scholarly journals The Cauchy problem for the radially symmetric homogeneous Boltzmann equation with Shubin class initial datum and Gelfand–Shilov smoothing effect

2017 ◽  
Vol 263 (8) ◽  
pp. 5120-5150 ◽  
Author(s):  
Hao-Guang Li ◽  
Chao-Jiang Xu
Keyword(s):  
Cauchy Problem ◽  
Boltzmann Equation ◽  
Smoothing Effect ◽  
Homogeneous Boltzmann Equation ◽  
The Cauchy Problem

scholarly journals Gevrey Regularity for the Noncutoff Nonlinear Homogeneous Boltzmann Equation with Strong Singularity

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Shi-you Lin
Keyword(s):  
Cauchy Problem ◽  
Boltzmann Equation ◽  
Gevrey Regularity ◽  
Homogeneous Boltzmann Equation ◽  
Strong Singularity ◽  
The Cauchy Problem ◽  
Spatially Homogeneous ◽  
Spatially Homogeneous Boltzmann Equation

The Cauchy problem of the nonlinear spatially homogeneous Boltzmann equation without angular cutoff is studied. By using analytic techniques, one proves the Gevrey regularity of theC∞solutions in non-Maxwellian and strong singularity cases.

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.

Smoothing effect on the damping mechanism for an inviscid two-phase gas—liquid model

2014 ◽  
Vol 144 (2) ◽  
pp. 351-362 ◽  
Author(s):  
Lizhi Ruan
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
External Force ◽  
Smooth Solution ◽  
Smoothing Effect ◽  
Two Phase ◽  
Global Smooth Solution ◽  
Liquid Model ◽  
The Cauchy Problem

In this paper, we consider the Cauchy problem for an inviscid two-phase gas—liquid model with external force, in order to demonstrate the smoothing effect on the damping mechanism. It is shown that the Cauchy problem admits a unique global smooth solution provided that the initial data are smooth and the C0-norm of the derivative of the initial data are small.

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