scholarly journals Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules

2015 ◽  
Vol 9 (1) ◽  
pp. 1-49 ◽  
Author(s):  
Kleber Carrapatoso
Keyword(s):  
Landau Equation ◽  
Propagation Of Chaos ◽  
Spatially Homogeneous ◽  
Maxwellian Molecules

DECREASE OF THE FISHER INFORMATION FOR SOLUTIONS OF THE SPATIALLY HOMOGENEOUS LANDAU EQUATION WITH MAXWELLIAN MOLECULES

2000 ◽  
Vol 10 (02) ◽  
pp. 153-161 ◽  
Author(s):  
C. VILLANI
Keyword(s):  
Numerical Simulation ◽  
Boltzmann Equation ◽  
Plasma Physics ◽  
Fisher Information ◽  
Landau Equation ◽  
Direct Proof ◽  
Homogeneous Boltzmann Equation ◽  
Spatially Homogeneous ◽  
Maxwellian Molecules ◽  
Spatially Homogeneous Boltzmann Equation

We give a direct proof of the fact that, in any dimension of the velocity space, Fisher's quantity of information is nonincreasing with time along solutions of the spatially homogeneous Landau equation for Maxwellian molecules. This property, which was first seen in numerical simulation in plasma physics, is linked with the theory of the spatially homogeneous Boltzmann equation.

ON THE SPATIALLY HOMOGENEOUS LANDAU EQUATION FOR MAXWELLIAN MOLECULES

1998 ◽  
Vol 08 (06) ◽  
pp. 957-983 ◽  
Author(s):  
C. VILLANI
Keyword(s):  
Cauchy Problem ◽  
Landau Equation ◽  
Explicit Solutions ◽  
The Cauchy Problem ◽  
Spatially Homogeneous ◽  
Maxwellian Molecules ◽  
Simplified Form

We establish a simplified form for the Landau equation with Maxwellian-type molecules. We study in details the Cauchy problem associated to this equation, and some qualitative features of the solution. Explicit solutions are given.

scholarly journals SECOND-ORDER SCHEME FOR THE SPATIALLY HOMOGENEOUS BOLTZMANN EQUATION WITH MAXWELLIAN MOLECULES

1996 ◽  
Vol 06 (01) ◽  
pp. 137-147 ◽  
Author(s):  
JENS STRUCKMEIER ◽  
KONRAD STEINER
Keyword(s):  
Boltzmann Equation ◽  
Particle Method ◽  
Time Discretization ◽  
Second Order ◽  
Particle Methods ◽  
Homogeneous Boltzmann Equation ◽  
The Boltzmann Equation ◽  
Spatially Homogeneous ◽  
Maxwellian Molecules ◽  
Spatially Homogeneous Boltzmann Equation

In the standard approach particle methods for the Boltzmann equation are obtained using an explicit time discretization of the spatially homogeneous Boltzmann equation. This kind of discretization leads to a restriction on the discretization parameter as well as on the differential cross-section in the case of the general Boltzmann equation. Recently, construction of an implicit particle scheme for the Boltzmann equation with Maxwellian molecules was shown. This paper combines both approaches using a linear combination of explicit and implicit discretizations. It is shown that the new method leads to a second-order particle method when using an equiweighting of explicit and implicit discretization.

scholarly journals Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules

2020 ◽  
Vol 13 (5) ◽  
pp. 951-978
Author(s):  
Yoshinori Morimoto ◽  
◽  
Chao-Jiang Xu ◽  
◽  
Keyword(s):  
Landau Equation ◽  
Smoothing Effect ◽  
Analytic Smoothing Effect ◽  
Maxwellian Molecules

scholarly journals A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation

2013 ◽  
Vol 6 (4) ◽  
pp. 715-727 ◽  
Author(s):  
Yoshinori Morimoto ◽  
◽  
Karel Pravda-Starov ◽  
Chao-Jiang Xu ◽  
◽  
...  
Keyword(s):  
Landau Equation ◽  
Spatially Homogeneous

scholarly journals KINETIC LIMITS FOR PAIR-INTERACTION DRIVEN MASTER EQUATIONS AND BIOLOGICAL SWARM MODELS

2013 ◽  
Vol 23 (07) ◽  
pp. 1339-1376 ◽  
Author(s):  
ERIC CARLEN ◽  
PIERRE DEGOND ◽  
BERNT WENNBERG
Keyword(s):  
Kinetic Theory ◽  
Particle System ◽  
Pair Interaction ◽  
Master Equations ◽  
Propagation Of Chaos ◽  
Invariant Density ◽  
Homogeneous Case ◽  
Kac Model ◽  
Spatially Homogeneous ◽  
Definition Of

We consider a class of stochastic processes modeling binary interactions in an N-particle system. Examples of such systems can be found in the modeling of biological swarms. They lead to the definition of a class of master equations that we call pair-interaction driven master equations. In the spatially homogeneous case, we prove a propagation of chaos result for this class of master equations which generalizes Mark Kac's well-known result for the Kac model in kinetic theory. We use this result to study kinetic limits for two biological swarm models. We show that propagation of chaos may be lost at large times and we exhibit an example where the invariant density is not chaotic.

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