CPAA
Compressible Navier-Stokes equations with vacuum state in one dimension
Daoyuan Fang Ting Zhang
Communications on Pure & Applied Analysis 2004, 3(4): 675-694 doi: 10.3934/cpaa.2004.3.675
In this paper, we consider the one-dimensional compressible Navier-Stokes equations for isentropic flow connecting to vacuum state with a continuous density when viscosity coefficient depends on the density. Precisely, the viscosity coefficient $\mu$ is proportional to $\rho^\theta$ and $0<\theta<1/2$, where $\rho$ is the density. The global existence of weak solutions is proved.
keywords: vacuum global existence of weak solutions Navier-Stokes equations
DCDS
Zero-electron-mass limit of Euler-Poisson equations
Jiang Xu Ting Zhang
Discrete & Continuous Dynamical Systems - A 2013, 33(10): 4743-4768 doi: 10.3934/dcds.2013.33.4743
We study the limit of vanishing ratio of the electron mass to the ion mass (zero-electron-mass limit) in the scaled Euler-Poisson equations. As the first step of this justification, we construct the uniform global classical solutions in critical Besov spaces with the aid of ``Shizuta-Kawashima" skew-symmetry. Then we establish frequency-localization estimates of Strichartz-type for the equation of acoustics according to the semigroup formulation. Finally, it is shown that the uniform classical solutions converge towards that of the incompressible Euler equations (for ill-preparedinitial data) in a refined way as the scaled electron-mass tends to zero. In comparison with the classical zero-mach-number limit in [7,23], we obtain different dispersive estimates due to the coupled electric field.
keywords: Zero-electron-mass limits Euler-Poisson equations Strichartz-type estimate. critical Besov spaces skew-symmetry
DCDS
Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations
Lihuai Du Ting Zhang
Discrete & Continuous Dynamical Systems - A 2018, 38(9): 4745-4765 doi: 10.3934/dcds.2018209

Considering the stochastic 3-D incompressible anisotropic Navier-Stokes equations, we prove the local existence of strong solution in $H^2(\mathbb{T}^3)$. Moreover, we express the probabilistic estimate of the random time interval for the existence of a local solution in terms of expected values of the initial data and the random noise, and establish the global existence of strong solution in probability if the initial data and the random noise are sufficiently small.

keywords: Stochastic Navier-Stokes equations anisotropic local and global strong solution
CPAA
A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients
Ping Chen Ting Zhang
Communications on Pure & Applied Analysis 2008, 7(4): 987-1016 doi: 10.3934/cpaa.2008.7.987
Local solutions of the multidimensional Navier-Stokes equations for isentropic compressible flow are constructed with spherically symmetric initial data between a solid core and a free boundary connected to a surrounding vacuum state. The viscosity coefficients $\lambda, \mu$ are proportional to $\rho^\theta$, $0<\theta<\gamma$, where $\rho$ is the density and $\gamma > 1$ is the physical constant of polytropic fluid. It is also proved that no vacuum develops between the solid core and the free boundary, and the free boundary expands with finite speed.
keywords: Compressible Navier-Stokes equations uniqueness. free boundary density-dependent viscosity vacuum existence
DCDS
Well-posedness for the 3D incompressible nematic liquid crystal system in the critical $L^p$ framework
Qiao Liu Ting Zhang Jihong Zhao
Discrete & Continuous Dynamical Systems - A 2016, 36(1): 371-402 doi: 10.3934/dcds.2016.36.371
In this paper, we consider the well-posedness of the Cauchy problem of the 3D incompressible nematic liquid crystal system with initial data in the critical Besov space $\dot{B}^{\frac{3}{p}-1}_{p,1}(\mathbb{R}^{3})\times \dot{B}^{\frac{3}{q}}_{q,1}(\mathbb{R}^{3})$ with $1< p<\infty$, $1\leq q<\infty$ and \begin{align*} -\min\{\frac{1}{3},\frac{1}{2p}\}\leq \frac{1}{q}-\frac{1}{p}\leq \frac{1}{3}. \end{align*} In particular, if we impose the restrictive condition $1< p<6$, we prove that there exist two positive constants $C_{0}$ and $c_{0}$ such that the nematic liquid crystal system has a unique global solution with initial data $(u_{0},d_{0}) = (u^{h}_{0}, u^{3}_{0}, d_{0})$ which satisfies \begin{align*} ((1+\frac{1}{\nu\mu})\|d_{0}-\overline{d}_{0}\|_{\dot{B}^{\frac{3}{q}}_{q,1}}+ \frac{1}{\nu}\|u_{0}^{h}\|_{\dot{B}^{\frac{3}{p}-1}_{p,1}}) \exp\left\{\frac{C_{0}}{\nu^{2}}(\|u_{0}^{3}\|_{\dot{B}^{\frac{3}{p}-1}_{p,1}}+\frac{1}{\mu})^{2}\right\}\leq c_{0}, \end{align*} where $\overline{d}_{0}$ is a constant vector with $|\overline{d}_{0}|=1$. Here $\nu$ and $\mu$ are two positive viscosity constants.
keywords: Nematic liquid crystal flows well-posedness. Navier-Stokes equations
DCDS
Regularity of 3D axisymmetric Navier-Stokes equations
Hui Chen Daoyuan Fang Ting Zhang
Discrete & Continuous Dynamical Systems - A 2017, 37(4): 1923-1939 doi: 10.3934/dcds.2017081

In this paper, we study the three-dimensional axisymmetric Navier-Stokes system with nonzero swirl. By establishing a new key inequality for the pair $(\frac{ω^{r}}{r},\frac{ω^{θ}}{r})$, we get several Prodi-Serrin type regularity criteria based on the angular velocity, $u^θ$. Moreover, we obtain the global well-posedness result if the initial angular velocity $u_{0}^{θ}$ is appropriate small in the critical space $L^{3}(\mathbb{R}^{3})$. Furthermore, we also get several Prodi-Serrin type regularity criteria based on one component of the solutions, say $ω^3$ or $u^3$.

keywords: Navier-Stokes equations regularity criteria global well-posedness axisymmetric swirl
CPAA
Free boundary problem for compressible flows with density--dependent viscosity coefficients
Ping Chen Daoyuan Fang Ting Zhang
Communications on Pure & Applied Analysis 2011, 10(2): 459-478 doi: 10.3934/cpaa.2011.10.459
In this paper, we consider the free boundary problem of the spherically symmetric compressible isentropic Navier--Stokes equations in $R^n (n \geq 1)$, with density--dependent viscosity coefficients. Precisely, the viscosity coefficients $\mu$ and $\lambda$ are assumed to be proportional to $\rho^\theta$, $0 < \theta < 1$, where $\rho$ is the density. We obtain the global existence, uniqueness and continuous dependence on initial data of a weak solution, with a Lebesgue initial velocity $u_0\in L^{4 m}$, $4m>n$ and $\theta<\frac{4m-2}{4m+n}$. We weaken the regularity requirement of the initial velocity, and improve some known results of the one-dimensional system.
keywords: density-dependent viscosity coefficients. Compressible Navier-Stokes equations
DCDS
Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space
Jihong Zhao Ting Zhang Qiao Liu
Discrete & Continuous Dynamical Systems - A 2015, 35(1): 555-582 doi: 10.3934/dcds.2015.35.555
In this paper, we are concerned with a model arising from electro-hydrodynamics, which is a coupled system of the Navier-Stokes equations and the Poisson-Nernst-Planck equations through charge transport and external forcing terms. The local well-posedness and global well-posedness with small initial data to the 3-D Cauchy problem of this system are established in the critical Besov space $\dot{B}^{-1+\frac{3}{p}}_{p,1}(\mathbb{R}^{3})\times(\dot{B}^{-2+\frac{3}{q}}_{q,1}(\mathbb{R}^{3}))^{2}$ with suitable choices of $p, q$. Especially, we prove that there exist two positive constants $c_{0}, C_{0}$ depending on the coefficients of system except $\mu$ such that if \begin{equation*} \big(\|u_{0}^{h}\|_{\dot{B}^{-1+\frac{3}{p}}_{p,1}}+(\mu+1)\|(v_{0},w_{0})\|_{\dot{B}^{-2+\frac{3}{q}}_{q,1}} \big) \exp\Big\{\frac{C_{0}}{\mu^{2}}(\|u_{0}^{3}\|_{\dot{B}^{-1+\frac{3}{p}}_{p,1}}^{2}+1)\Big\}\leq c_{0}\mu, \end{equation*} then the above local solution can be extended to the global one. This result implies the global well-posedness of this system with large initial vertical velocity component.
keywords: electro-hydrodynamics Navier-Stokes equations well-posedness Poisson-Nernst-Planck equations Besov space.
DCDS
Dispersive effects of the incompressible viscoelastic fluids
Daoyuan Fang Ting Zhang Ruizhao Zi
Discrete & Continuous Dynamical Systems - A 2018, 38(10): 5261-5295 doi: 10.3934/dcds.2018233

We consider the Cauchy problem of the $N$-dimensional incompressible viscoelastic fluids with $N≥2$. It is shown that, in the low frequency part, this system possesses some dispersive properties derived from the one parameter group $e^{± it\Lambda}$. Based on this dispersive effect, we construct global solutions with large initial velocity concentrating on the low frequency part. Such kind of solution has never been seen before in the literature even for the classical incompressible Navier-Stokes equations.

keywords: Viscoelastic fluids global well-posedness large data
JIMO
Duality formulations in semidefinite programming
Qinghong Zhang Gang Chen Ting Zhang
Journal of Industrial & Management Optimization 2010, 6(4): 881-893 doi: 10.3934/jimo.2010.6.881
In this paper, duals for standard semidefinite programming problems from both the primal and dual sides are studied. Explicit expressions of the minimal cones and their dual cones are obtained under closeness assumptions of certain sets. As a result, duality formulations resulting from regularizations for both primal and dual problems can be expressed explicitly in terms of equality and inequality constraints involving three vector and matrix variables under such assumptions. It is proved in this paper that these newly developed duals can be cast as the Extended Lagrange-Slater Dual (ELSD) and the Extended Lagrange-Slater Dual of the Dual (ELSDD) with one reduction step. Therefore, the duals formulated in this paper guarantee strong duality, i.e., a zero duality gap and dual attainment.
keywords: duality theory Extended Lagrange-Slater Dual. Semidefinite programming

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