scholarly journals On the pressureless damped Euler–Poisson equations with quadratic confinement: Critical thresholds and large-time behavior

2016 ◽  
Vol 26 (12) ◽  
pp. 2311-2340 ◽  
Author(s):  
José A. Carrillo ◽  
Young-Pil Choi ◽  
Ewelina Zatorska
Keyword(s):  
Large Time Behavior ◽  
Classical Solutions ◽  
Nonlocal Interaction ◽  
Time Behavior ◽  
Critical Thresholds ◽  
Poisson Equations ◽  
Subcritical Region ◽  
Linear Damping ◽  
The One ◽  
Time Existence

We analyze the one-dimensional pressureless Euler–Poisson equations with linear damping and nonlocal interaction forces. These equations are relevant for modeling collective behavior in mathematical biology. We provide a sharp threshold between the supercritical region with finite-time breakdown and the subcritical region with global-in-time existence of the classical solution. We derive an explicit form of solution in Lagrangian coordinates which enables us to study the time-asymptotic behavior of classical solutions with the initial data in the subcritical region.

scholarly journals Global Existence and Large Time Behavior of Solutions to the Bipolar Nonisentropic Euler-Poisson Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Min Chen ◽  
Yiyou Wang ◽  
Yeping Li
Keyword(s):  
Large Time ◽  
Large Time Behavior ◽  
Negative Ion ◽  
Initial Mass ◽  
Far Field ◽  
Time Behavior ◽  
Poisson Equations ◽  
Global Smooth Solutions ◽  
The Difference ◽  
The One

We study the one-dimensional bipolar nonisentropic Euler-Poisson equations which can model various physical phenomena, such as the propagation of electron and hole in submicron semiconductor devices, the propagation of positive ion and negative ion in plasmas, and the biological transport of ions for channel proteins. We show the existence and large time behavior of global smooth solutions for the initial value problem, when the difference of two particles’ initial mass is nonzero, and the far field of two particles’ initial temperatures is not the ambient device temperature. This result improves that of Y.-P. Li, for the case that the difference of two particles’ initial mass is zero, and the far field of the initial temperature is the ambient device temperature.

scholarly journals A Sharp Estimate of Entropy Solution to Euler-Poisson System for Semiconductors in the Whole Domain

2019 ◽  
pp. 1-12
Author(s):  
Yanqiu Cheng ◽  
Xixi Fang ◽  
Huimin Yu
Keyword(s):  
Large Time ◽  
Approximate Solutions ◽  
Entropy Inequality ◽  
Large Time Behavior ◽  
Doping Profile ◽  
Entropy Solutions ◽  
Time Behavior ◽  
One Dimensional ◽  
Poisson Equations ◽  
The One

In this paper, we are concerned with the global existence, large time behavior, and timeincreasing-rate of entropy solutions to the one-dimensional unipolar hydrodynamic model for semiconductors in the form of Euler-Poisson equations. When the adiabatic index γ > 2, the L∞ estimates of artificial viscosity approximate solutions are obtained by using entropy inequality and maximum principle. Then the L∞ compensated compactness framework demonstrates theconvergence of approximate solutions. Finally, the global entropy solutions are proved to decay exponentially fast to the stationary solution, without any assumption on the smallness of initial data and doping profile.

Initial Value Problems in Viscoelasticity

1988 ◽  
Vol 41 (10) ◽  
pp. 371-378 ◽  
Author(s):  
W. J. Hrusa ◽  
J. A. Nohel ◽  
M. Renardy
Keyword(s):  
Classical Solutions ◽  
Linear Wave ◽  
Time Behavior ◽  
Initial Value ◽  
One Dimensional ◽  
Nonlinear Viscoelastic ◽  
Linear Wave Propagation ◽  
Long Time ◽  
Nonlinear Viscoelastic Materials ◽  
Time Existence

We review some recent mathematical results concerning integrodiff erential equations that model the motion of one-dimensional nonlinear viscoelastic materials. In particular, we discuss global (in time) existence and long-time behavior of classical solutions, as well as the formation of singularities in finite time from smooth initial data. Although the mathematical theory is comparatively incomplete, we make some remarks concerning the existence of weak solutions (i e, solutions with shocks). Some relevant results from linear wave propagation will also be discussed.

scholarly journals The delayed Cucker-Smale model with short range communication weights

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zili Chen ◽  
Xiuxia Yin
Keyword(s):  
Initial Data ◽  
Short Range ◽  
Large Time ◽  
Large Time Behavior ◽  
Classical Solutions ◽  
Time Behavior ◽  
Potential Applications ◽  
Communication Ability ◽  
General Communication ◽  
Flocking Behavior

<p style='text-indent:20px;'>Various flocking results have been established for the delayed Cucker-Smale model, especially in the long range communication case. However, the short range communication case is more realistic due to the limited communication ability. In this case, the non-flocking behavior can be frequently observed in numerical simulations. Furthermore, it has potential applications in many practical situations, such as the opinion disagreement in society, fish flock breaking and so on. Therefore, we firstly consider the non-flocking behavior of the delayed Cucker<inline-formula><tex-math id="M2">\begin{document}$ - $\end{document}</tex-math></inline-formula>Smale model. Based on a key inequality of position variance, a simple sufficient condition of the initial data to the non-flocking behavior is established. Then, for general communication weights we obtain a flocking result, which also depends upon the initial data in the short range communication case. Finally, with no restriction on the initial data we further establish other large time behavior of classical solutions.</p>

scholarly journals A hydrodynamic model for synchronization phenomena

2020 ◽  
Vol 30 (11) ◽  
pp. 2175-2227
Author(s):  
Young-Pil Choi ◽  
Jaeseung Lee
Keyword(s):  
Hydrodynamic Model ◽  
Finite Time ◽  
Natural Frequencies ◽  
Blow Up ◽  
Classical Model ◽  
Kuramoto Model ◽  
Classical Solutions ◽  
Nonlocal Interaction ◽  
Phase Oscillators ◽  
Time Existence

We present a new hydrodynamic model for synchronization phenomena which is a type of pressureless Euler system with nonlocal interaction forces. This system can be formally derived from the Kuramoto model with inertia, which is a classical model of interacting phase oscillators widely used to investigate synchronization phenomena, through a kinetic description under the mono-kinetic closure assumption. For the proposed system, we first establish local-in-time existence and uniqueness of classical solutions. For the case of identical natural frequencies, we provide synchronization estimates under suitable assumptions on the initial configurations. We also analyze critical thresholds leading to finite-time blow-up or global-in-time existence of classical solutions. In particular, our proposed model exhibits the finite-time blow-up phenomenon, which is not observed in the classical Kuramoto models, even with a smooth distribution function for natural frequencies. Finally, we numerically investigate synchronization, finite-time blow-up, phase transitions, and hysteresis phenomena.

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