CHARACTERISTICS METHOD FOR SOLVING THE SPATIALLY INHOMOGENEOUS BOLTZMANN EQUATION

1993 ◽  
Vol 12 (2) ◽  
pp. 125-141
Author(s):  
B.J. GEURTS
Keyword(s):  
Boltzmann Equation ◽  
Characteristics Method ◽  
Spatially Inhomogeneous ◽  
Spatially Inhomogeneous Boltzmann Equation

UNIFORM Lp-STABILITY ESTIMATE FOR THE SPATIALLY INHOMOGENEOUS BOLTZMANN EQUATION NEAR VACUUM

2008 ◽  
Vol 05 (04) ◽  
pp. 713-739 ◽  
Author(s):  
SEUNG-YEAL HA ◽  
MITSURU YAMAZAKI ◽  
SEOK-BAE YUN
Keyword(s):  
Stability Analysis ◽  
Boltzmann Equation ◽  
Stability Estimate ◽  
Functional Approach ◽  
Classical Solutions ◽  
Nonlinear Functional ◽  
Spatially Inhomogeneous ◽  
Nonlinear Functionals ◽  
Spatially Inhomogeneous Boltzmann Equation

We present a new uniform Lp-stability theory for the spatially inhomogeneous Boltzmann equation near vacuum via the nonlinear functional approach proposed by the first author. Our stability analysis is based on new nonlinear functionals which are equivalent to the pth power of Lp-distance. The L1-nonlinear functionals play the key role of "modulators" which make the accumulative functional be non-increasing in time t along classical solutions.

THE BOLTZMANN EQUATION AND THE GROUP TRANSFORMATIONS

1993 ◽  
Vol 03 (04) ◽  
pp. 443-476 ◽  
Author(s):  
A.V. BOBYLEV
Keyword(s):  
Boltzmann Equation ◽  
Special Role ◽  
Spatially Inhomogeneous ◽  
The Boltzmann Equation ◽  
Symmetry Properties ◽  
Full Equation ◽  
Spatially Inhomogeneous Boltzmann Equation ◽  
Spatially Homogeneous ◽  
Spatially Homogeneous Boltzmann Equation

This paper is devoted to the investigation of group properties of the nonlinear Boltzmann equation. The complete Lie group of invariant transformations for the spatially inhomogeneous Boltzmann equation is constructed. The generalization to the Lie-Backlund groups is given for the spatially homogeneous case. It is shown that there are only two non-trivial group transformations for the Boltzmann equation in the wide class of Lie and Lie-Backlund transformations. Some consequences of these symmetry properties are discussed. The special role of Galileo group and the analogy between the spatially homogeneous Boltzmann equation and the full equation are also investigated.

scholarly journals PROPAGATION OF SINGULARITIES IN THE SOLUTIONS TO THE BOLTZMANN EQUATION NEAR EQUILIBRIUM

2008 ◽  
Vol 18 (07) ◽  
pp. 1093-1114 ◽  
Author(s):  
RENJUN DUAN ◽  
MENG-RONG LI ◽  
TONG YANG
Keyword(s):  
Cauchy Problem ◽  
Boltzmann Equation ◽  
Initial Data ◽  
Cross Section ◽  
The Other ◽  
Spatially Inhomogeneous ◽  
The Cauchy Problem ◽  
Spatially Inhomogeneous Boltzmann Equation ◽  
Propagation Of Singularities ◽  
Derivatives Of

This paper is about the propagation of the singularities in the solutions to the Cauchy problem of the spatially inhomogeneous Boltzmann equation with angular cutoff assumption. It is motivated by the work of Boudin–Desvillettes on the propagation of singularities in solutions near vacuum. It shows that for the solution near a global Maxwellian, singularities in the initial data propagate like the free transportation. Precisely, the solution is the sum of two parts in which one keeps the singularities of the initial data and the other one is regular with locally bounded derivatives of fractional order in some Sobolev space. In addition, the dependence of the regularity on the cross-section is also given.

Global well-posedness theory for the spatially inhomogeneous Boltzmann equation without angular cutoff

2010 ◽  
Vol 348 (15-16) ◽  
pp. 867-871 ◽  
Author(s):  
Radjesvarane Alexandre ◽  
Y. Morimoto ◽  
S. Ukai ◽  
Chao-Jiang Xu ◽  
T. Yang
Keyword(s):  
Boltzmann Equation ◽  
Well Posedness ◽  
Spatially Inhomogeneous ◽  
Spatially Inhomogeneous Boltzmann Equation

RENORMALIZATION AND BOLTZMANN EQUATIONS IN THERMAL QUANTUM FIELD THEORY

1995 ◽  
Vol 10 (11) ◽  
pp. 1693-1700 ◽  
Author(s):  
H. CHU ◽  
H. UMEZAWA
Keyword(s):  
Boltzmann Equation ◽  
Renormalization Scheme ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Boltzmann Equations ◽  
Self Energy ◽  
Spatially Inhomogeneous ◽  
Thermo Field Dynamics ◽  
Self Consistent ◽  
Consistent Manner

The renormalization scheme in nonequilibrium thermal quantum field theories is reexamined. Instead of the self-energy diagonalization scheme, we propose to diagonalize Green’s function at equal time. This eliminates the problem of on-shell definition related to time-dependent energies and spatially inhomogeneous situations, and yields a Boltzmann equation that contains memory effect. The new diagonalization scheme and the derivation of the Boltzmann equation from it can be applied to any thermal situation. It allows the treatment of a nonequilibrium problem beyond perturbational calculations in a self-consistent manner. The results are applicable to both thermo field dynamics and the closed time path formalism.

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