THE EXISTENCE AND UNIQUENESS OF TIME-PERIODIC SOLUTIONS TO THE NON-ISOTHERMAL MODEL FOR COMPRESSIBLE NEMATIC LIQUID CRYSTALS IN A PERIODIC DOMAIN

In this paper,we are concerned with a three-dimensional non-isothermal model for the compressible nematic liquid crystal flows in a periodic domain.Under some smallness and structural assumptions imposed on the time-periodic force,we establish the existence of the time-periodic solutions to the system by using a regularized approximation scheme and the topological degree theory.We also prove a uniqueness result via energy estimates.

Secondary steady-state and time-periodic flows from a basic flow with square array of odd number of vortices

In a magnetohydrodynamic(MHD)driven fluid cell,a plane non-parallel flow in a square domain satisfying a free-slip boundary condition is examined.The energy dissipation of the flow is controlled by the viscosity and linear friction.The latter arises from the influence of the Hartmann bottom boundary layer in a three-dimensional(3D)MHD experiment in a square bottomed cell.The basic flow in this fluid system is a square eddy flow exhibiting a network of N~2 vortices rotating alternately in clockwise and anticlockwise directions.When N is odd,the instability of the flow gives rise to secondary steady-state flows and secondary time-periodic flows,exhibiting similar characteristics to those observed when N=3.For this reason,this study focuses on the instability of the square eddy flow of nine vortices.It is shown that there exist eight bi-critical values corresponding to the existence of eight neutral eigenfunction spaces.Especially,there exist non-real neutral eigenfunctions,which produce secondary time-periodic flows exhibiting vortices merging in an oscillatory manner.This Hopf bifurcation phenomenon has not been observed in earlier investigations.

TIME-PERIODIC ISENTROPIC SUPERSONIC EULER FLOWS IN ONE-DIMENSIONAL DUCTS DRIVING BY PERIODIC BOUNDARY CONDITIONS

We show existence of time-periodic supersonic solutions in a finite interval, after certain start-up time depending on the length of the interval, to the one space-dimensional isentropic compressible Euler equations, subjected to periodic boundary conditions. Both classical solutions and weak entropy solutions, as well as high-frequency limiting behavior are considered. The proofs depend on the theory of Cauchy problems of genuinely nonlinear hyperbolic systems of conservation laws.

TIME-PERIODIC SOLUTIONS OF THE VLASOV-POISSON-FOKKER-PLANCK SYSTEM

In this note it is shown that the Vlasov-Poisson-Fokker-Planck system in the three-dimensional whole space driven by a time-periodic background profile near a positive constant state admits a time-periodic small-amplitude solution with the same period. The proof follows by the Serrin＇s method on the basis of the exponential time-decay property of the linearized system in the case of the constant background profile.

Electrokinetic energy conversion of electro-magneto-hydro-dynamic nanofluids through a microannulus under the time-periodic excitation

In this work,the effects of externally applied axial pressure gradients and transverse magnetic fields on the electrokinetic energy conversion(EKEC)efficiency and the streaming potential of nanofluids through a microannulus are studied.The analytical solution for electro-magneto-hydro-dynamic(EMHD)flow is obtained under the condition of the Debye-Huuckel linearization.Especially,Green’s function method is used to obtain the analytical solutions of the velocity field.The result shows that the velocity distribution is characterized by the dimensionless frequency?,the Hartmann number Ha,the volume fraction of the nanoparticlesφ,the geometric radius ratio a,and the wallζpotential ratio b.Moreover,the effects of three kinds of periodic excitations are compared and discussed.The results also show that the periodic excitation of the square waveform is more effective in increasing the streaming potential and the EKEC efficiency.It is worth noting that adjusting the wallζpotential ratio and the geometric radius ratio can affect the streaming potential and the EKEC efficiency.

WEAK TIME-PERIODIC SOLUTIONS TO THE COMPRESSIBLE NAVIER-STOKES EQUATIONS

The compressible Navier-Stokes equations driven by a time-periodic external force are considered in this article. We establish the existence of weak time-periodic solutions and improve the result from [3] in the following sense： we extend the class of pressure functions, that is, we consider lower exponent γ.

Time-Periodic Solutions of the Hydrodynamic Equations for a Reacting Mixture in n-Dimension

In this paper, we investigate the existence of time-periodic solutions to the n-dimension hydrodynamic model for a reacting mixture with a time-periodic external force when the dimension is under some smallness assumption. The energy method combined with the spectral analysis is used to obtain the optimal decay estimates on the linearized solution operator. We study the existence and uniqueness of the time-periodic solution in some suitable function space by using a fixed point method and the decay estimates. Furthermore, we obtain the time asymptotic stability of the time-periodic solution.

Time-Periodic Solutions to Quasilinear Hyperbolic Systems on General Networks

For quasilinear hyperbolic systems on general networks with time-periodic boundary-interface conditions with a dissipative structure,the existence and stability of the time-periodic classical solutions are discussed.

Traveling Fronts for a Time-periodic Population Model with Dispersal

In this paper,we study a class of time-periodic population model with dispersal.It is well known that the existence of the periodic traveling fronts has been established.However,the uniqueness and stability of such fronts remain unsolved.In this paper,we first prove the uniqueness of non-critical periodic traveling fronts.Then,we show that all non-critical periodic traveling fronts are exponentially asymptotically stable.

Time-periodic solutions of the Einstein's field equations Ⅰ:general framework

In this paper,we develop a new algorithm to find the exact solutions of the Einstein's field equations.Time-periodic solutions are constructed by using the new algorithm.The singularities of the time-periodic solutions are investigated and some new physical phenomena,such as degenerate event horizon and time-periodic event horizon,are found.The applications of these solutions in modern cosmology and general relativity are expected.

Time-periodic solutions of the Einstein's field equations II:geometric singularities

In this paper,we construct several kinds of new time-periodic solutions of the vacuum Einstein's field equations whose Riemann curvature tensors vanish,keep finite or take the infinity at some points in these space-times,respectively.The singularities of these new time-periodic solutions are investigated and some new physical phenomena are discovered.

Physical characters of KLS time-periodic universe

In this paper we consider the physical characters of the new time-periodic solution of the vacuum Einstein’s field equations which was constructed by Kong, Liu and Shen. By the analysis of the Penrose diagram associated with this solution, we find that this solution is very different from the other solutions.

Time-periodic solutions of the Einstein's field equations Ⅲ:physical singularities

In this paper we construct a new time-periodic solution of the vacuum Einstein's field equations, this solution possesses physical singularities, i.e., the norm of the solution's Riemann curvature tensor takes the infinity at some points. We show that this solution is intrinsically time-periodic and describes a time-periodic universe with the "time-periodic physical singularity". By calculating the Weyl scalars of this solution, we investigate new physical phenomena and analyze new singularities for this universal model.

Time-Periodic Solution to the Compressible Navier–Stokes/Allen–Cahn System

In this paper,we investigate the time-periodic solution to a coupled compressible Navier–Stokes/Allen–Cahn system which describes the motion of a mixture of two viscous compressible fluids with a time periodic external force in a periodic domain in R^N.The existence of the time-periodic solution to the system is established by using an approach of parabolic regularization and combining with the topology degree theory,and then the uniqueness of the period solution is obtained under some smallness and symmetry assumptions on the external force.

Coherence resonance and bi-resonance by time-periodic coupling strength in Hodgkin-Huxley neuron networks

We study the effect of time-periodic coupling strength on the spiking coherence of Newman-Watts networks of Hodgkin-Huxley(HH) neurons with non-Gaussian noise.It is found that the spiking can exhibit coherence resonance(CR) when the extent of deviation of non-Gaussian noise from Gaussian noise and the amplitude of the coupling strength are varied.In particular,coherence bi-resonance(CBR) is observed when the frequency of the coupling strength is varied,and the CBR is always observed when the frequency is equal to,or a multiple of,the spiking period,manifesting as the locking between the frequencies of the spiking and the coupling strength.The results show that a time-periodic coupling strength may play a more constructive and efficient role in enhancing the spiking coherence of the neuronal networks than a constant coupling strength.These findings provide insight into the role of time-periodic coupling strength for enhancing the time precision of information processing in neuronal networks.

Time-periodic Solution to a Nonlinear Parabolic Type Equation of Higher Order

In this paper, the existence and uniqueness of time-periodic generalized solutions and time-periodic classical solutions to a class of parabolic type equation of higher order are proved by Galerkin method.

The Boltzmann Equation with Time-periodic Boundary Temperature

This paper is concerned with the boundary-value problem on the Boltzmann equation in bounded domains with diffuse-reflection boundary where the boundary temperature is time-periodic. We establish the existence of time-periodic solutions with the same period for both hard and soft potentials, provided that the time-periodic boundary temperature is sufficiently close to a stationary one which has small variations around a positive constant. The dynamical stability of time-periodic profiles is also proved under small perturbations, and this in turn yields the non-negativity of the profile. For the proof, we develop new estimates in the time-periodic setting.

Time-Periodic Fitzhugh-Nagumo Equation and the Optimal Control Problems

In this work, the authors considered the periodic optimal control problem of Fitzhugh-Nagumo equation. They firstly prove the existence of time-periodic solution to Fitzhugh-Nagumo equation. Then they show the existence of optimal solution to the optimal control problem, and finally the first order necessary condition is obtained by constructing an appropriate penalty function.

Meandering Spiral Waves Induced by Time-Periodic Coupling Strength

Effects of time-periodic coupling strength （TPCS） on spiral waves dynamics are studied by numerical computations and mathematical analyses. We find that meandering or drifting spirals waves, which are not observed for the case of constant coupling strength, can be induced by TPCS. In particular, a transition between outward petal and inward petal meandering spirals is observed when the period of TPCS is varied. These two types of meandering spirals are separated by a drifting spiral, which can be induced by TPCS when the period of TPCS is very close to that of rigidly rotating spiral. Similar results can be obtained if the coupling strength is modulated by a rectangle wave. Furthermore, a kinetic model for spiral movement suggested by Diet al., [Phys. Rev. E 85 （2012） 046216] is applied for explaining the above findings. The theoretical results are in good qualitative agreement with numerical simulations.

CONCERNING TIME-PERIODIC SOLUTIONS OF THE NAVIER-STOKES EQUATIONS IN CYLINDRICAL DOMAINS UNDER NAVIER BOUNDARY CONDITIONS

The problem of the existence of time-periodic flows in infinite cylindrical pipes in correspondence to any given, time-periodic, total flux, was solved only quite recently in [1]. In this last reference we solved the above problem for flows under the non-slip boundary condition as a corollary of a more general result. Here we want to show that the abstract theorem proved in [1] applies as well to the solutions of the well known slip （or Navier） boundary condition （1.7） or to the mixed boundary condition （1.14）. Actually, the argument applies for solutions of many other boundary value problems. This paper is a continuation of reference [1], to which the reader is referred for some notation and results.