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 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 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 Blow up for the solutions of the pressureless Euler-Poisson equations with time-dependent damping

2021 ◽  
Author(s):  
Jianli Liu ◽  
Jingwei Wang ◽  
Lining Tong
Keyword(s):  
Plasma Physics ◽  
Formation Mechanism ◽  
Blow Up ◽  
Time Dependent ◽  
Singularity Formation ◽  
Poisson Equations ◽  
Repulsive Forces ◽  
Physical Phenomena ◽  
Semiconductor Modeling

The Euler-Poisson equations can be used to describe the important physical phenomena in many areas, such as semiconductor modeling and plasma physics. In this paper, we show the singularity formation mechanism for the solutions of the pressureless Euler-Poisson equations with time-dependent damping for the attractive forces in R^n (n ≧1) and the repulsive forces in R. We obtain the blow up of the derivative of the velocity under the appropriate assumptions.

scholarly journals Blowup Phenomena for the Compressible Euler and Euler-Poisson Equations with Initial Functional Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Sen Wong ◽  
Manwai Yuen
Keyword(s):  
Boundary Condition ◽  
Life Span ◽  
Finite Time ◽  
Blow Up ◽  
Symmetric Case ◽  
Governing Equations ◽  
Pressure Function ◽  
Poisson Equations ◽  
Compressible Euler ◽  
Blowup Result

We study, in the radial symmetric case, the finite time life span of the compressible Euler or Euler-Poisson equations inRN. For timet≥0, we can define a functionalH(t)associated with the solution of the equations and some testing functionf. When the pressure functionPof the governing equations is of the formP=Kργ, whereρis the density function,Kis a constant, andγ>1, we can show that the nontrivialC1solutions with nonslip boundary condition will blow up in finite time ifH(0)satisfies some initial functional conditions defined by the integrals off. Examples of the testing functions includerN-1ln(r+1),rN-1er,rN-1(r3-3r2+3r+ε),rN-1sin((π/2)(r/R)), andrN-1sinh r. The corresponding blowup result for the 1-dimensional nonradial symmetric case is also given.

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.

scholarly journals Blow-up criterion for the density dependent inviscid Boussinesq equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Li Li ◽  
Yanping Zhou
Keyword(s):  
Velocity Field ◽  
Energy Method ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Boussinesq Equations ◽  
Smooth Solutions ◽  
Density Dependent ◽  
Priori Estimates ◽  
Blow Up Criterion

Abstract In this work, we consider the density-dependent incompressible inviscid Boussinesq equations in $\mathbb{R}^{N}\ (N\geq 2)$ R N ( N ≥ 2 ) . By using the basic energy method, we first give the a priori estimates of smooth solutions and then get a blow-up criterion. This shows that the maximum norm of the gradient velocity field controls the breakdown of smooth solutions of the density-dependent inviscid Boussinesq equations. Our result extends the known blow-up criteria.

Blow-up for the compressible isentropic Navier-Stokes-Poisson equations

2019 ◽  
Vol 70 (1) ◽  
pp. 9-19
Author(s):  
Jianwei Dong ◽  
Junhui Zhu ◽  
Yanping Wang
Keyword(s):  
Blow Up ◽  
Navier Stokes ◽  
Poisson Equations
Sign in / Sign up

Export Citation Format

Share Document