DCDS
Stability of stationary solutions to the compressible bipolar Euler-Poisson equations
Hong Cai Zhong Tan
Discrete & Continuous Dynamical Systems - A 2017, 37(9): 4677-4696 doi: 10.3934/dcds.2017201

In this paper, we study the compressible bipolar Euler-Poisson equations with a non-flat doping profile in three-dimensional space. The existence and uniqueness of the non-constant stationary solutions are established under the smallness assumption on the gradient of the doping profile. Then we show the global existence of smooth solutions to the Cauchy problem near the stationary state provided the $H^3$ norms of the initial density and velocity are small, but the higher derivatives can be arbitrarily large.

keywords: Bipolar Euler-Poisson equations stability global solution energy method
KRM
Asymptotic behavior of the compressible non-isentropic Navier-Stokes-Maxwell system in $\mathbb{R}^3$
Zhong Tan Leilei Tong
Kinetic & Related Models 2018, 11(1): 191-213 doi: 10.3934/krm.2018010

The compressible non-isentropic Navier-Stokes-Maxwell system is investigated in $\mathbb{R}^3$ and the global existence and large time behavior of solutions are established by pure energy method provided the initial perturbation around a constant state is small enough. We first construct the global unique solution under the assumption that the $H^3$ norm of the initial data is small, but the higher order derivatives can be arbitrarily large. If further the initial data belongs to $\dot{H}^{-s}$ ($0≤ s<3/2$) or $\dot{B}_{2, ∞}^{-s}$ ($0< s≤3/2$), by a regularity interpolation trick, we obtain the various decay rates of the solution and its higher order derivatives. As an immediate byproduct, the $L^p$-$L^2$ $(1≤ p≤ 2)$ type of the decay rates follows without requiring that the $L^p$ norm of initial data is small.

keywords: Compressible Navier-Stokes-Maxwell system global solution time decay rate energy method interpolation
KRM
Energy dissipation for weak solutions of incompressible liquid crystal flows
Shanshan Guo Zhong Tan
Kinetic & Related Models 2015, 8(4): 691-706 doi: 10.3934/krm.2015.8.691
In this paper, we are concerned with the simplified Ericksen-Leslie system (1)--(3), modeling the flow of nematic liquid crystals for any initial and boundary (or Cauchy) data $(u_0, d_0)\in {\bf H}\times H^1(\Omega, \mathbb{S}^2)$, with $d_0(\Omega)\subset\mathbb{S}^2_+$. We define a dissipation term $D(u,d)$ that stems from an eventual lack of smoothness in the solutions, and then obtain a local equation of energy for weak solutions of liquid crystals in dimensions three. As a consequence, we consider the 2D case and obtain $D(u,d)=0$.
keywords: incompressible energy dissipation energy equation. liquid crystals Weak solution
DCDS
Asymptotic stability of stationary solutions for magnetohydrodynamic equations
Zhong Tan Leilei Tong
Discrete & Continuous Dynamical Systems - A 2017, 37(6): 3435-3465 doi: 10.3934/dcds.2017146

In this paper, we are concerned with the compressible magnetohydrodynamic equations with Coulomb force in three-dimensional space. We show the asymptotic stability of solutions to the Cauchy problem near the non-constant equilibrium state provided that the initial perturbation is sufficiently small. Moreover, the convergence rates are obtained by combining the linear Lp-Lq decay estimates and the higher-order energy estimates.

keywords: MHD equations global solutions stability energy method time decay rate
DCDS
Optimal partial regularity results for nonlinear elliptic systems in Carnot groups
Shuhong Chen Zhong Tan
Discrete & Continuous Dynamical Systems - A 2013, 33(8): 3391-3405 doi: 10.3934/dcds.2013.33.3391
In this paper, we consider partial regularity for weak solutions of second-order nonlinear elliptic systems in Carnot groups. By the method of A-harmonic approximation, we establish optimal interior partial regularity of weak solutions to systems under controllable growth conditions with sub-quadratic growth in Carnot groups.
keywords: controllable growth condition sub-quadratic growth. Nonlinear elliptic systems A-harmonic approximation technique Carnot group
DCDS
Optimal interior partial regularity for nonlinear elliptic systems
Shuhong Chen Zhong Tan
Discrete & Continuous Dynamical Systems - A 2010, 27(3): 981-993 doi: 10.3934/dcds.2010.27.981
We consider interior regularity for weak solutions of nonlinear elliptic systems with subquadratic under controllable growth condition. By $\mathcal{A}$-harmonic approximation technique, we obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particularly, the regular result is optimal.
keywords: Controllable growth condition Nonlinear elliptic systems Optimal partial regularity. $\mathcal{A}$-harmonic approximation technique
CPAA
Higher integrability of weak solution of a nonlinear problem arising in the electrorheological fluids
Zhong Tan Jianfeng Zhou
Communications on Pure & Applied Analysis 2016, 15(4): 1335-1350 doi: 10.3934/cpaa.2016.15.1335
In this paper, we study the Dirichlet problem arising in the electrorheological fluids \begin{eqnarray} \begin{cases} -{\rm div}\ a(x,Du)=k(u^{\gamma-1}-u^{\beta-1}) & x\in \Omega, \\ u=0 & x\in \partial \Omega, \end{cases} \end{eqnarray} where $\Omega$ is a bounded domain in $R^n$ and ${\rm div}\ a(x,Du)$ is a $p(x)$-Laplace type operator with $1<\beta<\gamma<\inf_{x\in \Omega} p(x)$, $p(x)\in(1,2]$. By establish a reversed Hölder inequality, we show that for any suitable $\gamma,\beta$, the weak solution of previous equation has bounded $p(x)$ energy satisfies $|Du|^{p(x)}\in L_{\text{loc}}^{\delta}$ with some $\delta>1$.
keywords: reversed Hölder inequality Higher integrability weak solution electrorheological fluids $p(x)$-Laplace.
KRM
Time decay of solutions to the compressible Euler equations with damping
Qing Chen Zhong Tan
Kinetic & Related Models 2014, 7(4): 605-619 doi: 10.3934/krm.2014.7.605
We consider the time decay rates of the solution to the Cauchy problem for the compressible Euler equations with damping. We prove the optimal decay rates of the solution as well as its higher-order spatial derivatives. The damping effect on the time decay estimates of the solution is studied in details.
keywords: optimal decay rates Euler equations with damping energy method damping effect. Compressible flow
CPAA
The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources
Zhong Tan Zheng-An Yao
Communications on Pure & Applied Analysis 2004, 3(3): 475-490 doi: 10.3934/cpaa.2004.3.475
In this paper we consider the existence, nonexistence and the asymptotic behavior of the global solutions of the quasilinear parabolic equation of the following form:

$u_t-\Delta_pu=|u|^{q-2}u, \quad (x,t)\in\Omega\times (0,T),$

$u(x,t)=0,\quad (x,t)\in\partial\Omega\times (0,T), $

$ u(x,0)=u_0(x), \quad u_0(x)\geq 0, u_0(x)$ ≠ $0, $

where $\Omega$ is a smooth bounded domain in $R^N(N\geq 3)$, $\Delta_pu=$ div$(|\nabla u|^{p-2}\nabla u )$, $\frac{2N}{N+2}$ < $p$ < $N$, $q=p^\star=\frac{pN}{N-p}$ is the critical Sobolev exponent. In particular, we employ the concentration-compactness principle to prove that the global solutions with the initial data in "stable set" converge strongly to zero in $W_0^{1,p}(\Omega)$.

keywords: critical Sobolev exponent p-Laplace evolution equation Finite time blowup. Asymptotic behavior
KRM
Global existence in critical spaces for the compressible magnetohydrodynamic equations
Qing Chen Zhong Tan
Kinetic & Related Models 2012, 5(4): 743-767 doi: 10.3934/krm.2012.5.743
In this paper, we are concerned with the global existence and uniqueness of the strong solutions to the compressible Magnetohydrodynamic equations in $\mathbb{R}^N(N\ge3)$. Under the condition that the initial data are close to an equilibrium state with constant density, temperature and magnetic field, we prove the global existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations.
keywords: global existence Besov space. Magnetohydrodynamics critical space compressible

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