张祖锦论文
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# 2022
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Zhang, Zujin; Zeng, Caiyang. Global regularity for a family of models of the axisymmetric Navier-Stokes system. Z. Angew. Math. Phys. 73 (2022), no. 1, Paper No. 32, 13 pp. (http://www.ams.org/mathscinet-getitem?mr=4356477)
In this paper, we investigate a family of 3D models for the axisymmetric incompressible Navier–Stokes system. The models are derived by changing the strength of convection and adding an stirring force. We show the global regularity when the strength of convection is stronger than that of the original Navier–Stokes system, which reveals the potential stabilization effect of convection. Our result answers an open problem in Hou–Liu–Wang (Nonlinearity 31:1940-1954, 2018). 链接: https://pan.baidu.com/s/1n5D-KHTuW7f8xSnAfunLig?pwd=jun4 提取码: jun4
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Zhang, Zujin; Tong, Chenxuan. Remarks on the a priori bound for the vorticity of the axisymmetric Navier-Stokes equations. Appl. Math. 67 (2022), no. 4, 485--507. (http://www.ams.org/mathscinet-getitem?mr=4444789)
We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used refined test function and re-scaling scheme, and showed that
$$\begin{aligned} \|\omega^r(x,t)\|+\|\omega^z(r,t)\|\leq \frac{C}{r^{10}},\quad0 < r\leq \frac{1}{2}. \end{aligned}$$
By invoking the dimension reduction technique by Lei-Navas-Zhang, and analyzing $\omega^r, \omega^z$ and $\frac{\omega^\theta}{r}$ on different hollowed cylinders, we could be able to improve it as
$$\begin{aligned} \|\omega^r(x,t)\|+\|\omega^z(r,t)\|\leq \frac{C\|\ln r\|}{r^\frac{17}{2}},\quad0 < r\leq \frac{1}{2}. \end{aligned}$$
链接: https://pan.baidu.com/s/1AEB_R3dxJfi1E29_6cJM_A?pwd=gvzh 提取码: gvzh
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Zhang, Zujin; Rao, Jinfan. Global well-posedness of 3D axisymmetric MHD system with large swirl magnetic field. J. Math. Anal. Appl. 516 (2022), no. 1, Paper No. 126483.(http://www.ams.org/mathscinet-getitem?mr=4454289)
In this work, we investigate the axisymmtric MHD system with nearly critical initial data having the special structure: $u_0=u_0^r e_r+u_0^\theta e_\theta+u_0^z e_z, b_0=b_0^\theta e_\theta$. It is proved that if the scaling-invariant norm $\left\Vert ru_0^\theta\right\Vert _{L^\infty}$ is sufficiently small, then this system is globally well-posed. 链接: https://pan.baidu.com/s/1Faa7s6GkXFZh1iAuiSLtww?pwd=2yj8 提取码: 2yj8
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# 2021
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Zhang, Zujin; Zhang, Yali. On regularity criteria for the Navier-Stokes equations based on one directional derivative of the velocity or one diagonal entry of the velocity gradient. Z. Angew. Math. Phys. 72 (2021), no. 1, 24.(http://www.ams.org/mathscinet-getitem?mr=4198773)
It is proved that if the solution of the Navier-Stokes system satisfies
$$\begin{aligned} \partial_3u\in L^p(0,T;L^q(\mathbb{R}^3)),\quad\frac{2}{p}+\frac{3}{q} =\frac{22}{13}+\frac{3}{13q},\quad3 < q < 4, \end{aligned}$$
or
$$\begin{aligned} \partial_3u_3\in L^\beta(0,T;L^\alpha(\mathbb{R}^3)),\quad\frac{2}{\beta}+\frac{3}{\alpha} =\frac{3(\sqrt{65\alpha^2-78\alpha+49}+7-\alpha)}{16\alpha},\quad\frac{3+\sqrt{17}}{4}\leq \alpha\leq\infty, \end{aligned}$$
then the solution is smooth on $(0,T]$. These two improve many previous results. 链接: https://pan.baidu.com/s/1iNJnifWaaMYJ6MXQhvqdDg?pwd=auzv 提取码: auzv
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Zhang, Zujin; Wang, Sinan. Serrin type regularity criterion for the shear thinning fluids via the velocity field. Appl. Math. Lett. 116 (2021), 107011.(http://www.ams.org/mathscinet-getitem?mr=4201470)
By estimating the nonlinear term in an innovative way, we could obtain the optimal regularity criterion for the shear thinning fluids via the velocity field. 链接: https://pan.baidu.com/s/1rcE71Sovg_Tr1D_iNsx4FQ?pwd=3zb2 提取码: 3zb2
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Fan, Jishan; Zhang, Zujin. Regularity Criteria of the Density-Dependent Incompressible Ideal Boussinesq and Liquid Crystals Model. Acta Appl. Math. 173 (2021), 3. (http://www.ams.org/mathscinet-getitem?mr=4253396)
In this paper, we prove some regularity criteria for the density-dependent incompressible Boussinesq and liquid crystals model. The Kato-Ponce type commutator estimates play a key role. 链接: https://pan.baidu.com/s/1F5vDp-HXzovd_I1F-IWKZg?pwd=mh3a 提取码: mh3a
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Alghamdi, Ahmad Mohammad; Gala, Sadek; Ragusa, Maria Alessandra; Zhang, Zujin. A Regularity Criterion for the 3D Density-Dependent MHD Equations. Bull. Braz. Math. Soc. (N.S.) 52 (2021), no. 2, 241--251. (http://www.ams.org/mathscinet-getitem?mr=4253404)
In this paper, we consider the $3$D density-dependent magnetohydrodynamic equations with vacuum in the whole space $\mathbb{R}^3$, and provide a regularity criterion involving the velocity field in BMO space norm. This work generalizes the regularity criterion of the constant density MHD equations to the density-dependent one. 链接: https://pan.baidu.com/s/1VybsEEg8iWdVeWIXCDLZOw?pwd=7qtw 提取码: 7qtw
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Zhang, Zujin; Zhang, Yali. An Improved Regularity Criteria for the MHD System Based on Two Components of the Solution. Appl. Math. 66 (2021), no. 3, 451--460.(http://www.ams.org/mathscinet-getitem?mr=4263161)
As observed by Yamazaki, the third component b3 of the magnetic field can be estimated by the corresponding component $u_3$ of the velocity field in $L^\lambda\ (2\leq \lambda\leq 6)$ norm. This leads him to establish regularity criterion involving $u_3,j_3$ or$u_3,\omega_3$. Noticing that $\lambda$ can be greater than $6$ in this paper, we can improve previous results. 链接: https://pan.baidu.com/s/1vDPZ2Ejocoq-_YdrBPd6NQ?pwd=cequ 提取码: cequ
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Zhang, Zujin; Zhang, Yali. Remarks on the regularity criteria for the axisymmetric MHD system. Ann. Polon. Math. 126 (2021), no. 1, 79--95.(http://www.ams.org/mathscinet-getitem?mr=4241180)
In this article, we show several weighted regularity criteria for the axisymmetric solutions to the three-dimensional magnetohydrodynamic equations, involving $u^r, u^z$; $u^r, b^r, \partial_ru^z,\partial_rb^z$; $u^r,b^r, \partial_zu^\theta, \partial_zb^\theta$; $u^r, b^r, \partial_ru^\theta, \partial_rb^\theta$; or $\omega^\theta$, where $u^r,u^\theta,u^z$ are the angular, swirl and axial components of the velocity respectively and $\omega^\theta$ denotes the swirl component of the vorticity. 链接: https://pan.baidu.com/s/14H2IMop8fiAl-sIbFlXS-g 提取码: bqmt
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# 2020
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Zhang, Zujin; Wang, Sinan. Weighted a priori estimates for the swirl component of the vorticity of the axisymmetric Navier–Stokes system. Appl. Math. Lett. 104 (2020), 106275.(http://www.ams.org/mathscinet-getitem?mr=4067918)
We study the axisymmetric Navier–Stokes equations. By invoking a two-dimensional Gagliardo–Nirenberg inequality, we could show
$$\begin{aligned} r^{\frac{7}{2}-\frac{4}{p}}\omega^\theta\in L^\infty \left(0,T; L^p(\mathbb{R}^3)\right), \end{aligned}$$
for any $\frac{3}{2}\leq p\leq \frac{8}{5}$. 链接: https://pan.baidu.com/s/1K38rMoGSZyl0EJ8Fd9WH2g?pwd=pkt4 提取码: pkt4
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# 2019
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Tang, Tong; Zhang, Zujin. A remark on the global existence of weak solutions to the compressible quantum Navier-Stokes equations. Nonlinear Anal. Real World Appl. 45 (2019), 255--261. (http://www.ams.org/mathscinet-getitem?mr=3854305)
In this paper, we obtain the global existence of weak solutions to the compressible quantum Navier–Stokes equations. By virtue of a useful identity and an interesting estimate, we solve the critical case that the viscosity equals the dispersive coefficient. This result removes the restrictions on the coefficients and improves the recent work of Antonell and Spirito (2017) in some senses. 链接: https://pan.baidu.com/s/1pW5xDn3hNfyBXpev07LK6A?pwd=5ddi 提取码: 5ddi
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Zhang, Zujin; Wang, Weihua; Yao, Zheng-an. Components reduction regularity results for the Navier-Stokes equations in general dimensions. J. Math. Anal. Appl. 469 (2019), no. 2, 827--840.(http://www.ams.org/mathscinet-getitem?mr=3860448)
We consider the Cauchy problem of the Navier–Stokes equations in arbitrary dimensions, and establish several new components reduction regularity criteria. 链接: https://pan.baidu.com/s/1IcMb0We5svwZy-BCpph9kQ?pwd=5796 提取码: 5796
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Zhang, Zujin. New a priori estimates for the axisymmetric Navier-Stokes system. Appl. Math. Lett. 92 (2019), 139--143. (http://www.ams.org/mathscinet-getitem?mr=3903956)
We show in this paper that $r^d\omega^\theta$ belongs to
$$\begin{aligned} L^\infty\left(0,\infty;L^2(\mathbb{R}^3)\right)\cap L^2\left(0,\infty; H^1(\mathbb{R}^3)\right) \end{aligned}$$
for any $2 < d\leq 3$. This improves previous a priori estimate significantly. 链接: https://pan.baidu.com/s/1sXnRLm-Hxt9vlQaQNcbOfQ?pwd=3xv4 提取码: 3xv4
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Zhang, Zujin; Wang, Weihua; Yang, Xian. An extension and simpler proof of Berselli–C\'ordoba's geometric regularity condition for the Navier–Stokes system. Comput. Math. Appl. 77 (2019), no. 3, 765--769.(http://www.ams.org/mathscinet-getitem?mr=3913628)
In this short note, we provide an simpler proof of Berselli–Cordoba’s geometric regularity condition for the Navier–Stokes system. Moreover, some interesting extensions are obtained. 链接: https://pan.baidu.com/s/1ysfItpqNxcrnynSjS1ZE9Q?pwd=y9u3 提取码: y9u3
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Zhang, Zujin; Yuan, Weijun; Zhou, Yong. Some remarks on the Navier-Stokes equations with regularity in one direction. Appl. Math. 64 (2019), no. 3, 301--308.(http://www.ams.org/mathscinet-getitem?mr=3956174)
We review the developments of the regularity criteria for the Navier-Stokes equations, and make some further improvements. 链接: https://pan.baidu.com/s/1SoGjfk6VUJu-HEhe2xohXQ?pwd=h8my 提取码: h8my
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谢元福, 张祖锦. 拓扑空间中子集的导集的计算. 赣南师范大学学报, 2019, 03: 7-8.(https://kns.cnki.net/KCMS/detail/detail.aspx?dbcode=CJFQ&dbname=CAPJLAST&filename=GNSY20190423001&uid=WEEvREcwSlJHSldRa1FhcEE0QVN2K1BzZm1vakgwaVU1RENPdk1SQVpTQT0=$9A4hF_YAuvQ5obgVAqNKPCYcEjKensW4IQMovwHtwkF4VYPoHbKxJw!!&v=MDU1ODBxNDFHWk9zT1l3OU16bVJuNmo1N1QzZmxxV00wQ0xMN1I3cWVidWR0RmlEbFZyN1BKVjA9SWlQWWQ3RzRIOWpN)
本文讨论拓扑空间中子集的导集的计算问题，给出了导集的一个新的简单计算方法．链接: https://pan.baidu.com/s/12wonBUkqRu8_nE_is1A0tg?pwd=199s 提取码: 199s
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Zhang, Zujin. Regularity Criteria for the Axisymmetric Navier–Stokes System with Negative Weights. Results Math. 74 (2019), no. 4, 74:134.(http://www.ams.org/mathscinet-getitem?mr=3974041)
As is well-known, for the axisymmetric Navier-Stokes equations, $ru^\theta$ obeys a maximal principle, $\left\Vert ru^\theta(t)\right\Vert _{L^\infty} \leq \left\Vert ru^\theta_0\right\Vert _{L^\infty}$. Utilizing this fact, we can establish some more weighted regularity criteria involving $u^r$ ($u^z$), the radial (axial) component of the velocity. As a by-product, weighted regularity condition via $\omega^\theta$ is also established. These extend previous results significantly. 链接: https://pan.baidu.com/s/132ZO6HWsvHDDuX-rLO5a2w?pwd=wq7p 提取码: wq7p
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Zhang, Zujin; Pan, Jian; Qiu, Shulin. Extended Regularity Criteria for the Navier–Stokes–Maxwell system. Bull. Malays. Math. Sci. Soc. 42 (2019), no. 5, 2039--2046.(http://www.ams.org/mathscinet-getitem?mr=3984407)
We consider the Cauchy problem of the Navier–Stokes–Maxwell system in three dimensions and establish two regularity criteria involving $u$ (or $\nablau$) in Besov spaces and $\nablaB$ in more flexible Lebesgue spaces. This improves and extends recent results of Ma, Jiang and Zhu. 链接: https://pan.baidu.com/s/1do8oXygClUh5wqk3_9PBIg?pwd=6f9k 提取码: 6f9k
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Zhang, Zujin. Remarks on the energy equality for the non-Newtonian fluids. J. Math. Anal. Appl. 480 (2019), no. 2, 123443.(http://www.ams.org/mathscinet-getitem?mr=4000107)
By establishing some new trilinear estimates, we could show that a weak solution to the non-Newtonian fluids satisfies the energy equality, under some integrability conditions on the velocity or velocity gradient. This significantly extends (Yang (2019) )for non-Newtonian fluids, and improves (Berselli and Chiodaroli (2018))for Newtonian fluids. 链接: https://pan.baidu.com/s/1qUkRk51sMoUWhgn6djjhgg?pwd=wak3 提取码: wak3
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Zhang, Zujin; Wu, Chupeng; Zhou, Yong. On Ratio Improvement of Prodi-Serrin-Ladyzhenskaya Type Regularity Criteria for the Navier-Stokes System. Czechoslovak Math. J. 69 (2019), no. 4, 1165--1175.(http://www.ams.org/mathscinet-getitem?mr=4039628)
This paper concerns improving Prodi-Serrin-Ladyzhenskaya type regularity criteria for the Navier-Stokes system, in the sense of multiplying certain negative powers of scaling invariant norms. 链接: https://pan.baidu.com/s/1IQSBWaJldBqoyuXUnnNXlA?pwd=6i78 提取码: 6i78
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Zhang, Zujin; Wang, Weihua; Zhou, Yong. Global regularity criterion for the Navier-Stokes equations based on the direction of vorticity. Math. Methods Appl. Sci. 42 (2019), no. 18, 7126--7134. (http://www.ams.org/mathscinet-getitem?mr=4037956)
We study the regularity criterion for the Navier-Stokes equations and show that the $\beta_1,\beta_2,\beta_3$-Holder continuity assumption in $x_1,x_2,x_3$ on the direction of the vorticity ensures the regularity of the solution. This may be viewed as an extension ofmany previous results, since some of the $\beta_i$ can be arbitrarily small. 链接: https://pan.baidu.com/s/1R5XlolzhT1I9VJ4P6hk2qA?pwd=4pyy 提取码: 4pyy
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张祖锦; 郭雅妮; 杨娴;.一类广义积分与无穷级数的条件收敛性.赣南师范大学学报, 2019, 06:11-13.(gnsy.cbpt.cnki.net/WKE/WebPublication/paperDigest.aspx?paperID=b922a5b8-d19d-4118-b050-9d7dfe015ffb)
本文对数学分析中的一类广义积分与无穷级数的条件收敛性给出了统一处理，证明了两个广义积分与无穷级数条件收敛的审敛方法． 链接: https://pan.baidu.com/s/13jhh26hC_8mpcdwfFuz0bw?pwd=w4g3 提取码: w4g3
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# 2018
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Zhang, Zujin. A pointwise regularity criterion for axisymmetric Navier-Stokes system. J. Math. Anal. Appl. 461 (2018), no. 1, 1--6. (http://www.ams.org/mathscinet-getitem?mr=3759525)
In this paper, we consider the axisymmetric Navier–Stokes equations with non-zero swirl component. It is proved that if $ru^r\geq M$ for some $M>-2$, then the solution actually is smooth. This extends the result in Pan (2017) . 链接: https://pan.baidu.com/s/1o48kl2eQgXE0VdazjyLwiQ?pwd=czsu 提取码: czsu
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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.(http://www.ams.org/mathscinet-getitem?mr=3766500)
In this paper, we consider the blow-up criterion for the quasi-geostrophic equations with dissipation $\varLambda^\gamma$ ($0 < \gamma < 1$). By establishing a new trilinear estimate, we show that if
$$\begin{aligned} \theta\in L^\frac{\gamma}{\gamma+s-1}(0,T;\dot B^s_{\infty,\infty}(\mathbb{R}^2)) \end{aligned}$$
for some $s\in\left(1-\frac{\gamma}{2},1\right)$, then the solution can be extended smoothly past $T$. This improves and extends the corresponding results in Dong and Pavlovic (2009) and Yuan (2010). 链接: https://pan.baidu.com/s/1KV8ZyA_GMXzuC0OWFLUBhg?pwd=bmp8 提取码: bmp8
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Zhang, Zujin. Regularity criteria for the three dimensional Ericksen-Leslie system in homogeneous Besov spaces. Comput. Math. Appl. 75 (2018), no. 3, 1060--1065. (http://www.ams.org/mathscinet-getitem?mr=3766502)
In this paper, we establish some regularity criteria involving homogeneous Besov spaces for both the simplified and the general three dimensional Ericksen–Leslie system. This improves many previous results, and can be viewed as the ultimate optimal regularity criterion in the Besov space framework.链接: https://pan.baidu.com/s/1Q1SJUKUqUXef5lzPXyz9YA?pwd=x325 提取码: x325
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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. (http://www.ams.org/mathscinet-getitem?mr=3775266)
In this paper, we establish a new multiplicative Sobolev inequality. As applications, we refine and extend the results in Kukavica and Ziane (J Math Phys 48:065203, 2007) and Cao (Discrete Contin Dyn Syst 26:1141–1151, 2010) simultaneously.链接: https://pan.baidu.com/s/1Nsu6AlbFL11ckEHPPInlvw?pwd=97gp 提取码: 97gp
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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. (http://www.ams.org/mathscinet-getitem?mr=3783594)
We consider the Cauchy problem for the three-dimensional Navier-Stokes equations, and provide an optimal regularity criterion in terms of $u_3$ and $\omega_3$, which are the third components of the velocity and vorticity, respectively. This gives an affirmative answer to an open problem in the paper by P. Penel, M.Pokorny (2004). 链接: https://pan.baidu.com/s/1SB7yssSxWskbtuT27j6QQQ?pwd=a645 提取码: a645
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Zhang, Zujin; Li, Jinlu; Yao, Zheng-an. A remark on the global regularity criterion for the 3D Navier-Stokes equations based on end-point Prodi-Serrin conditions. Appl. Math. Lett. 83 (2018), 182--187.(http://www.ams.org/mathscinet-getitem?mr=3795689)
By taking full advantage of the regularity of the vertical velocity component, we could be able to show a regularity criterion in terms of the vertical velocity component, and the vertical derivative of the horizontal velocity components, which extends (Qian, 2016) significantly. 链接: https://pan.baidu.com/s/10kkx5Axvxy8JQDkDKZyBvw?pwd=72hx 提取码: 72hx
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Zhang, Zujin; Wu, Chupeng; Yao, Zheng-an. Remarks on global regularity for the 3D MHD system with damping. Appl. Math. Comput. 333 (2018), 1--7.(http://www.ams.org/mathscinet-getitem?mr=3796318)
We investigate the Cauchy problem for the 3D MHD system with damping terms $\|u\|^{\alpha-1}u$ and $\|b\|^{\beta-1}b$ ($\alpha,\beta\geq 1$), and show that the strong solution exists globally and uniquely if one of the following four conditions holds, (1) $3\leq \alpha\leq \frac{27}{8}, \beta\geq 4$; (2) $\frac{27}{8} < \alpha\leq\frac{7}{2}, \beta\geq \frac{7}{2\alpha-5}$; (3) $\frac{7}{2} < \alpha < 4, \beta\geq \frac{5\alpha+7}{2\alpha}$; (4) $\alpha\geq 4, \beta\geq 1$. This improves the previous results significantly. 链接: https://pan.baidu.com/s/1piLxzO6YpxYUG2lMKZoUPw?pwd=sndy 提取码: sndy
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张祖锦, 张程荣, 陈媛, 胡燕玲. 一个数学分析定理在点集拓扑中的推广. 赣南师范大学学报, 2018, 03 : 8--9.(https://kns.cnki.net/KCMS/detail/detail.aspx?dbcode=CJFQ&dbname=CAPJLAST&filename=GNSY2018042400X&uid=WEEvREcwSlJHSldRa1FhcEE0QVN2K1BzZm1vakgwaVU1RENPdk1SQVpTQT0=$9A4hF_YAuvQ5obgVAqNKPCYcEjKensW4IQMovwHtwkF4VYPoHbKxJw!!&v=MDgzNTMzZmxxV00wQ0xMN1I3cWVidWR0RmlEbFZyN1BJVms9SWlQWWQ3RzRIOW5NcTQxQlpPdG5ZdzlNem1SbjZqNTdU)
实数空间到实数空间的两个连续映射，如果在有理数集上相等，则恒等． 上述定理可推广到适当的拓扑空间之间的连续映射． 链接: https://pan.baidu.com/s/1ZOcwiOPSdWR-uG9RTooJzQ?pwd=tz3i 提取码: tz3i
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Zhang, Zujin. 3D Hall-MHD system with vorticity in Besov spaces. Ann. Polon. Math. 121 (2018), no. 1, 91--98.(http://www.ams.org/mathscinet-getitem?mr=3816175)
By introducing some new ideas, using the methods from Z. Ye and Z. Ye and Z. J. Zhang and the result of Z. J. Zhang , we establish the regularity criterion
$$\begin{aligned} \omega\in L^\frac{2}{s}\left(0,T;\dot B^s_{\infty,\infty}(\mathbb{R}^3)\right), 0 < s < 1, \end{aligned}$$
for the 3D Hall-MHD system. This improves several previous results. 链接: https://pan.baidu.com/s/1_Qb6KrZGLNs-c6aGe5pL5A?pwd=t3ek 提取码: t3ek
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Pan, Jian; Zhang, Zujin; Zhou, Xiangying. Optimal dynamic mean-variance asset-liability management under the Heston model. Adv. Difference Equ. 2018, 2018:258.(http://www.ams.org/mathscinet-getitem?mr=3833843)
This paper studies a continuous-time mean-variance asset-liability management problem under the Heston model. Specifically, an asset-liability manager is allowed to invest in a risk-free asset and a risky asset whose price process is governed by the Heston model. By applying the Lagrange duality theorem and stochastic control theory, we derive the closed-form expressions of the efficient investment strategy and the efficient frontier. Moreover, we provide numerical experiments to analyze the sensitivity of the efficient frontier with respect to the relevant parameters in the Heston model. 链接: https://pan.baidu.com/s/15QuJYJ8Ls0rBtrBjNHdDMg?pwd=nj4w 提取码: nj4w
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Zhang, Zujin. Several new regularity criteria for the axisymmetric Navier-Stokes equations with swirl. Comput. Math. Appl. 76 (2018), no. 6, 1420--1426.(http://www.ams.org/mathscinet-getitem?mr=3850659)
In this paper, we consider the axisymmetric Navier–Stokes equations with swirl, and show that the global regularity is ensured if we add some (weighted) integrable conditions on $\omega^\theta=\partial_zu^r-\partial_ru^z, \partial_ru^r, \partial_zu^z$ or $\partial_ru^\theta$. 链接: https://pan.baidu.com/s/1yqYytrlZw8840L9amk-oYg?pwd=45uc 提取码: 45uc
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Zhang, Zujin; Tang, Tong. Global regularity for a special family of axisymmetric solutions to the three-dimensional magnetic Bénard problem. Appl. Anal. 97 (2018), no. 14, 2533--2543. (http://www.ams.org/mathscinet-getitem?mr=3859619)
We study the three-dimensional magnetic Bénard problem, and establish the global regularity for a special family of axisymmetric solutions.链接: https://pan.baidu.com/s/1QUmOGdDPqBclhk6AK1muRA?pwd=aa5n 提取码: aa5n
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Zhang, Zujin; Wu, Chupeng. Some new multiplicative Sobolev inequalities with applications to the Navier-Stokes equations. Ann. Polon. Math. 121 (2018), no. 3, 279--290.(http://www.ams.org/mathscinet-getitem?mr=3878626)
We establish some new multiplicative Sobolev inequalities. As applications, we refine and extend some known regularity criteria for the Navier-Stokes equations. 链接: https://pan.baidu.com/s/1zN8FrgRlz8sOlDaTbFtcvQ?pwd=7nvr 提取码: 7nvr
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Zhang, Zujin. Remarks on regularity criteria for the 2D generalized MHD system in Besov spaces. Rocky Mountain J. Math. 48 (2018), no. 8, 2785--2795. (http://www.ams.org/mathscinet-getitem?mr=3895003)
This paper concerns regularity criteria for the $2$D generalized MHD system and shows that, if we can control the Besov norm of the vorticity and/or the current density, then the solution is, in fact, smooth. This improves the recent result . 链接: https://pan.baidu.com/s/1lBPLyFElGv1dj2HrdSY2gw?pwd=qjch 提取码: qjch
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邱树林, 张祖锦, 刘智广, 范丽娜. 函数在多个点Taylor展开. 赣南师范大学学报, 2018, 06: 13-14.(https://kns.cnki.net/KCMS/detail/detail.aspx?dbcode=CJFQ&dbname=CAPJLAST&filename=GNSY2018110600W&uid=WEEvREcwSlJHSldRa1FhcEE0QVN2K1BzZm1vakgwaVU1RENPdk1SQVpTQT0=$9A4hF_YAuvQ5obgVAqNKPCYcEjKensW4IQMovwHtwkF4VYPoHbKxJw!!&v=MDgzMjNZZDdHNEg5bk5ybzlEWk90b1l3OU16bVJuNmo1N1QzZmxxV00wQ0xMN1I3cWVidWR0RmlEbFZyN0tKRjQ9SWlQ)
本文将数学分析中的Taylor 定理推广到在多个点Taylor 展开． 链接: https://pan.baidu.com/s/1JdUGVz945ujYy_lu4O-I4g?pwd=82ia 提取码: 82ia
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# 2017
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Zhang, Zujin. On weighted regularity criteria for the axisymmetric Navier-Stokes equations. Appl. Math. Comput. 296 (2017), 18--22.(http://www.ams.org/mathscinet-getitem?mr=3572774)
In this paper, we consider the axisymmetric Navier–Stokes equations with swirl. By invoking the magic identity of Miao and Zheng, the symmetry properties of Riesz transforms and the Hardy–Sobolev inequality, we establish regularity criterion involving $r^d\omega^\theta$ with $-1\leq d < 0$. This improves and extends the results of . 链接: https://pan.baidu.com/s/1wiR73DIQcDgfbsUsRx6uqg?pwd=tb6a 提取码: tb6a
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Ye, Zhuan; Zhang, Zujin. A remark on regularity criterion for the 3D Hall-MHD equations based on the vorticity. Appl. Math. Comput. 301 (2017), 70--77.(http://www.ams.org/mathscinet-getitem?mr=3598586)
In this paper we investigate the regularity criterion for the local-in-time classical solution to the three-dimensional ($3$D) incompressible Hall-magnetohydrodynamic equations (Hall- MHD). It is proved that the control of the vorticity alone can ensure the smoothness of the solution. 链接: https://pan.baidu.com/s/1Fll_plZlufG_ZSB2j_irgw?pwd=7d7f 提取码: 7d7f
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Zhang, Zujin; Ouyang, Xiqin; Yang, Xian. Refined a priori estimates for the axisymmetric Navier-Stokes equations. J. Appl. Anal. Comput. 7 (2017), no. 2, 554--558.(http://www.ams.org/mathscinet-getitem?mr=3602437)
In this paper, we consider the axisymmetric Navier-Stokes equations, and provide a refined a priori estimate for the swirl component of the vorticity. This extends Theorem 2 of . 链接: https://pan.baidu.com/s/1P5Hmruxf9iqi4ubHAS9FvQ?pwd=ra5q 提取码: ra5q
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Zhang, Zujin. Generalized MHD System with Velocity Gradient in Besov Spaces of Negative Order. Acta Appl. Math. 149 (2017), 139--144. (http://www.ams.org/mathscinet-getitem?mr=3647036)
This paper studies the $3$D generalized MHD system with fractional diffusion terms $(-\Delta)^\alpha u$ and $(-\Delta)^\beta b$ with $0 < \alpha < \frac{5}{4}\leq \beta$, and establishes a regularity criterion involving the velocity gradient in Besov spaces of negative order. This improves Fan et al. (Math. Phys. Anal. Geom. 17: 333-340, 2014) a lot. 链接: https://pan.baidu.com/s/1VPEephS6pT6A_DH1kHr9YQ?pwd=mb4w 提取码: mb4w
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Zhang, Zujin; Zhong, Dingxing; Huang, Xiantong. A refined regularity criterion for the Navier-Stokes equations involving one non-diagonal entry of the velocity gradient. J. Math. Anal. Appl. 453 (2017), no. 2, 1145--1150.(http://www.ams.org/mathscinet-getitem?mr=3648279)
This paper concerns about the regularity criteria for the three-dimensional Navier–Stokes equations. By establishing a more subtle estimate of the crucial convective term in Navier–Stokes equations, we improve and a lot. 链接: https://pan.baidu.com/s/1LcW6dJgK_JCHR5jxez44aw?pwd=24j4 提取码: 24j4
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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.(http://www.ams.org/mathscinet-getitem?mr=3649997)
This paper concerns with the regularity criteria for the 3D axisymmetric MHD system. It is proved that the control of swirl component of vorticity can ensure the smoothness of the solution. 链接: https://pan.baidu.com/s/1FrSHLCGuVcIRDi1UMHz8nQ?pwd=e94e 提取码: e94e
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张祖锦, 杨兰萍, 李文鑫. 拓扑学中凝聚点的几个等价定义. 赣南师范大学学报, 2017 , 03 : 6--7.(https://kns.cnki.net/KCMS/detail/detail.aspx?dbcode=CJFQ&dbname=CJFDLAST2017&filename=GNSY201703002&uid=WEEvREcwSlJHSldRa1FhcEE0QVN2K1BzZm1vakgwaVU1RENPdk1SQVpTQT0=$9A4hF_YAuvQ5obgVAqNKPCYcEjKensW4IQMovwHtwkF4VYPoHbKxJw!!&v=MDUyOTFNckk5RlpvUjhlWDFMdXhZUzdEaDFUM3FUcldNMUZyQ1VSTE9lWmVabUZ5M2tWN3JJSWlQWWQ3RzRIOWI=)
类比于数学分析中凝聚点的定义，给出了 $A_1$ 且 $T_1$ 拓扑空间中凝聚点的等价刻画，并通过例子说明了 $A_1$ 性和 $T_1$ 性不可或缺．链接: https://pan.baidu.com/s/1Wi5d5Wc-mfJhp4oTLgRQ4w?pwd=ki67 提取码: ki67
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Zhang, Zujin; Zhong, Dingxing; Gao, Shujing; Qiu, Shulin. Fundamental Serrin type regularity criteria for 3D MHD fluid passing through the porous medium. Filomat 31 (2017), no. 5, 1287--1293.(http://www.ams.org/mathscinet-getitem?mr=3630046)
In this paper, we consider the Cauchy problem for the $3$D MHD fluid passing through the porous medium, and provide some fundamental Serrin type regularity criteria involving the velocity or its gradient, the pressure or its gradient. This extends and improves . 链接: https://pan.baidu.com/s/1AszfYlwGoZ773obs6BZHKQ?pwd=y89g 提取码: y89g
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Zhang, Zujin. Refined regularity criteria for the MHD system involving only two components of the solution. Appl. Anal. 96 (2017), no. 12, 2130--2139.(http://www.ams.org/mathscinet-getitem?mr=3667850)
We focus on the Cauchy problem for the MHD system, and refine three previous regularity conditions involving two components of the solutions. 链接: https://pan.baidu.com/s/1KwiqqRFT2E9FrTnrfrUWBA?pwd=gpn6 提取码: gpn6
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Gao, Shujing; Xia, Lijun; Wang, Jialin; Zhang, Zujin. Modeling the effects of cross-protection control in plant disease with seasonality. Int. J. Biomath. 10 (2017), no. 6, 1750088, 24 pp.(http://www.ams.org/mathscinet-getitem?mr=3668116)
Cross-protection in plants has been widely used to control losses caused by virus diseases in the world. Here, a non-autonomous plant-virus disease model was developed including cross-protection. Global dynamics of the model was discussed. Under the quite weak assumptions, integral form conditions were resolved for permanence of the system and extinction of diseases. Furthermore, we looked into the sufficient conditions that plants could be protected against the detrimental effects of infection by an infection with the mild virus isolates. Last, we performed numerical simulations. Our investigations suggested that cross-protection played an important role in controlling the spread of the challenging virus in plants. 链接: https://pan.baidu.com/s/1rCTRbMA3O5XQE5r4_KLlHQ?pwd=1yts 提取码: 1yts
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Zhang, Zujin. Blow-up criterion of strong solutions to the 3D ghost effect system in Besov spaces with negative indices. ZAMM Z. Angew. Math. Mech. 97 (2017), no. 5, 576--585.(http://www.ams.org/mathscinet-getitem?mr=3655250 )
In this paper, we investigate the $3$D ghost effect system, and provide a blow up criterion involving $u$ and $\nabla\vartheta$ in Besov spaces with negative indices. This could be viewed as a generalization of . 链接: https://pan.baidu.com/s/13gCULALP9Qz_Lv2nX8ToDg?pwd=pwea 提取码: pwea
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Gala, Sadek; Ragusa, Maria Alessandra; Zhang, Zujin. A Regularity Criterion in Terms of Pressure for the 3D Viscous MHD Equations. Bull. Malays. Math. Sci. Soc. 40 (2017), no. 4, 1677--1690. (http://www.ams.org/mathscinet-getitem?mr=3712578)
In this note, we are concerned with the regularity of solutions of the MHD equation in terms of the pressure. More precisely, it is proved that if the pressure satisfies the critical growth condition
$$\begin{aligned} \pi(x,t)\in L^\frac{2}{2+r}\left(0,T; \dot B^r_{\infty,\infty}(\mathbb{R}^3)\right), \end{aligned}$$
then the solution remains smooth on $(0,T]$. The finding is mainly based on the innovative function decomposition methods together with Besov space techniques. Here $\dot B^r_{\infty,\infty}$ denotes the homogeneous Besov space. 链接: https://pan.baidu.com/s/1CFcFqeFJy5vnmwlGcTRTaA?pwd=mdpj 提取码: mdpj
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# 2016
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Tang, Tong; Zhang, Zujin. Blow-Up of Smooth Solution to the Compressible Navier-Stokes-Poisson Equations. Bull. Malays. Math. Sci. Soc. 39 (2016), no. 4, 1487--1497.(http://www.ams.org/mathscinet-getitem?mr=3549976)
In this paper, we show the blow-up phenomenon of smooth solutions to the compressible Navier–Stokes–Poisson (N-S-P) equations in $\mathbb{R}^2$, under the assumption that the initial density has compact support. The proof is based on some useful physical quantities. In particular, our result is valid for both isentropic and isothermal cases. 链接: https://pan.baidu.com/s/180AvGyMBFbUS7vvjlwr6ig?pwd=mwyi 提取码: mwyi
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Zhang, Zujin; Yang, Xian; Qiu, Shulin. A regularity condition for the density-dependent magnetohydrodynamic equations in BMO space. Proc. Jangjeon Math. Soc. 19 (2016), no. 1, 125--133.(http://www.ams.org/mathscinet-getitem?mr=3469658)
In this paper, we consider the density-dependent magnetohydrodynamic equations, and provide a regularity criterion involving the velocity field in BMO space, which extends . Moreover, we applied our method of proof to the density-dependent Hall-MHD system. 链接: https://pan.baidu.com/s/1aakV4m76skzC5Wz9UG7-Ag?pwd=jdsy 提取码: jdsy
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Zhang, Zujin; Ouyang Xiqin; Yang, Xian. Navier-Stokes equations with anisotropic regularity in one velocity component. Proc. Jangjeon Math. Soc. 19 (2016), 465--476.(http://www.ams.org/mathscinet-getitem?mr=3617691)
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# 2015
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Zhang, Zujin. Regularity criteria for the 3D MHD equations involving one current density and the gradient of one velocity component. Nonlinear Anal. 115 (2015), 41--49.(http://www.ams.org/mathscinet-getitem?mr=3305136)
In this paper, we consider the Cauchy problem for the 3D MHD equations, and provide a regularity criterion involving $\nablau_3$ and $j_3$, where $u_3$ and $j_3$ are the third component of the velocity and the current density, respectively. As by-products, we show that assumptions on two vorticity and one current density component are enough to ensure the smoothness of the solution, and improve some previous results (Skalak, 2014; Zhou and Pokorny, 2009, 2010) for the Navier–Stokes equations. 链接: https://pan.baidu.com/s/12UzdrTm38XxQ3ivzNtvuyw?pwd=329c 提取码: 329c
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Zhang, Zujin; Yang, Xian. On the regularity criterion for the Navier-Stokes equations involving the diagonal entry of the velocity gradient. Nonlinear Anal. 122 (2015), 169--175.(http://www.ams.org/mathscinet-getitem?mr=3348069)
In this paper, we provide a regularity criterion for the Navier-Stokes equations involving the diagonal entry of the velocity gradient, which says that if
$$\begin{aligned} \partial_3u_3\in L^\infty\left(0,T; L^2(\mathbb{R}^3)\right), \end{aligned}$$
then the weak solution is smooth on $(0,T)$. This is an important case in the sense that the strong solution is in this class. Morover, we verity the limiting case of one regularity criterion in Cao and Titi (2011). 链接: https://pan.baidu.com/s/1obMJz-2jVe1qD7bfVmEO2Q?pwd=wtam 提取码: wtam
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Zhang, Zujin. Remarks on the global regularity criteria for the 3D MHD equations via two components. Z. Angew. Math. Phys. 66 (2015), no. 3, 977--987.(http://www.ams.org/mathscinet-getitem?mr=3347420)
In this paper, we would like to improve the recent results of Yamazaki on the regularity criteria for the threedimensional magentohydrodynamic equations (Yamazaki in J Math Phys 55:031505, 2014; J Math Fluid Mech. 2014. doi:10. 1007/s00021-014-0178-1). 链接: https://pan.baidu.com/s/10RYuJe76D-IxYHCwuVlnRw?pwd=e9dr 提取码: e9dr
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Zhang, Zujin. An almost Serrin-type regularity criterion for the Navier-Stokes equations involving the gradient of one velocity component. Z. Angew. Math. Phys. 66 (2015), no. 4, 1707--1715.(http://www.ams.org/mathscinet-getitem?mr=3377710)
This paper concerns about the Cauchy problem for the three-dimensional Navier–Stokes equations and provides a regularity criterion in terms of the gradient of one velocity component. This improves previous results. 链接: https://pan.baidu.com/s/11vOaanlHv5NZQI1HqGOhYw?pwd=57uj 提取码: 57uj
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Zhang, Zujin; Yang, Xian. A note on the regularity criterion for the 3D Navier-Stokes equations via the gradient of one velocity component. J. Math. Anal. Appl. 432 (2015), no. 1, 603--611.(http://www.ams.org/mathscinet-getitem?mr=3371254)
In this paper, we consider the regularity criterion for weak solutions to the $3$D Navier-Stokes equations. We show that if
$$\begin{aligned} \nablau_3\in L^\frac{32}{7}\left(0,T; L^2(\mathbb{R}^3)\right), \end{aligned}$$
then the solution is smooth on $(0,T)$, which improves the regularity criterion $\nablau_3\in L^\frac{24}{5}\left(0,T;L^2(\mathbb{R}^3)\right)$. 链接: https://pan.baidu.com/s/1rDTMoU69jmt64nAB0ggBYw?pwd=89jh 提取码: 89jh
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Zhang, Zujin; Yang, Xian. A regularity criterion for the 3D density-dependent incompressible flow of liquid crystals with vacuum. Ann. Polon. Math. 115 (2015), no. 2, 165--177.(http://www.ams.org/mathscinet-getitem?mr=3404272)
We consider the Cauchy problem for the $3$D density-dependent incompressible flow of liquid crystals with vacuum, and provide a regularity criterion in terms of $u$ and $\nablad$ in the Besov spaces of negative order. This improves recent result of Fan-Li . 链接: https://pan.baidu.com/s/1ELaYeNg7Pr3HrLl4HCuKvg?pwd=sssa 提取码: sssa
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Zhang, Zujin. Regularity criteria for the 3D magneto-micropolar fluid equations via the direction of the velocity. Proc. Indian Acad. Sci. Math. Sci. 125 (2015), no. 1, 37--43.(http://www.ams.org/mathscinet-getitem?mr=3331910)
We consider sufficient conditions to ensure the smoothness of solutions to $3$D magneto-micropolar fluid equations. It involves only the direction of the velocity and the magnetic field. Our result extends to the cases of Navier–Stokes and MHD equations. 链接: https://pan.baidu.com/s/11fmU_860TdPqNBiOlyRheA?pwd=ifs8 提取码: ifs8
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Zhang, Zujin. Navier-Stokes equations with regularity in one directional derivative of the pressure. Math. Methods Appl. Sci. 38 (2015), no. 17, 4019--4023.(http://www.ams.org/mathscinet-getitem?mr=3434338)
This paper concerns the $3$D Navier–Stokes equations and prove an almost Serrin-type regularity criterion in terms of one directional derivative of the pressure. 链接: https://pan.baidu.com/s/183UsLX-QqzCQorBd6UpShQ?pwd=9kg9 提取码: 9kg9
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Zhang, Zujin; Hong, Pingzhou; Zhong, Dingxing; Qiu, Shulin. A regularity criterion for the 3D MHD equations in terms of the gradient of the pressure in the multiplier spaces. Arab. J. Math. (Springer) 4 (2015), no. 2, 153--157.(http://www.ams.org/mathscinet-getitem?mr=3344165)
In this paper, we consider the regularity criterion for the 3D MHD equations and prove that if the gradient of the pressure belongs to $L^\frac{2}{2-r}\left(0,T;\dot X_r(\mathbb{R}^3)\right)$ with $0\leq r\leq 1$, then the solution is smooth. Notice that we extend the result given by Gala (Appl Anal 92:96–103, 2013). 链接: https://pan.baidu.com/s/1hSyNKXiUxWr683i5RppdwQ?pwd=k4k2 提取码: k4k2
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Zhang, Zujin; Yang, Xian; Qiu, Shulin. Remarks on Liouville type result for the 3D Hall-MHD system. J. Partial Differ. Equ. 28 (2015), no. 3, 286--290.(http://www.ams.org/mathscinet-getitem?mr=3444413)
In this paper, we consider the 3D Hall-MHD system, and provide an improved Liouville type result for its stationary version.链接: https://pan.baidu.com/s/10m9Yncub-VDBJnaPfSokOw?pwd=59c6 提取码: 59c6
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Xu, Xiaojing; Ye, Zhuan; Zhang, Zujin. Remark on an improved regularity criterion for the 3D MHD equations. Appl. Math. Lett. 42 (2015), 41--46. (http://www.ams.org/mathscinet-getitem?mr=3294367)
In this paper we investigate the regularity criterion for the local in time classical solution to the $3$D incompressible magnetohydrodynamic equations. We prove that if $\nabla \times u$ belongs to $L^2\left(0,T;\dot B^{-1}_{\infty,\infty}\right)$, then the local solution (u, B) can be extended beyond time $T$. As a consequence, this result extends several previous works. 链接: https://pan.baidu.com/s/1c3NMyEAEubAO35n9ADxjEg?pwd=m1wm 提取码: m1wm
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Hu, Lin; Wu, Qiang; Xu Qingcui; Zhang, Zujin; Li, Huacan. Numerical Analysis of Balanced Methods for the Impulsive Stochastic Differential Equations, Journal of Donghua University (English Edition), 32 (2015), no. 4, 626—635.(http://www.cnki.net/KCMS/detail/detail.aspx?QueryID=6&CurRec=7&recid=&filename=DHDY201504019&dbname=CJFDLAST2016&dbcode=CJFQ&pr=&urlid=&yx=&uid=WEEvREcwSlJHSldRa1FhdXNXZjNlaHFkRDRGTlhKQ0NQcVEvUlFCY2ZVVT0=$9A4hF_YAuvQ5obgVAqNKPCYcEjKensW4IQMovwHtwkF4VYPoHbKxJw!!&v=MDcyNjJYUGQ3RzRIOVRNcTQ5RWJZUjhlWDFMdXhZUzdEaDFUM3FUcldNMUZyQ1VSTHlmWWVkbUZ5L2xVTDNQSVM=)
Positive results are proved here about the ability of balanced methods to reproduce the mean square stability of the impulsive stochastic differential equations． It is shown that the balanced methods with strong convergence can preserve the mean square stability with the sufficiently small stepsize． Weak variants and their mean square stability are also considered． Several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean-square stabilities for relatively large stepsizes especially. 链接: https://pan.baidu.com/s/19xTWfnzk9htCigQw2YDtMg?pwd=cvgn 提取码: cvgn
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# 2014
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Zhang, Zujin; Li, Peng; Zhong, Dingxing. Navier-Stokes equations with regularity in two entries of the velocity gradient tensor. Appl. Math. Comput. 228 (2014), 546--551.(http://www.ams.org/mathscinet-getitem?mr=3151939)
This paper concerns itself the regularity criteria for the three-dimensional Navier–Stokes equations. In particular, it is proved that if
$$\begin{aligned} \partial_1u_3, \partial_3u_3\in L^\frac{16}{3}\left(0,T;L^2(\mathbb{R}^3)\right), \end{aligned}$$
or
$$\begin{aligned} \partial_1u_3, \partial_2u_3\in L^8\left(0,T; L^2(\mathbb{R}^3)\right), \end{aligned}$$
then the solution is in fact smooth on $(0,T)$. 链接: https://pan.baidu.com/s/1oVZvqhK39NOE3am-PozL1g?pwd=7dig 提取码: 7dig
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Zhang, Zujin; Zhong, Dingxing; Hu, Lin. A new regularity criterion for the 3D Navier-Stokes equations via two entries of the velocity gradient tensor. Acta Appl. Math. 129 (2014), 175--181.(http://www.ams.org/mathscinet-getitem?mr=3152082)
We consider the Cauchy problem for the incompressible Navier-Stokes equations in $\mathbb{R}^3$, and provide a new regularity criterion involving only two entries of the Jacobian matrix of the velocity field. 链接: https://pan.baidu.com/s/1LiJ4UAQ1NOvMSbYSoEI86Q?pwd=pvnu 提取码: pvnu
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Zhang, Zujin. A remark on the regularity criterion for the 3D Boussinesq equations involving the pressure gradient. Abstr. Appl. Anal. 2014, Art. ID 510924, 4 pp.(http://www.ams.org/mathscinet-getitem?mr=3166623)
We consider the three-dimensional Boussinesq equations and obtain a regularity criterion involving the pressure gradient in the Morrey-Companato space $M_{p,q}$. This extends and improves the result of Gala (Gala 2013) for the Navier-Stokes equations. 链接: https://pan.baidu.com/s/1LiJ4UAQ1NOvMSbYSoEI86Q?pwd=pvnu 提取码: pvnu
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Zhang, Zujin. A remark on the regularity criterion for the 3D Navier-Stokes equations involving the gradient of one velocity component. J. Math. Anal. Appl. 414 (2014), no. 1, 472--479.(http://www.ams.org/mathscinet-getitem?mr=3165323)
In thispaper, we consider the regularity criterion for weak solutions to the 3D Navier-Stokes equations. We show that if $\nablau_3$ belongs to some multiplier spaces, then the solution actually is smooth on $(0,T)$. In particular, we have the regularity criterion in the BMO spaces: $\nablau_3\in L^\frac{4}{3}(0,T; BMO)$, which improves previous results. 链接: https://pan.baidu.com/s/1q1zXQSj-2CGauHRAhoUZ_w?pwd=u43y 提取码: u43y
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Zhang, Zujin; Tang, Tong; Liu, Lihan. An Osgood type regularity criterion for the liquid crystal flows. NoDEA Nonlinear Differential Equations Appl. 21 (2014), no. 2, 253--262.(http://www.ams.org/mathscinet-getitem?mr=3180883 )
In this paper, we prove an Osgood type regularity criterion for the model of liquid crystals, which says that the condition
$$\begin{aligned} \sup_{2\leq q < \infty}\int_0^T \frac{\left\Vert \bar S_q\nablau(t)\right\Vert _{L^\infty}}{q\ln q}\mathrm{ d} t < \infty \end{aligned}$$
implies the smoothness of the solution. Here,
$$\begin{aligned} \bar S_q=\sum_{k=-q}^q \dot \Delta_k \end{aligned}$$
with $\dot \Delta_k$ being the frequency localization operator. 链接: https://pan.baidu.com/s/1N0kFBhqIxc3Q3hmdH7aBDQ?pwd=2u74 提取码: 2u74
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Zhang, Zujin; Wang, Xiaofeng; Yao, Zheng-an. On the weak solution to a fractional nonlinear Schr\"odinger equation. Abstr. Appl. Anal. 2014, Art. ID 569693, 6 pp.(http://www.ams.org/mathscinet-getitem?mr=3193522)
We obtain the existence of a global weak solution to a fractional nonlinear Schrodinger equation by the Galerkin method. Its uniqueness is also discussed. In our proof, we use harmonic analysis techniques and compactness arguments. 链接: https://pan.baidu.com/s/1P_RMul_S_9FFGdinRR48nw?pwd=ta45 提取码: ta45
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Zhang, Zujin. A logarithmically improved regularity criterion for the 3D Boussinesq equations via the pressure. Acta Appl. Math. 131 (2014), 213--219. (http://www.ams.org/mathscinet-getitem?mr=3207709)
In this paper, we consider the three-dimensional Boussinesq equations, and obtain a logarithmically improved regularity criterion in terms of pressure. 链接: https://pan.baidu.com/s/1nEGmmxnZkIIJvxGpWq8aaA?pwd=489a 提取码: 489a
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Zhang, Zujin. Global regularity for the 2D micropolar fluid flows with mixed partial dissipation and angular viscosity. Abstr. Appl. Anal. 2014, Art. ID 709746, 6 pp.(http://www.ams.org/mathscinet-getitem?mr=3212444)
This paper establishes the global existence and uniqueness of classical solutions to the 2D micropolar fluid flows with mixed partial dissipation and angular viscosity. 链接: https://pan.baidu.com/s/1q6aFo25M8FM4oQXRyS2FDg?pwd=37ch 提取码: 37ch
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Zhang, Zujin; Alzahrani, Faris; Hayat, Tasawar; Zhou, Yong. Two new regularity criteria for the Navier-Stokes equations via two entries of the velocity Hessian tensor. Appl. Math. Lett. 37 (2014), 124--130.(http://www.ams.org/mathscinet-getitem?mr=3231739)
We consider the Cauchy problem for the incompressible Navier–Stokes equations in $\mathbb{R}^3$, and provide two sufficient conditions to ensure the smoothness of solutions. Both of them only involve two entries of the velocity Hessian tensor. 链接: https://pan.baidu.com/s/1cFrj6WFxsr-7NfaapePzsw?pwd=7wjf 提取码: 7wjf
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Zhang, Zujin. An improved regularity criterion for the 3D Navier-Stokes equations in terms of two entries of the velocity gradient. (Chinese) Acta Math. Sci. Ser. A Chin. Ed. 34 (2014), no. 5, 1327--1335.(http://www.ams.org/mathscinet-getitem?mr=3310237)
该文考虑三维 Navier-Stokes 方程组的 Cauchy 问题, 得到了一个改进的、各向异性的、关于速度梯度的两个分量的正则性准则. 此准则改进了文献 中的结果. 链接: https://pan.baidu.com/s/11o-8LYC6wAGEV6hBtFHAxA?pwd=jfsq 提取码: jfsq
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Zhang, Zujin. Some regularity criteria for the 3D Boussinesq equations in the class $L^2(0,T;\dot B^{-1}_{\infty,\infty})$. ISRN Appl. Math. 2014, Art. ID 564758, 4 pp.(http://www.ams.org/mathscinet-getitem?mr=3176809)
We consider the three-dimensional Boussinesq equations and obtain some regularity criteria via the velocity gradient (or the vorticity, or the deformation tensor) and the temperature gradient. 链接: https://pan.baidu.com/s/1UBxPqmKWqCCNhn-ys5HsLA?pwd=bpn3 提取码: bpn3
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Zhang, Zujin; Tang, Tong; Zhang, Fumin. A remark on the regularity criterion for the MHD equations via two components in Morrey-Campanato spaces. Journal of Difference Equations, 2014, Art. ID364269, 5 pp.(http://www.hindawi.com/journals/jde/2014/364269/)
We consider the regularity criterion for the $3$D MHD equations. It is proved that if the horizontal components of the velocity and magnetic fields satisfy
$$\begin{aligned} \tilde u, \tilde b\in L^\frac{2}{1-r}\left(0,T;\dot M_{2,\frac{3}{r}}\right)\mbox{with}0 < r < 1, \end{aligned}$$
then the solution smooth. This improves the result given by Gala(2012). 链接: https://pan.baidu.com/s/19VJahtj1h7yGPX47wnRTlQ?pwd=suyv 提取码: suyv
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Zhang, Zujin; Qiu, Shulin; Pan, Jian; Ma, Li. A refined blow up criterion for the nematic liquid crystals. International Journal of Contemporary Mathematical Sciences. 9 (2014), 441--446.(http://www.m-hikari.com/ijcms/ijcms-2014/9-12-2014/4438.html)
In this paper, we refine the blow-up criterion of Huang-Wang to BMO spaces. 链接: https://pan.baidu.com/s/1ywBBDECE6cjIbWI5fGvy1Q?pwd=x84q 提取码: x84q
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Zhang, Zujin. A smallness regularity criterion for the 3D Navier-Stokes equations in the largest class. Journal of Mathematical and Computational Science. 4 (2014), 587--593.(http://scik.org/index.php/jmcs/article/view/1854)
In this paper, we consider the three-dimensional Navier-Stokes equations, and show that if the $\dot B^{-1}_{\infty,\infty}$ norm of the velocity field is sufficiently small, then the solution is in fact classical. 链接: https://pan.baidu.com/s/1TAKWxEsUXs0ArPm26bpSuw?pwd=4s62 提取码: 4s62
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Zhang, Zujin. Liquid crystal flows with regularity in one direction. J. Partial Differ. Equ. 27 (2014), no. 3, 245--250.(http://www.ams.org/mathscinet-getitem?mr=3243985)
In this paper, we consider the Cauchy problem for the model of liquid crystal. We show that if the velocity field $u$ satisfies
$$\begin{aligned} \partial_3u\in L^p\left(0,T;L^q(\mathbb{R}^3)\right), \frac{2}{p}+\frac{3}{q}=1+\frac{1}{q}, 2 < q\leq\infty, \end{aligned}$$
then the solution is in fact smooth. 链接: https://pan.baidu.com/s/1tUi4OL5KfcdhxEc18gsw_Q?pwd=vmfj 提取码: vmfj
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Zhang, Zujin. MHD equations with regularity in one direction. International Journal of Partial Differential Equations. 2014,Art. ID213083, 6 pp.(https://www.hindawi.com/journals/ijpde/2014/213083/)
We consider the 3D MHD equations and prove that if one directional derivative of the fluid velocity, say,
$$\begin{aligned} \partial_3u\in L^p\left(0,T;L^q(\mathbb{R}^3)\right), \mbox{with} \frac{2}{p}+\frac{3}{q}=\gamma\in \left[1,\frac{3}{2}\right), \frac{3}{\gamma}\leq q\leq\frac{1}{\gamma-1}, \end{aligned}$$
then the solution is in fact smooth. This improves previous results greatly. 链接: https://pan.baidu.com/s/1FulC237Lw73CUPpBVtTSnA?pwd=6ua4 提取码: 6ua4
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Wu, Qiang; Hu, Lin; Zhang, Zujin. Convergence and stability of balanced methods for stochastic delay integro-differential equations. Appl. Math. Comput. 237 (2014), 446--460.(http://www.ams.org/mathscinet-getitem?mr=3201145)
This paper deals with a family of balanced implicit methods for the stochastic delay integro-differential equations. It is shown that the balanced methods, which own the implicit iterative scheme in the diffusion term, give strong convergence rate of at least $\frac{1}{2}$. It proves that the mean-square stability for the stochastic delay integro-differential equations is inherited by the strong balanced methods and the weak balanced methods with sufficiently small stepsizes. Several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean-square stabilities. 链接: https://pan.baidu.com/s/1RMzimBMWtPn5V1wcKK8I5w?pwd=mqc1 提取码: mqc1
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Benbernou, Samia; Terbeche, Mekki; Ragusa, Maria Alessandra; Zhang, Zujin. A note on the regularity criterion for 3D MHD equations in $\dot B^{-1}_{\infty,\infty}$ space. Appl. Math. Comput. 238 (2014), 245--249.(http://www.ams.org/mathscinet-getitem?mr=3209631)
In this note, we consider sufficient conditions for the regularity of Leray-Hopf solutions of the $3$D incompressible magnetohydrodynamic equations via the velocity and magnetic fields in terms of $\dot B^{-1}_{\infty,\infty}$ spaces. We prove that if $(\nabla\times u, \nabla\times B)$ belongs to the space $L^2\left(0,T;\dot B^{-1}_{\infty,\infty}\right)$, then the solution$(u,B)$ is regular. This extends recent results contained in Gala (2011) . 链接: https://pan.baidu.com/s/1brlxYFt8lUdnwMazXJPgtw?pwd=2dva 提取码: 2dva
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Zhong, Dingxing; Zhang, Zujin; Tao, Lingyang. The hypersurfaces with parallel Lagueree form in $\mathbb{R}^n$. Acta Mathematica Sinica, Chinese Series, 57 (2014), 21--32.(http://123.57.41.99/Jwk_sxxb_cn//CN/abstract/abstract22359.shtml)
设 $x: M\to\mathbb{R}^n$ 是 $\mathbb{R}^n$ 中定向无脐超曲面, 主曲率非零, 那么在 $\cup\mathbb{R}^n$ 的 Laguerre 变换群下超曲面的四个 Laguerre 不变量是 Laguerre 不变度量 $g$, Laguerre 第二基本形式形式 $B$, Laguerre 形式 $C$和 Laguerre 张量 L. 本文研究 Laguerre 形式 $C$ 平行与 Laguerre 形式 $C$ 为零之间的关系. 链接: https://pan.baidu.com/s/1MDk0IFfFEmoquY9HqdGepQ?pwd=5cne 提取码: 5cne
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# 2013
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Zhang, Zujin. A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component. Commun. Pure Appl. Anal. 12 (2013), no. 1, 117--124.(http://www.ams.org/mathscinet-getitem?mr=2972425)
We study the Cauchy problem for the 3D Navier-Stokes equations, and prove some scalaring-invariant regularity criteria involving only one velocity component. 链接: https://pan.baidu.com/s/1_tux2VKkqDQhmJs6FP5ZQw?pwd=8c3i 提取码: 8c3i
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Zhang, Zujin; Yao, Zheng-an; Li, Peng; Guo, Congchong; Lu, Ming. Two new regularity criteria for the 3D Navier-Stokes equations via two entries of the velocity gradient tensor. Acta Appl. Math. 123 (2013), 43--52.(http://www.ams.org/mathscinet-getitem?mr=3010223)
We consider the Cauchy problem for the incompressible Navier-Stokes equations in $\mathbb{R}^3$, and provide two new regularity criteria involving only two entries of the Jacobian matrix of the velocity field. 链接: https://pan.baidu.com/s/1N7gJXXPtEqBb_c4xNBYmRw?pwd=8d8u 提取码: 8d8u
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Zhang, Zujin; Li, Peng; Yu, Gaohang. Regularity criteria for the 3D MHD equations via one directional derivative of the pressure. J. Math. Anal. Appl. 401 (2013), no. 1, 66--71.(http://www.ams.org/mathscinet-getitem?mr=3011247)
In this paper, we consider the Cauchy problem for the 3D viscous MHD equations, and provide some regularity criteria involving only one directional derivative of the pressure, say $\partial_3p$. In particular, if
$$\begin{aligned} \partial_3p\in L^\alpha\left(0,T;L^\beta(\mathbb{R}^3)\right), \frac{2}{\alpha}+\frac{3}{\beta}=2, \frac{3}{2}\leq \beta\leq 3, \end{aligned}$$
then the solution remains smooth on $$. 链接: https://pan.baidu.com/s/1ZMpXIS_fSp00_eibHthyGw?pwd=i8u5 提取码: i8u5
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Zhang, Zujin; Gala, Sadek. Osgood type regularity criterion for the 3D Newton-Boussinesq equation. Electron. J. Differential Equations 2013, No. 223, 6 pp.(http://www.ams.org/mathscinet-getitem?mr=3119077)
In this article, we show an Osgood type regularity criterion for the three-dimensional Newton-Boussinesq equations, which improves the recent results in . 链接: https://pan.baidu.com/s/1Ypfrh9ed8n8aM7NJ3tQUVQ?pwd=qifx 提取码: qifx
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Zhang, Zujin; Wang, Xiaofeng; Yao, Zheng-an. On a fractional nonlinear hyperbolic equation arising from relative theory. Abstr. Appl. Anal. 2013, Art. ID 548562, 6 pp.(http://www.ams.org/mathscinet-getitem?mr=3132564)
We obtain the existence of a weak solution to a fractional nonlinear hyperbolic equation arising from relative theory by the Galerkin method. Its uniqueness is also discussed. Furthermore, we show the regularity of the obtained solution. In our proof, we use harmonic analysis techniques and compactness arguments. 链接: https://pan.baidu.com/s/1YP0-CWb_KFkg5Kuuzxj1iA?pwd=vv81 提取码: vv81
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Zhang, Zujin; Ouyang, Xiqin; Zhong, Dingxing; Qiu, Shulin. Remarks on the regularity criteria for the 3D MHD equations in the multiplier spaces. Bound. Value Probl. 2013, 2013:270, 7 pp.(http://www.ams.org/mathscinet-getitem?mr=3341364)
In this paper, we consider the regularity criteria for the 3D MHD equations. It is proved that if
$$\begin{aligned} \partial_3(u+b)\in L^\frac{2}{1-r}(0,T; \dot X_r), 0\leq r\leq 1, \end{aligned}$$
or
$$\begin{aligned} \partial_3(u-b)\in L^\frac{2}{1-r}(0,T; \dot X_r), 0\leq r\leq 1, \end{aligned}$$
then the solution actually is smooth. This extends the previous results given by Guo and Gala (Anal. Appl. 10:373-380, 2013), Gala (Math. Methods Appl. Sci. 33:1496-1503, 2010). 链接: https://pan.baidu.com/s/1MjaIVqqM2MDM53jEca7DUw?pwd=si7v 提取码: si7v
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Zhong, Ding Xing; Sun, Hong An; Zhang, Zujin. Hypersurfaces in $S^{n+1}$ with three distinct constant para-Blaschke eigenvalues. (Chinese) Acta Math. Sinica (Chin. Ser.) 56 (2013), no. 5, 751--766.(http://www.ams.org/mathscinet-getitem?mr=3154481)
设 $x: M\to S^{n+1}$ 是 $(n+1)-$ 维单位球面上不含脐点的超曲面. 在 $S^{n+1}$ 的 Mobius 变换群下浸入 $x$ 的四个基本不变量是: Mobius 度量 $g$; Mobius 第二基本形式 $B$; Mobius 形式 $\varPhi$ 和 Blaschke 张量 $A$. 对称的 $(0,2)$ 张量 $D=A+\lambda B$ 也是 Mobius 不变量, 其中 $\lambda$ 是常数. $D$ 称为浸入 $x$ 的仿 Blaschke 张量, 仿 Blaschke 张量的特征值称为浸入 $x$ 的仿 Blaschke 特征值. 如果 $\varPhi=0$, 对某常数 $\lambda$, 仿 Blaschke 特征值为常数, 那么超曲面 $x: M\to S^{n+1}$ 称为仿 Blaschke 等参超曲面. 本文对具有三个互异仿 Blaschke 特征值 (其中有一个重数为 $1$) 的仿 Blaschke 等参超曲面进行了分类. 链接: https://pan.baidu.com/s/1SFzy2griaN2_bjYtsrWksw?pwd=3394 提取码: 3394
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Li, Peng; Li, Shuai-Jie; Yao, Zheng-An; Zhang, Zujin. Two anisotropic fourth-order partial differential equations for image inpainting. IET Image Process. 7 (2013), no. 3, 260--269.(http://www.ams.org/mathscinet-getitem?mr=3098363)
In this study, the authors propose two fourth-order partial differential equations (PDEs) to inpaint the image. By analysing those anisotropic fourth-order PDEs and comparing their diffusion images, the authors confirm they are forward diffusion or backward diffusion. A numerical algorithm is presented using a finite-difference method and analyse the stability of discretisation. Finally, they show various experimental results and conclude that the proposed new models are better than the second-order and third-order PDEs, especially for weakening the blocky effects. 链接: https://pan.baidu.com/s/1_GBLOw6lEHuiFVVWItzIRg?pwd=jetj 提取码: jetj
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# 2012
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Zhang, Zujin; Wang, Xiaofeng; Yao, Zheng-an. Remarks on regularity criteria for the weak solutions of liquid crystals. J. Evol. Equ. 12 (2012), no. 4, 801--812.(http://www.ams.org/mathscinet-getitem?mr=3000456)
In this paper, we consider the regularity criteria for weak solutions of liquid crystals. It is proved that the solution is in fact smooth if the velocity or the velocity gradient belongs to some critical multiplier spaces or Tribel–Lizorkin spaces. As a corollary, we obtain the Beal–Kato–Majda criteria for liquid crystals. 链接: https://pan.baidu.com/s/1JZ-ha1GCw-Uo1G7iFq6m_w?pwd=ii1z 提取码: ii1z
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Lu, Ming; Du, Yi; Yao, Zheng-an; Zhang, Zujin. A blow-up criterion for the 3D compressible MHD equations. Commun. Pure Appl. Anal. 11 (2012), no. 3, 1167--1183.(http://www.ams.org/mathscinet-getitem?mr=2968615)
In this paper, we study the 3D compressible magneto-hydrodynamic equations. We extend the well-known Serrin's blow-up criterion(see ) for the $3$D incompressible Navier-Stokes equations to the $3$D compressible magnetohydrodynamic equations. In addition, initial vacuum is allowed in our case. 链接: https://pan.baidu.com/s/1OOuzzVLlvLjB2osFztSIJg?pwd=tqdp 提取码: tqdp
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Guo, Congchong; Zhang, Zujin; Wang, Jialin. Regularity criteria for the 3D magneto-micropolar fluid equations in Besov spaces with negative indices. Appl. Math. Comput. 218 (2012), no. 21, 10755--10758.(http://www.ams.org/mathscinet-getitem?mr=2927090)
We consider the Cauchy problem of the magneto-micropolar fluid equations in three space dimensions. It is proved that if the velocity, magnetic field and the micro-rotational velocity belong to some critical Besov space with negative indices, then the solution is in fact smooth. 链接: https://pan.baidu.com/s/1KPpIils0dQ4xY3MRK3kCPQ?pwd=2hwt 提取码: 2hwt
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# 2011
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Zhang, Zujin. Remarks on the regularity criteria for generalized MHD equations. J. Math. Anal. Appl. 375 (2011), no. 2, 799--802.(http://www.ams.org/mathscinet-getitem?mr=2735616)
We study the Cauchy problem for the generalized MHD equations, and prove some regularity criteria involving the integrability of $\nablau$ in the Morrey, multiplier spaces. 链接: https://pan.baidu.com/s/1k9XUGyB9QblNRwKoRrfbaA?pwd=kznj 提取码: kznj
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Zhang, Zujin; Wu, Xinglong; Lu, Ming. On the uniqueness of strong solution to the incompressible Navier-Stokes equations with damping. J. Math. Anal. Appl. 377 (2011), no. 1, 414--419.(http://www.ams.org/mathscinet-getitem?mr=2754840)
In this paper, we show that the Cauchy problem of the incompressible Navier–Stokes equations with damping $\alpha \|u\|^{\beta-1}u$ ($\alpha > 0$) has global strong solution for any $\beta>3$ and the strong solution is unique when $3 < \beta\leq 5$. This improves earlier results. 链接: https://pan.baidu.com/s/1VjTqmqAdUh3cg0pZxkvMZg?pwd=3f32 提取码: 3f32
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Zhang, Zujin; Yao, Zheng-an; Wang, Xiaofeng. A regularity criterion for the 3D magneto-micropolar fluid equations in Triebel-Lizorkin spaces. Nonlinear Anal. 74 (2011), no. 6, 2220--2225.(http://www.ams.org/mathscinet-getitem?mr=2781751)
We consider the regularity criteria for the $3$D magneto-micropolar fluid equations in Triebel-Lizorkin spaces. It is proved that if
$$\begin{aligned} \nablau\in L^p\left(0,T; \dot F^0_{q,\frac{2q}{3}}\right) \end{aligned}$$
with
$$\begin{aligned} \frac{2}{p}+\frac{3}{q}=2, \quad\frac{3}{2} < q\leq \infty, \end{aligned}$$
then the solution remains smooth in $(0,T)$. As a corollary, we obtain the classical Beale-Kato-Majda criteria, that is, the condition
$$\begin{aligned} \nabla\times u\in L^1(0,T; \dot B^0_{\infty,\infty}) \end{aligned}$$
ensures the smoothness of the solution. 链接: https://pan.baidu.com/s/1TIcTqIBSCSEPIlnTVqQ_Xg?pwd=khk4 提取码: khk4
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Zhang, Zujin; Yao, Zheng-an; Lu, Ming; Ni, Lidiao. Some Serrin-type regularity criteria for weak solutions to the Navier-Stokes equations. J. Math. Phys. 52 (2011), no. 5, 053103, 7 pp.(http://www.ams.org/mathscinet-getitem?mr=2839081)
We consider the regularity criteria for a weak solution $u=(u_1,u_2,u_3)$ to the Navier-Stokes equations in $\mathbb{R}^3$. Denoting by $\omega=(\omega_1,\omega_2,\omega_3)=\mathrm{ curl} u$ the vorticity, we then prove that $u$ is smooth, provided $u_3,\omega_3$; or $\partial_3u_3,\omega_3$ are in some Serrin-type integrability classes. 链接: https://pan.baidu.com/s/1qM3IuBsbswHy75UwPK5_9A?pwd=d88c 提取码: d88c
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# 2010
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Zhang, Zujin. Regularity criterion for the system modeling the flow of liquid crystals via the direction of velocity. Comm. Appl. Nonlinear Anal. 17 (2010), no. 3, 55--60.(http://www.ams.org/mathscinet-getitem?mr=2721921)
We consider sufficient conditions for the regularity of weak solutions to the system modeling the flow of liquid crystals. It is proved that we can control the direction of velocity and the direction of liquid crystals to ensure smoothness. Our result extend the case for incompressible Navier-Stokes equations. 链接: https://pan.baidu.com/s/1La7qUSN6u9gt5JGwpsj4Xg?pwd=thy1 提取码: thy1
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