Existence of steady subsonic Euler flows through infinitely long periodic nozzles

In this paper, we study the global existence of steady subsonic Euler flows through infinitely long nozzles which are periodic in $x_1$-direction with the period L. It is shown that when the variation of Bernoulli function at some given section is small and mass flux is in a suitable regime, there exists a unique global subsonic flow in the nozzle. Furthermore, the flow is also periodic in $x_1$-direction with the period L. If, in particular, the Bernoulli function is a constant, we also get the existence of subsonic-sonic flows when the mass flux takes the critical value.     

Global subsonic Euler flows in an infinitely long axisymmetric nozzle

In this paper, we consider global subsonic compressible flows through an infinitely long axisymmetric nozzle. The flow is governed by the steady Euler equations and has boundary conditions on the nozzle walls. Existence and uniqueness of global subsonic solution are established for an infinitely long axisymmetric nozzle, when the variation of Bernoulli's function in the upstream is sufficiently small and the mass flux of the incoming flow is less than some critical value. The results give a strictly mathematical proof to the assertion in Bers (1958) [2]: there exists a critical value of the incoming mass flux such that a global subsonic flow exists uniquely in a nozzle, provided that the incoming mass flux is less than the critical value. The existence of subsonic flow is obtained by the precisely a priori estimates for the elliptic equation of two variables. With the assumptions on the nozzle in the far fields, the asymptotic behavior can be derived by a blow-up argument for the infinitely long nozzle. Finally, we obtain the uniqueness of uniformly subsonic flow by energy estimate and derive the existence of the critical value of incoming mass flux.     

Periodic Traveling Waves in Diatomic Granular Chains

We study bifurcations of periodic traveling waves in diatomic granular chains from the anti-continuum limit, when the mass ratio between the light and heavy beads is zero. We show that every limiting periodic wave is uniquely continued with respect to the mass ratio parameter, and the periodic waves with a wavelength larger than a certain critical value are spectrally stable. Numerical computations are developed to study how this solution family is continued to the limit of equal mass ratio between the beads, where periodic traveling waves of homogeneous granular chains exist.     

Existence of doubly periodic vortices in a generalized Chern–Simons model

We establish an existence theorem for the doubly periodic vortices in a generalized self-dual Chern–Simons model. We show that there exists a critical value of the coupling parameter such that there exist self-dual doubly periodic vortex solutions for the generalized self-dual Chern–Simons equation if and only if the coupling parameter is less than or equal to the value. The energy, magnetic flux, and electric charge associated to the field configurations are all specifically quantized. By the solutions obtained for this generalized self-dual Chern–Simons equation we can also construct doubly periodic vortex solutions to a related generalized self-dual Abelian Higgs equation.     

Effect of the potential shape and of a Brownian particle mass on noise-induced transport

The noise-induced transport of a Brownian particle with regard to its mass is considered. The results of approximate analytical calculations for the averaged probability flux in periodic ratchet-like potentials are presented. The influence of the potential shape on the mean particle velocity is traced. In the case of sufficiently small particle mass, it is shown that with increase in mass the reversal of flux is possible.     

Global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles

In this paper, we establish existence of global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles by combining variational method, various elliptic estimates and a compensated compactness method. More precisely, it is shown that there exist global subsonic flows in nozzles for incoming mass flux less than a critical value; moreover, uniformly subsonic flows always approach to uniform flows at far fields when nozzle boundaries tend to be flat at far fields, and flow angles for axially symmetric flows are uniformly bounded away from π/2; finally, when the incoming mass flux tends to the critical value, subsonic-sonic flows exist globally in nozzles in the weak sense by using angle estimate in conjunction with a compensated compactness framework.     

On the neutron flux flattening problem

In this paper, we present the critical neutron flux flattening problem governed by the critical transport equation in a nonuniform slab with periodic boundary conditions. Existence and uniqueness theorem of the optimal solution is shown in continuous function space.     

Catastrophe of coronal magnetic flux ropes in fully open magnetic field

The catastrophe of coronal magnetic flux ropes is closely related to solar explosive phenomena, such as prominence eruptions, coronal mass ejections, and two-ribbon solar flares. Using a 2-dimensional, 3-component ideal MHD model in Cartesian coordinates, numerical simulations are carried out to investigate the equilibrium property of a coronal magnetic flux rope which is embedded in a fully open background magnetic field. The flux rope emerges from the photosphere and enters the corona with its axial and annular magnetic fluxes controlled by a single “emergence parameter”. For a flux rope that has entered the corona, we may change its axial and annular fluxes artificially and let the whole system reach a new equilibrium through numerical simulations. The results obtained show that when the emergence parameter, the axial flux, or the annular flux is smaller than a certain critical value, the flux rope is in equilibrium and adheres to the photosphere. On the other hand, if the critical value is exceeded, the flux rope loses equilibrium and erupts freely upward, namely, a catastrophe takes place. In contrast with the partly-opened background field, the catastrophic amplitude is infinite for the case of fully-opened background field     

Catastrophe of coronal magnetic flux ropes in fully open magnetic field

The catastrophe of coronal magnetic flux ropes is closely related to solar explosive phenomena, such as prominence eruptions, coronal mass ejections, and two-ribbon solar flares. Using a 2-dimensional, 3-component ideal MHD model in Cartesian coordinates, numerical simulations are carried out to investigate the equilibrium property of a coronal magnetic flux rope which is embedded in a fully open background magnetic field. The flux rope emerges from the photosphere and enters the corona with its axial and annular magnetic fluxes controlled by a single "emergence parameter". For a flux rope that has entered the corona, we may change its axial and annular fluxes artificially and let the whole system reach a new equilibrium through numerical simulations. The results obtained show that when the emergence parameter, the axial flux, or the annular flux is smaller than a certain critical value, the flux rope is in equilibrium and adheres to the photosphere. On the other hand, if the critical value is exceeded, the flux rope loses equilibrium and erupts freely upward, namely, a catastrophe takes place. In contrast with the partly-opened background field, the catastrophic amplitude is infinite for the case of fully-opened background field.     

Subsonic non‐isentropic ideal gas with large vorticity in nozzles

This paper investigates the steady subsonic inviscid flows with large vorticities through two‐dimensional infinitely long nozzles. We establish the existence and uniqueness of the smooth subsonic ideal flows, which are governed by a two‐dimensional complete Euler system. More precisely, given the horizontal velocity with possible large oscillation and the entropy of the incoming flows at the entrance of the nozzles, it was shown that there exists a critical value; if the mass flux of the incoming flows is larger than the critical one, then there exists a unique smooth subsonic polytropic gas through the given smooth infinitely long nozzles. Furthermore, the maximal speed of the flows approaches to the sonic speed, as the mass flux goes to the critical value. The results improve the previous work for steady subsonic flows with small vorticities and for subsonic irrotational flows and indicate that the large vorticity is admissible for the smooth subsonic ideal flows in nozzles. This paper gives a rigorous proof to the well posedness of the smooth subsonic problem first posed back in the basic survey of Lipman Bers for inviscid flows with large vorticities. John Wiley & Sons, Ltd.     

Extinction and permanence of a two-prey two-predator system with impulsive on the predator

In this paper, the dynamic behaviors of a two-prey two-predator system with impulsive effect on the predator of fixed moment are investigated. By applying the Floquet theory of liner periodic impulsive equation, we show that there exists a globally asymptotically stable two-prey eradication periodic solution when the impulsive period is less than some critical value. Further, we prove that the system is permanent if the impulsive period is large than some critical value, and meanwhile the conditions for the extinction of one of the two prey and permanence of the remaining three species are given. Finally, numerical simulation shows that there exists a stable positive periodic solution with a maximum value no larger than a given level. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels.     

周期载荷下超弹性圆柱壳的动力响应

研究了不可压超弹性圆柱壳在内表面周期载荷及突加常值载荷作用下的运动与破坏等动力响应问题．通过对所得描述圆柱壳内表面运动的非线性常微分方程解的数值计算和动力学定性分析,发现存在一个临界载荷;当突加常值载荷或周期载荷的平均载荷值小于这一临界值时,圆柱壳的运动随时间的演化是周期性的或拟周期性的非线性振动,而当其大于这一临界值时,圆柱壳将被破坏．另外,准静态问题的解可作为突加常值载荷作用下系统动力响应解的不动点,且不动点的性质与动力响应解及圆柱壳运动的性质有关．讨论了圆柱壳的厚度和载荷等参数对临界载荷值和圆柱壳运动特性的影响．     

p-q-Laplace系统周期解的存在性

利用临界点理论研究了p-q-Laplace系统的周期边值问题.首先定义p-q-Laplace系统的弱周期解;其次给出一些引理;然后用临界点理论中的极小极大方法得到关于p-q-Laplace系统弱周期解的一个存在性定理;最后讨论了p-q-Laplace系统的相关问题.本文使用的主要方法是临界点理论中的环绕定理.     

自全国各地100多位理事Road Machinery Branch of China Highway and Transportation Society He

In this paper, a three dimensional ratio-dependent chemostat model with periodically pulsed input is considered. By using the discrete dynamical system determined by the stroboscopic map and Floquet theorem, an exact periodic solution with positive concentrations of substrate and predator in the absence of prey is obtained. When β is less than some critical value the boundary periodic solution (xs(t), 0, zs(t)) is locally stable, and when β is larger than the critical value there are periodic oscillations in substrate, prey and predator. Increasing the impulsive period τ, the system undergoes a series of period-doubling bifurcation leading to chaos, which implies that the dynamical behaviors of the periodically pulsed ratio-dependent predator-prey ecosystem are very complex.     

A note on hysteresis in glaciology

Recent studies on the mechanism governing the Laurentide ice sheet oscillations of the Last Ice Age focus on the most critical effect of the basal hydraulic processes enhanced when the ice is sliding along soft deformable beds. To understand the import of this, we consider Fowler and Johnson's 0-D hydrological flow model describing the sudden and rapid movements forward (surges) of a till-based 1-D ice sheet sliding on a flat soft bed. The basic idea is that the interplay between the ice sheets dynamics, the basal drainage system, and the sliding law can generate a surging behaviour. Mathematically this means that a multiple valued relationship between the ice flux and the ice thickness arises and the mass conservation equation turns out to be of multivalued type for some special values of the dimensionless parameters involved in the model. Assuming that a multiple valued ice flux law of the Fowler and Johnson type holds, we prove the existence of a weak bounded discontinuous solution to the system which becomes periodic after a suitable time.     

Dynamic analysis of a pest management SEI model with saturation incidence concerning impulsive control strategy

According to biological strategy for pest control, we investigate the dynamic behavior of a pest management SEI model with saturation incidence concerning impulsive control strategy-periodic releasing infected pests at fixed times. We prove that all solutions of the system are uniformly ultimately bounded and there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. When the impulsive period is larger than some critical value, the stability of the pest-eradication periodic solution is lost; the system is uniformly permanent. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels. Numerical results show that the system we consider can take on various kinds of periodic fluctuations and several types of attractor coexistence and is dominated by period-doubling cascade, symmetry-breaking pitchfork bifurcation, quasi-periodic oscillate, chaos, and non-unique dynamics.     

On the approach to the critical solution in leading order thin-film coating and rimming flow

The approach to the critical solution in leading order coating and rimming flow of a thin fluid film on a uniformly rotating horizontal cylinder is investigated. In particular, it is shown that the weight of the leading order “full film” solution approaches its critical maximum value with logarithmically infinite slope as the volume flux approaches its critical value.     

Critical Quenching Exponents for Heat Equations Coupled with Nonlinear Boundary Flux

We discuss the quenching phenomena for a system of heat equations coupled with nonlinear boundary flux. We determine a critical value for the exponents in the boundary flux, such that only in the super critical case the simultaneous quenching can happen for any solution.