个人简历

[1]   Dingxing Zhong, Zujin Zhang, Lingyang Tao, The hypersurfaces with parallel Lagueree form in $\bbR^n$, Acta Mathematica Sinica, Chinese Series, 57 (2014), 21—32.

Abstract: Let $x:M\to \bbR^n$ be an umbilical free oriented hypersurface with non-zero principal curvatures, four basic Laguerre invariants under the Laguerre transformation group are the Laguerre metric $g$, the Laguerre second fundamental form ${\bf B}$, the Laguerre form ${\bf C}$ and the Laguerre tensor ${\bf L}$. In this paper, we study the relationships between parallel Laguerre form and vanishing metric Laguerre form. [Link]

 

[2]   Samia Benbernou, Mekki Terbeche, Maria Alessandra Ragusa, Zujin Zhang, A note on the regularity criterion for the $3$D MHD equations in  space, Applied Mathematics and Computation, 238 (2014),  245—249. [SCI 收录]

Abstract: In this note, we consider sufficient conditions for the regularity of Leray Hoph Hopf solutions of the 3D incompressible magnetohydrodynamic equations via the velocity and magnetic fields in terms of $\dot B^{-1}_{\infty,\infty}$ spaces. We prove that if $(\n\times{\bf u},\n\times {\bf B})$ belongs to the space $L^2(0,T;\dot B^{-1}_{\infty,\infty}(\bbR^3))$, then the solution $({\bf u},{\bf B})$ is regular. This extends recent results contained in Gala (2011) [3]. [Link]

 

[3]   Qiang Wu, Lin Hu, Zujin Zhang, Convergence and stability of balanced methods for stochastic delay integro-differential equations, Applied Mathematics and Computation, 237 (2014), 446—460. [SCI 收录]

Abstract: 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 $1/2$. 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. [Link]

 

[4]   Zujin Zhang, MHD equations with regularity in one direction, International Journal of Partial Differential Equations, 2014 (2014), 213083.

Abstract: We consider the 3D MHD equations and prove that if one directional derivative of the fluid velocity, say, $\p_3{\bf u}\in L^p(0,T;L^q(\bbR^3))$, with $2/p+3/q=\gamma\in [1,3/2]$, $3/\gamma\leq q\leq 1/(\gamma-1)$, then the solution is in fact smooth.  This improves previous results greatly. [Link]

 

[5]   Zujin Zhang, Liquid crystal flows with regularity in one direction, Journal of Partial Differential Equations, 27 (2014), 245—250.

Abstract: In this paper, we consider the Cauchy problem for the model of liquid crystal. We show that if the velocity field ${\bf u}$ satisfies $$\p_3{\bf u}\in L^p(0,T;L^q(\bbR^3)),\quad \frac{2}{p}+\frac{3}{q}=1+\frac{1}{q},\quad 2<q\leq\infty,$$ then the solution is in fact smooth. [Link]

 

[6]   Zujin Zhang, A smallness regularity criterion for the $3$D Navier-Stokes equations in the largest class, Journal of Mathematical and Computational Science, 4  (2014), 587—593.

Abstract: 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. [Link]

 

[7]   Zujin Zhang, Shulin Qiu, Jian Pan, Li Ma, A refined blow up criterion for the nematic liquid crystals, International Journal of Contemporary Mathematical Sciences,  9 (2014), 441—446.

Abstract: In this paper, we refine the blow-up criterion of Huang-Wang [T. Huang, C.Y. Wang, Blow up criterion for the nematic liquid crystal flows, Comm. Partial Differential Equations, 37 (2012), 875-884] to BMO spaces. [Link]

 

[8]   Zujin Zhang, Tong Tang, Fumin Zhang, A remark on the regularity criterion for the MHD equations via two components in Morrey-Campanato spaces, Journal of Difference Equations, 2014 (2014), 364269.

Abstract: 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 $$\tilde {\bf u}, \tilde {\bf b} \in L^\frac{2}{1-r}(0,T;\dot M_{2,\frac{3}{r}})$$ with $0<r<1$, then the solution smooth. This improves the result given by Gala (2012). [Link]

 

[9]   Zujin Zhang, Some regularity criteria for the $3$D Boussinesq equations in the class $L^2(0,T;\dot B^{-1}_{\infty,\infty})$, ISRN Applied Mathematics, 2014 (2014), 564758.

Abstract: 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. [Link]

 

[10]     Zujin Zhang, An improved regularity criterion for the $3$D Navier-Stokes equations in terms of two entries of the velocity gradient, Acta Mathematica Scientia, Chinese Series, 34 (2014), 1327—1335.

Abstract: We consider the Cauchy problem for the three-dimensional Navier-Stokes equations. An improved anisotropic regularity criterion via any two entries of the velocity gradient is given. This improves the result of [24]. [Link]

 

[11]     Zujin Zhang, Faris Alzahrani, Tasawar Hayat, Yong Zhou, Two new regularity criteria for the Navier-Stokes equations via two entries of the velocity Hessian tensor, Applied Mathematics Letters, 37 (2014), 124—130. [SCI 收录]

Abstract: We consider the Cauchy problem for the incompressible Navier-Stokes equations in $\bbR^3$, and provide two sufficient conditions to ensure the smoothness of solutions. Both of them only involve tow entries of the velocity Hessian tensor. [Link]

 

[12]     Zujin Zhang, Global regularity for the 2D micropolar fluid flows with mixed partial dissipation and angular viscosity, Abstract and Applied Analysis, 2014 (2014), 709746. [SCI 收录]

Abstract: This paper establishes the global existence and uniqueness of classical solutions to the $2$D micropolar fluid flows with mixed partial dissipation and angular viscosity. [Link]

 

[13]     Zujin Zhang, A logarithmically improved regularity criterion for the $3$D Boussinesq equations via the pressure, Acta Applicandae Mathematicae, 131  (2014), 213—219. [SCI 收录]

Abstract: In this paper, we consider the three-dimensional Boussinesq equation, and obtain a logarithmically improved regularity criterion in terms of pressure. [Link]

 

[14]     Zujin Zhang, Xiaofeng Wang, Zheng-an Yao, On the weak solution to a fractional nonlinear Schrödinger equation, Abstract and Applied Analysis, 2014 (2014), 569693. [SCI 收录]

Abstract: 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 aguments. [Link]

 

[15]     Zujin Zhang, Tong Tang, Lihan Liu, An Osgood type regularity criterion for the liquid crystal flows, Nonlinear Differential Equations and Applications, 21 (2014), 253—262. [SCI 收录]

Abstract: In this paper, we prove an Osgood type regularity criterion for the model of liquid crystals, which says that the condition $$\sup_{2\leq q<\infty}\int_0^T \frac{\sen{\bar S_q\n {\bf u}}_{L^\infty}}{q\ln q}\rd t<\infty$$ implies the smoothness of the solution. Here, $$\bar S_q=\sum_{k=-q}^q \dot \lap_k$$ with $\dot \lap_k$ being the frequency localization operator. [Link]

 

[16]     Zujin Zhang, A remark on the regularity criterion for the $3$D Navier-Stokes equations involving the gradient of one velocity component, Journal of Mathematical Analysis and Applications, 414 (2014), 472—479. [SCI 收录]

Abstract: In this paper, we consider the regularity criterion for weak solutions to the $3$D Navier-Stokes equations. We show that $\n u_3$ 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: $\n u_3\in L^\frac{4}{3}(0,T;BMO)$, which improves previous results. [Link]

 

[17]     Zujin Zhang, A remark on the regularity criterion for the $3$D Boussinesq equations via the pressure gradient, Abstract and Applied Analysis, 2014 (2014), 510924. [SCI 收录]

Abstract: We consider the three-dimensional Boussinesq equations and obtain a regularity criterion involving the pressure gradient in the Morrey-Companato space . This extends and improves the result of Gala (Gala 2013) for the Navier-Stokes equations. [Link]

 

[18]     Zujin Zhang, Dingxing Zhong, Lin Hu, A new regularity criterion for the $3$D Navier-Stokes equations via two entries of the velocity gradient tensor, Acta Applicandae Mathematicae, 129 (2014), 175—181. [SCI 收录]

Abstract: We consider the Cauchy problem for the incompressible Navier-Stokes equations in $\bbR^3$, and provide a new regularity criterion involving only two entries of the Jacobian matrix of the velocity field. [Link]

 

[19]     Zujin Zhang, Peng Li, Dingxing Zhong, Navier-Stokes equations with regularity in two entries of the velocity gradient tensor, Applied Mathematics and Computation, 228 (2014), 546—551. [SCI 收录]

Abstract: This paper concerns itself the regularity criteria for the three-dimensional Navier–Stokes equations. In particular, it is proved that if $$\p_1u_3,\p_3u_3\in L^\frac{16}{3}(0,T;L^2(\bbR^3)),$$ or $$\p_1u_3,\p_2u_3\in L^8(0,T;L^2(\bbR^3)),$$ then the solution is in fact smooth on $(0,T)$. [Link]

 

[20]     Peng Li, Shuaijie Li, Zheng-an Yao, Zujin Zhang, Two anisotropic fourth-order partial differential equations for image inpainting, IET Image Processing, 7 (2013), 260—269. [SCI 收录]

Abstract: 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. [Link]

 

[21]     Dingxing Zhong, Hong-an Sun, Zujin Zhang, The hypersurfaces in $S^{n+1}$ with three distinct constant Para-Blaschke Eigenvalues, Acta Mathematica Sinica, Chinese Series, 56 (2013), 735—750.

Abstract: Let  $x:M^n\to S^{n+1}$ be a hypersurface in the $(n+1)$-dimensional unit sphere $S^{n+1}$ without umbilics. Four basic invariants of $x$ under the Möbius transformation group in $S^{n+1}$ are Möbius metric $g$, Möbius second fundamental form $\bf{B}$; Möbius form $\bf{\varPhi}$; Blaschke tensor $\bf{A}$. Let $\bf{D}=\bf{A}+\lm \bf{B}$, where $\lm$ is a constant, then $\bf{D}$ is a symmetric $(0;2)$ tensor and a Möbius invariant. $\bf{D}$ is called para-Blaschke tensor of $x$, the eigenvalues of $\bf{D}$ is called para-Blaschke eingenvalues of $x$. If $\bf{\varPhi}=\bf{0}$; and the para-Blaschke eingenvalues are constant. Then the hypersurface  $x:M^n\to S^{n+1}$ is called para-Blaschke isoparametric hypersurface. In this paper, we classify the para-Blaschke isoparametric hypersurfaces with three distinct para-Blaschke eingenvalues such that one of them is simple. [Link]

 

[22]     Zujin Zhang, Xiqin Ouyang, Dingxing Zhong, Shulin Qiu, Remarks on the regularity criteria for the $3$D MHD equations in the multiplier spaces, Boundary Value Problems, 2013 (2013), 270.

Abstract: In this paper, we consider the regularity criteria for the $3$D MHD equations. It is proved that if $$\p_3({\bf u}+{\bf b})\in L^\frac{2}{1-r}(0,T;\dot X_r)\ (0\leq r\leq 1),$$ or $$\p_3({\bf u}-{\bf b})\in L^\frac{2}{1-r}(0,T;\dot X_r)\ (0\leq r\leq 1),$$ 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). [Link]

 

[23]     Zujin Zhang, Xiaofeng Wang, Zheng-an Yao, On a fractional nonlinear hyperbolic equation arising from relative theory, Abstract and Applied Analysis, 2013 (2013), 548562. [SCI 收录]

Abstract: We obtain the existence of a weak solution to a fractional nonlinear hyperbolic equations 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. [Link]

 

[24]     Zujin Zhang, Sadek Gala, Osgood type regularity criterion for the $3$D Newton-Boussinesq equations, Electronic Journal of Differential Equations, 223 (2013), 1—6. [SCI 收录]

Abstract: In this article, we show an Osgood type regularity criterion for the three-dimensional Nweton-Boussinesq equations, which improves the recent results in [4]. [Link]

 

[25]     Zujin Zhang, Peng Li, Gaohang Yu, Regularity criteria for the $3$D MHD equations via one directional derivative of the pressure, Journal of Mathematical Analysis and Applications, 401(2013), 66—71. [SCI 收录]

Abstract: In this paper, we consider the Cauchy problem for the $3$D viscous MHD equations, and provide some regularity criteria involving only one directional derivative of the pressure, say $\p_3p$. In particular, if $$\p_3p\in L^\al(0,T;L^\beta(\bbR^3)),\quad \mbox{with}\quad \frac{2}{\al}+\frac{3}{\beta}=2,\quad \frac{3}{2}\leq \beta\leq 3,$$ then the solution remains smooth on $[0,T]$. [Link]

 

[26]     Zujin Zhang, Zheng-an Yao, Peng Li, Congchong Guo, Ming Lu, Two new regularity criteria for the $3$D Navier-Stokes equations via two entries of the velocity gradient tensor, Acta Applicandae Mathematicae, 123 (2013), 43--52. [SCI 收录]

Abstract: We consider the Cauchy problem for the incompressible Navier-Stokes equations in $\bbR^3$, and provide two new regularity criteria involving only two entries of the Jacobian matrix of the velocity field. [Link]

 

[27]     Zujin Zhang, A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component, Communications on Pure and Applied Analysis, 12 (2013), 117—124. [SCI 收录]

Abstract: We study the Cauchy problem for the $3$D Navier-Stokes equations, and prove some scaling-invariant regularity criteria for involving only one velocity component. [Link]

 

[28]     Congchong Guo, Zujin Zhang, Jialin Wang, Regularity criteria for the $3$D magneto-micropolar fluid equations in Besov spaces with negative indices, Applied Mathematics and Computation, 218 (2012), 10755—10758. [SCI 收录]

Abstract: We consider the Cauchy problem of the magneto-micropolar fluid equations in three space dimension. It is proved that if the velocity, magnetic field and the micro-rotational velocity belong to some critical Besov spaces with negative indices, then the solution is in fact smooth. [Link]

 

[29]     Ming Lu, Yi Du, Zheng-an Yao, Zujin Zhang, A blow up criterion for the $3$D compressible MHD equations, Communications on Pure and Applied Analysis, 11 (2012), 1167—1183. [SCI 收录]

Abstract: In this paper, we study the $3$D compressible magneto-hydrodynamic equations. We extend the well-known Serrin’s blow-up criterion (see [32]) for the $3$D incompressible Navier-Stokes equations to the $3$D compressible magnetohydrodynamic equations. In addition, initial vacuum is allowed in our case. [Link]

 

[30]     Zujin Zhang, Xiaofeng Wang, Zheng-an Yao, Remarks on regularity criteria for the weak solutions of liquid crystals, Journal of Evolution Equations, 12 (2012), 801—812. [SCI 收录]

Abstract: 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 Triebel-Lizorkin spaces. As a corollary, we obtain the Beale-Kato-Majda criteria for liquid crystals. [Link]

 

[31]     Zujin Zhang, Zheng-an Yao, Ming Lu, Lidiao Ni, Some Serrin-type regularity criteria for weak solutions to the Navier-Stokes equations, Journal of Mathematical Physics, 52 (2011) , 053103. [SCI 收录]

Abstract: We consider the regularity criteria for a weak solution $\bbu=(u_1,u_2,u_3)$ to the Navier-Stokes equations in $\bbR^3$. Denoting by ${\bf \omega}=(\omega_1,\omega_2,\omega_3)$ the vorticity, we then prove that $\bbu$ is smooth, provided $u_3,\omega_3$; or $\p_3u_3, \omega_3$ are in some Serrin-type integrability classes. [Link]

 

[32]     Zujin Zhang, Zheng-an Yao, Xiaofeng Wang, A regularity criterion for the $3$D magneto-micropolar fluid equations in Triebel-Lizorkin spaces, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 2220—2225. [SCI 收录]

Abstract: We consider the regularity criterion for the $3$D magneto-micropolar fluid equations in Triebel-Lizorkin spaces. It is proved that if $\n\bbu\in L^p(0,T;\dot F^0_{q,\frac{2q}{3}})$ with $$\frac{2}{p}+\frac{3}{q}=2,\quad \frac{3}{2}<q\leq\infty,$$ then the solution remains smooth on $(0,T)$. As a corollary, we obtain the classical Beale-Kao-Majda criterion, that is, the condition $$\n\times \bbu\in L^1(0,T;\dot B^0_{\infty,\infty})$$ ensures the smoothness of the solution. [Link]

 

[33]     Zujin Zhang, Xinglong Wu, Ming Lu, On the uniqueness of strong solutions to the incompressible Navier-Stokes equations with damping, Journal of Mathematical Analysis and Applications, 377 (2011), 414—419. [SCI 收录]

Abstract: In this paper, we show that the Cauchy problem of the incompressible Navier-Stokes equations with damping $\al |\bbu|^{\beta-1}\bbu\ (\al>0)$ has global strong solution for any $\beta>3$ and the strong solution is unique when $3<\beta\leq 5$. This improves earlier results. [Link]

 

[34]     Zujin Zhang, Remarks on the regularity criteria for generalized MHD equations, Journal of Mathematical Analysis and Applications, 375 (2011), 799—802. [SCI 收录]

Abstract: We study the Cauchy problem for the generalized MHD equations, and prove some regularity criteria involving the integrability of $\n \bbu$ in the Morrey, multiplier spaces. [Link]

 

[35]     Zujin Zhang, Regularity criterion for the system modeling the flow of liquid crystals via the direction of velocity, Communications on Applied Nonlinear Analysis, 17 (2010), 55—60.

Abstract: We consider the sufficient conditions on 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 extends the case for the incompressible Navier-Stokes equations. [Link]