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.

scholarly journals Towards a further understanding of the dynamics in the excitatory NNLIF neuron model: Blow-up and global existence

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Pierre Roux ◽  
Delphine Salort
Keyword(s):  
Global Existence ◽  
Blow Up ◽  
Mean Field ◽  
Classical Solutions ◽  
Field Limit ◽  
Mean Field Limit ◽  
Diffusive Approximation ◽  
The Mean ◽  
Weak Connectivity ◽  
Time Existence

<p style='text-indent:20px;'>The Nonlinear Noisy Leaky Integrate and Fire (NNLIF) model is widely used to describe the dynamics of neural networks after a diffusive approximation of the mean-field limit of a stochastic differential equation. In previous works, many qualitative results were obtained: global existence in the inhibitory case, finite-time blow-up in the excitatory case, convergence towards stationary states in the weak connectivity regime. In this article, we refine some of these results in order to foster the understanding of the model. We prove with deterministic tools that blow-up is systematic in highly connected excitatory networks. Then, we show that a relatively weak control on the firing rate suffices to obtain global-in-time existence of classical solutions.</p>

scholarly journals Stabilities for Nonisentropic Euler-Poisson Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Ka Luen Cheung ◽  
Sen Wong
Keyword(s):  
Energy Method ◽  
Finite Time ◽  
Blow Up ◽  
Classical Solutions ◽  
Poisson Equations ◽  
Repulsive Forces

We establish the stabilities and blowup results for the nonisentropic Euler-Poisson equations by the energy method. By analysing the second inertia, we show that the classical solutions of the system with attractive forces blow up in finite time in some special dimensions when the energy is negative. Moreover, we obtain the stabilities results for the system in the cases of attractive and repulsive forces.

scholarly journals On a class of derivative Nonlinear Schrödinger-type equations in two spatial dimensions

2019 ◽  
Vol 53 (5) ◽  
pp. 1477-1505
Author(s):  
Jack Arbunich ◽  
Christian Klein ◽  
Christof Sparber
Keyword(s):  
Laser Pulse ◽  
Numerical Simulations ◽  
Finite Time ◽  
Blow Up ◽  
Dispersive Equations ◽  
Stationary States ◽  
Nonlinear Dispersive Equations ◽  
Nonlinear Schrödinger ◽  
Spatial Dimensions ◽  
Time Existence

We present analytical results and numerical simulations for a class of nonlinear dispersive equations in two spatial dimensions. These equations are of (derivative) nonlinear Schrödinger type and have recently been obtained by Dumaset al.in the context of nonlinear optics. In contrast to the usual nonlinear Schrödinger equation, this new model incorporates the additional effects of self-steepening and partial off-axis variations of the group velocity of the laser pulse. We prove global-in-time existence of the corresponding solution for various choices of parameters. In addition, we present a series of careful numerical simulations concerning the (in-)stability of stationary states and the possibility of finite-time blow-up.

scholarly journals Some Properties of Solutions for the Sixth-Order Cahn-Hilliard-Type Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Zhao Wang ◽  
Changchun Liu
Keyword(s):  
Boundary Value Problem ◽  
Finite Time ◽  
Blow Up ◽  
Type Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Sixth Order ◽  
Classical Solutions ◽  
Oil Water ◽  
Initial Boundary Value

We study the initial boundary value problem for a sixth-order Cahn-Hilliard-type equation which describes the separation properties of oil-water mixtures, when a substance enforcing the mixing of the phases is added. We show that the solutions might not be classical globally. In other words, in some cases, the classical solutions exist globally, while in some other cases, such solutions blow up at a finite time. We also discuss the existence of global attractor.

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 Local Well-Posedness and Blow-Up for the Solutions to the Axisymmetric Inviscid Hall-MHD Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-16
Author(s):  
Eunji Jeong ◽  
Junha Kim ◽  
Jihoon Lee
Keyword(s):  
Classical Solution ◽  
Finite Time ◽  
Blow Up ◽  
Mhd Equations ◽  
Well Posedness ◽  
Regular Solutions ◽  
Regularity Problem ◽  
Time Existence ◽  
Hall Mhd

In this paper, we consider the regularity problem of the solutions to the axisymmetric, inviscid, and incompressible Hall-magnetohydrodynamics (Hall-MHD) equations. First, we obtain the local-in-time existence of sufficiently regular solutions to the axisymmetric inviscid Hall-MHD equations without resistivity. Second, we consider the inviscid axisymmetric Hall equations without fluids and prove that there exists a finite time blow-up of a classical solution due to the Hall term. Finally, we obtain some blow-up criteria for the axisymmetric resistive and inviscid Hall-MHD equations.

scholarly journals Development of singularities in solutions of a hyperbolic system

2001 ◽  
Vol 28 (1) ◽  
pp. 1-7
Author(s):  
S. A. Messaoudi
Keyword(s):  
Initial Data ◽  
Hyperbolic System ◽  
Finite Time ◽  
Blow Up ◽  
Classical Solutions ◽  
Small Initial Data

We consider a special type of a hyperbolic system and show that classical solutions blow up in finite time even for small initial data.

scholarly journals Sharp thresholds of blow-up and global existence for the Schrödinger equation with combined power-type and Choquard-type nonlinearities

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yongbin Wang ◽  
Binhua Feng
Keyword(s):  
Global Existence ◽  
Finite Time ◽  
Critical Case ◽  
Blow Up ◽  
Critical Mass ◽  
Choquard Equation ◽  
Power Type ◽  
Scale Invariant ◽  
Nonlinear Schrödinger ◽  
Sharp Thresholds

AbstractIn this paper, we consider the sharp thresholds of blow-up and global existence for the nonlinear Schrödinger–Choquard equation $$ i\psi _{t}+\Delta \psi =\lambda _{1} \vert \psi \vert ^{p_{1}}\psi +\lambda _{2}\bigl(I _{\alpha } \ast \vert \psi \vert ^{p_{2}}\bigr) \vert \psi \vert ^{p_{2}-2}\psi . $$iψt+Δψ=λ1|ψ|p1ψ+λ2(Iα∗|ψ|p2)|ψ|p2−2ψ. We derive some finite time blow-up results. Due to the failure of this equation to be scale invariant, we obtain some sharp thresholds of blow-up and global existence by constructing some new estimates. In particular, we prove the global existence for this equation with critical mass in the $L^{2}$L2-critical case. Our obtained results extend and improve some recent results.

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