}$. Arch. Ration. Mech. Anal. 195 (2010), no. 1, 159--169.   Chen, Qionglei; Miao, Changxing; Zhang, Zhifei. On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations. Comm. Math. Phys. 284 (2008), no. 3, 919--930. 科研指导20180907 科研指导20180907: 二维电流满足的方程 20180905研讨班 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  http://www.cnblogs.com/zhangzujin/p/9035439.html. 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, Several new regularity criteria for the axisymmetric Navier-Stokes equations with swirl, 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 $1=1/2:om} \om\in L^\frac{2\be}{2\be-r}(0,T;\dot B^{-r}_{\infty,\infty}(\bbR^2))\mbox{ for some } 0=1/2:j} j\in L^\frac{2\be}{2\be-r}(0,T;\dot B^{-r}_{\infty,\infty}(\bbR^2)),\mbox{ for some }00}$ 时, 如果 $$\bee\label{thm:al,be>0:om,j} \om,j\in L^{\max\sed{\frac{2\al}{2\al-r},\frac{2\be}{2\be-r}}} (0,T;\dot B^{-r}_{\infty,\infty}(\bbR^2))\mbox{ for some } 01, 要 \om\in L^\infty(L^r), 都要 \be\geq 1/2. 对涡度做 L^p 估计根本没用啊. Vorticity directions 1: self-improving property of the vorticity 在 [Li, Siran. "On Vortex Alignment and Boundedness of  L^ q  Norm of Vorticity." arXiv preprint arXiv:1712.00551 (2017)] 中, 作者证明了$$\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 3-2$是一个常数, 那么解光滑. 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}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

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