Global weak solutions to the 1-D fractional Landau-Lifshitz equation
Xueke Pu Boling Guo Jingjun Zhang
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
Pullback attractors of FitzHugh-Nagumo system on the time-varying domains
Zhen Zhang Jianhua Huang Xueke Pu

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
Quasineutral limit for the quantum Navier-Stokes-Poisson equations
Min Li Xueke Pu Shu Wang

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
Quasineutral limit of the Euler-Poisson system under strong magnetic fields
Xueke Pu
The quasineutral limit of the three dimensional compressible Euler-Poisson (EP) system for ions in plasma under strong magnetic field is rigorously studied. It is proved that as the Debye length and the Larmor radius tend to zero, the solution of the compressible EP system converges strongly to the strong solution of the one-dimensional compressible Euler-equation in the external magnetic field direction. Higher order approximation and convergence rates are also given and detailed studied.
keywords: strong magnetic field. quasineutral limit Euler-Poisson equation
Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction
Xueke Pu Boling Guo
The hydrodynamic equations with quantum effects are studied in this paper. First we establish the global existence of smooth solutions with small initial data and then in the second part, we establish the convergence of the solutions of the quantum hydrodynamic equations to those of the classical hydrodynamic equations. The energy equation is considered in this paper, which added new difficulties to the energy estimates, especially to the selection of the appropriate Sobolev spaces.
keywords: global solutions Quantum hydrodynamics semiclassical limit.

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