scholarly journals Mathematical treatment of the homogeneous Boltzmann equation for Maxwellian molecules in the presence of singular kernels

2014 ◽  
Vol 194 (6) ◽  
pp. 1707-1732
Author(s):  
Emanuele Dolera
Keyword(s):  
Boltzmann Equation ◽  
Mathematical Treatment ◽  
Homogeneous Boltzmann Equation ◽  
Singular Kernels ◽  
Maxwellian Molecules

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.

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.

scholarly journals Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction

2017 ◽  
Vol 10 (4) ◽  
pp. 901-924 ◽  
Author(s):  
Jean-Marie Barbaroux ◽  
◽  
Dirk Hundertmark ◽  
Tobias Ried ◽  
Semjon Vugalter ◽  
...  
Keyword(s):  
Boltzmann Equation ◽  
Homogeneous Boltzmann Equation ◽  
Type Interaction ◽  
Maxwellian Molecules

LITTLEWOOD–PALEY THEORY AND REGULARITY ISSUES IN BOLTZMANN HOMOGENEOUS EQUATIONS I: NON-CUTOFF CASE AND MAXWELLIAN MOLECULES

2005 ◽  
Vol 15 (06) ◽  
pp. 907-920 ◽  
Author(s):  
RADJESVARANE ALEXANDRE ◽  
MOUHAMAD EL SAFADI
Keyword(s):  
Weak Solution ◽  
Boltzmann Equation ◽  
Initial Data ◽  
Cross Sections ◽  
Homogeneous Boltzmann Equation ◽  
Collision Cross Sections ◽  
Maxwellian Molecules

We use Littlewood–Paley theory for the analysis of regularization properties of solutions of the homogeneous Boltzmann equation. For non-cutoff Maxwellian molecules, we show that such solutions are smoother than the initial data. Although the recent and up to date results of Desvillettes and Wennberg10 assume much more general assumptions on the collision cross sections, our method seems to be simpler than those used earlier to show such properties and also applies to any weak solution.

scholarly journals SPATIALLY HOMOGENEOUS MAXWELLIAN MOLECULES IN A NEIGHBORHOOD OF THE EQUILIBRIUM

2014 ◽  
Author(s):  
Emanuele Dolera
Keyword(s):  
Boltzmann Equation ◽  
Exponential Convergence ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Homogeneous Boltzmann Equation ◽  
Long Time ◽  
Spatially Homogeneous ◽  
Maxwellian Molecules ◽  
Spatially Homogeneous Boltzmann Equation

This note deals with the long-time behavior of the solution to the spatially homogeneous Boltzmann equation for Maxwellian molecules, when the initial datum belongs to a suitable neighborhood of the Maxwellian equilibrium. In particular, it contains a quantification of the rate of exponential convergence, obtained by simple arguments.

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