The cauchy problem for the Boltzmann equation. A survey of recent results

1984 ◽  
pp. 202-206
Author(s):  
Andrzej Palczewski
Keyword(s):  
Cauchy Problem ◽  
Boltzmann Equation ◽  
The Boltzmann Equation ◽  
The Cauchy Problem

scholarly journals THE EINSTEIN–BOLTZMANN SYSTEM AND POSITIVITY

2013 ◽  
Vol 10 (01) ◽  
pp. 77-104 ◽  
Author(s):  
HO LEE ◽  
ALAN D. RENDALL
Keyword(s):  
Cauchy Problem ◽  
General Relativity ◽  
Boltzmann Equation ◽  
Cross Section ◽  
Collision Cross Section ◽  
Curved Spacetime ◽  
The Boltzmann Equation ◽  
The Cauchy Problem

The Einstein–Boltzmann (EB) system is studied, with particular attention to the non-negativity of the solution of the Boltzmann equation. A new parametrization of post-collisional momenta in general relativity is introduced and then used to simplify the conditions on the collision cross-section given by Bancel and Choquet-Bruhat. The non-negativity of solutions of the Boltzmann equation on a given curved spacetime has been studied by Bichteler and Tadmon. By examining to what extent the results of these authors apply in the framework of Bancel and Choquet-Bruhat, the non-negativity problem for the EB system is resolved for a certain class of scattering kernels. It is emphasized that it is a challenge to extend the existing theory of the Cauchy problem for the EB system so as to include scattering kernels which are physically well-motivated.

"ETERNAL" SOLUTIONS FOR A MODEL OF THE BOLTZMANN EQUATION

1999 ◽  
Vol 09 (01) ◽  
pp. 127-137 ◽  
Author(s):  
HENRI CABANNES
Keyword(s):  
Cauchy Problem ◽  
Boltzmann Equation ◽  
Plane Model ◽  
The Boltzmann Equation ◽  
Eternal Solutions ◽  
The Cauchy Problem

This paper deals with the analysis of the so-called "eternal" solutions to the Cauchy problem for a semidiscrete plane model of the Boltzmann equation. By eternal solutions we mean solutions existing globally for both positive and negative values of time.

scholarly journals THE BOLTZMANN EQUATION WITHOUT ANGULAR CUTOFF IN THE WHOLE SPACE: II, GLOBAL EXISTENCE FOR HARD POTENTIAL

2011 ◽  
Vol 09 (02) ◽  
pp. 113-134 ◽  
Author(s):  
R. ALEXANDRE ◽  
Y. MORIMOTO ◽  
S. UKAI ◽  
C.-J. XU ◽  
T. YANG
Keyword(s):  
Cauchy Problem ◽  
Boltzmann Equation ◽  
Global Existence ◽  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Existence Of Solution ◽  
The Boltzmann Equation ◽  
The Cauchy Problem ◽  
Whole Space ◽  
Global Equilibrium

As a continuation of our series works on the Boltzmann equation without angular cutoff assumption, in this part, the global existence of solution to the Cauchy problem in the whole space is proved in some suitable weighted Sobolev spaces for hard potential when the solution is a small perturbation of a global equilibrium.

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