Pullback attractors of FitzHugh-Nagumo system on the time-varying domains
Zhen Zhang Jianhua Huang Xueke Pu
Discrete & Continuous Dynamical Systems - B 2017, 22(10): 3691-3706 doi: 10.3934/dcdsb.2017150

The existence and uniqueness of solutions satisfying energy equality is proved for non-autonomous FitzHugh-Nagumo system on a special time-varying domain which is a (possibly non-smooth) domain expanding with time. By constructing a suitable penalty function for the two cases respectively, we establish the existence of a pullback attractor for non-autonomous FitzHugh-Nagumo system on a special time-varying domain.

keywords: Pullback attractor FitzHugh-Nagumo equation time-varying domain penalty function
Global weak solutions to the 1-D fractional Landau-Lifshitz equation
Xueke Pu Boling Guo Jingjun Zhang
Discrete & Continuous Dynamical Systems - B 2010, 14(1): 199-207 doi: 10.3934/dcdsb.2010.14.199
In this work, we generalize the idea of Ginzburg-Landau approximation to study the existence and asymptotic behaviors of global weak solutions to the one dimensional periodical fractional Landau-Lifshitz equation modeling the soft micromagnetic materials. We apply the Galerkin method to get an approximate solution and, to get the convergence of the nonlinear terms we introduce the commutator structure and take advantage of special structures of the equation.
keywords: Heat flow of harmonic maps Commutator estimate. Fractional Landau-Lifshitz equation Ginzburg-Landau approximation
Quasineutral limit for the quantum Navier-Stokes-Poisson equations
Min Li Xueke Pu Shu Wang
Communications on Pure & Applied Analysis 2017, 16(1): 273-294 doi: 10.3934/cpaa.2017013

In this paper, we study the quasineutral limit and asymptotic behaviors for the quantum Navier-Stokes-Possion equation. We apply a formal expansion according to Debye length and derive the neutral incompressible Navier-Stokes equation. To establish this limit mathematically rigorously, we derive uniform (in Debye length) estimates for the remainders, for well-prepared initial data. It is demonstrated that the quantum effect do play important roles in the estimates and the norm introduced depends on the Planck constant $\hbar>0$.

keywords: Quantum Navier-Stokes-Possion system quasineutral limit formal expansion well-prepared initial data uniform energy estimates

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