scholarly journals A stochastic particle system associated with the spatially inhomogeneous Boltzmann equation

1994 ◽  
Vol 23 (4) ◽  
pp. 455-477 ◽  
Author(s):  
W. Wagner
Keyword(s):  
Boltzmann Equation ◽  
Particle System ◽  
Spatially Inhomogeneous ◽  
Stochastic Particle System ◽  
Spatially Inhomogeneous Boltzmann Equation ◽  
Stochastic Particle

scholarly journals Derivation of a kinetic model from a stochastic particle system

2008 ◽  
Vol 1 (4) ◽  
pp. 557-572 ◽  
Author(s):  
Pierre Degond ◽  
◽  
Simone Goettlich ◽  
Axel Klar ◽  
Mohammed Seaid ◽  
...  
Keyword(s):  
Kinetic Model ◽  
Particle System ◽  
Stochastic Particle System ◽  
Stochastic Particle

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.

scholarly journals Propagation of chaos for the Vlasov–Poisson–Fokker–Planck equation with a polynomial cut-off

2019 ◽  
Vol 21 (04) ◽  
pp. 1850039 ◽  
Author(s):  
José A. Carrillo ◽  
Young-Pil Choi ◽  
Samir Salem
Keyword(s):  
Particle System ◽  
Planck Equation ◽  
Newtonian Potential ◽  
Empirical Measure ◽  
Propagation Of Chaos ◽  
Interaction Forces ◽  
Fokker Planck Equation ◽  
Stochastic Particle System ◽  
Stochastic Particle ◽  
Fokker Planck

We consider a [Formula: see text]-particle system interacting through the Newtonian potential with a polynomial cut-off in the presence of noise in velocity. We rigorously prove the propagation of chaos for this interacting stochastic particle system. Taking the cut-off like [Formula: see text] with [Formula: see text] in the force, we provide a quantitative error estimate between the empirical measure associated to that [Formula: see text]-particle system and the solutions of the [Formula: see text]-dimensional Vlasov–Poisson–Fokker–Planck (VPFP) system. We also study the propagation of chaos for the Vlasov–Fokker–Planck equation with less singular interaction forces than the Newtonian one.

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.

Well-Posedness of the Second-Order SDEs Describing an N-Particle System Interacting via Coulomb Interaction

2020 ◽  
Author(s):  
Rong Yang
Keyword(s):  
Finite Time ◽  
Singular Potential ◽  
Particle System ◽  
Second Order ◽  
Time Interval ◽  
Well Posedness ◽  
Finite Time Interval ◽  
Interacting Particle ◽  
Stochastic Particle System ◽  
Stochastic Particle

This paper concerns a second-order N interacting stochastic particle system with singular potential for any dimension n ≥ 2. By some estimates of total energy of the system, we prove that there is no collision among particles almost surely in any finite time interval, then the well-posedness of this interacting particle system can be established.

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