Well-Posedness and Asymptotics of Grazing Collisions Limit of Boltzmann Equation with Coulomb Interaction

2014 ◽  
Vol 46 (6) ◽  
pp. 4104-4165 ◽  
Author(s):  
Lingbing He ◽  
Xiongfeng Yang
Keyword(s):  
Boltzmann Equation ◽  
Coulomb Interaction ◽  
Well Posedness ◽  
Grazing Collisions

scholarly journals Local well-posedness of the Boltzmann equation with polynomially decaying initial data

2020 ◽  
Vol 13 (4) ◽  
pp. 837-867 ◽  
Author(s):  
Christopher Henderson ◽  
◽  
Stanley Snelson ◽  
Andrei Tarfulea ◽  
◽  
...  
Keyword(s):  
Boltzmann Equation ◽  
Initial Data ◽  
Well Posedness ◽  
The Boltzmann Equation

scholarly journals Linear Stability of Hyperbolic Moment Models for Boltzmann Equation

2017 ◽  
Vol 10 (2) ◽  
pp. 255-277 ◽  
Author(s):  
Yana Di ◽  
Yuwei Fan ◽  
Ruo Li ◽  
Lingchao Zheng
Keyword(s):  
Boltzmann Equation ◽  
Stability Condition ◽  
Hyperbolic Systems ◽  
Local Equilibrium ◽  
Cauchy Data ◽  
Binary Collision ◽  
Well Posedness ◽  
Convection Term ◽  
Collision Model ◽  
Moment Models

AbstractGrad's moment models for Boltzmann equation were recently regularized to globally hyperbolic systems and thus the regularized models attain local well-posedness for Cauchy data. The hyperbolic regularization is only related to the convection term in Boltzmann equation. We in this paper studied the regularized models with the presentation of collision terms. It is proved that the regularized models are linearly stable at the local equilibrium and satisfy Yong's first stability condition with commonly used approximate collision terms, and particularly with Boltzmann's binary collision model.

scholarly journals On the multi-species Boltzmann equation with uncertainty and its stochastic Galerkin approximation

2021 ◽  
Author(s):  
Shi Jin ◽  
Esther Daus ◽  
Liu Liu
Keyword(s):  
Boltzmann Equation ◽  
Galerkin Approximation ◽  
Single Species ◽  
Collision Kernel ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time ◽  
Stochastic Galerkin ◽  
Analytical Tools ◽  
Gap Estimate

In this paper the nonlinear multi-species Boltzmann equation with random uncertainty coming from the initial data and collision kernel is studied. Well-posedness and long-time behavior – exponential decay to the global equilibrium – of the analytical solution, and spectral gap estimate for the corresponding linearized gPC-based stochastic Galerkin system are obtained, by using and extending the analytical tools provided in [M. Briant and E. S. Daus, Arch. Ration. Mech. Anal., 3, 1367–1443, 2016] for the deterministic problem in the perturbative regime, and in [E. S. Daus, S. Jin and L. Liu, Kinet. Relat. Models, 12, 909–922, 2019] for the single-species problem with uncertainty. The well-posedness result of the sensitivity system presented here has not been obtained so far even for the single-species case.

scholarly journals Arrest of blow up for the 3D semi-relativistic Schrödinger–Poisson system with pseudo-relativistic diffusion

2015 ◽  
Vol 27 (10) ◽  
pp. 1550023
Author(s):  
W. Abou Salem ◽  
T. Chen ◽  
V. Vougalter
Keyword(s):  
Coulomb Interaction ◽  
Finite Time ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Negative Energy ◽  
Energy Norm ◽  
Well Posedness ◽  
System Of Equations ◽  
Initial Condition ◽  
Poisson System

We show global well-posedness in energy norm of the semi-relativistic Schrödinger–Poisson system of equations with attractive Coulomb interaction in [Formula: see text] in the presence of pseudo-relativistic diffusion. We also discuss sufficient conditions to have well-posedness in [Formula: see text]. In the absence of dissipation, we show that the solution corresponding to an initial condition with negative energy blows up in finite time, which is as expected, since the nonlinearity is critical.

LOCAL SMOOTH SOLUTIONS OF A THIN SPRAY MODEL WITH COLLISIONS

2010 ◽  
Vol 20 (02) ◽  
pp. 191-221 ◽  
Author(s):  
JULIEN MATHIAUD
Keyword(s):  
Boltzmann Equation ◽  
Initial Data ◽  
Euler Equations ◽  
Existence And Uniqueness ◽  
Well Posedness ◽  
Liquid Droplets ◽  
Smooth Solutions ◽  
Perfect Gas ◽  
Complex Flows

Sprays are complex flows made of liquid droplets surrounded by a gas. The aim of this paper is to study the local in time well-posedness of a collisional thin spray model, that is a coupling between Euler equations for a perfect gas and a Vlasov–Boltzmann equation for the droplets. We prove the existence and uniqueness of (local in time) solutions for this problem as soon as initial data are smooth enough.

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