The Boltzmann equation: Continuity of the solution with respect to initial data

1980 ◽  
Vol 4 (2) ◽  
pp. 123-130 ◽  
Author(s):  
R. Semenzato
Keyword(s):  
Boltzmann Equation ◽  
Initial Data ◽  
The Boltzmann Equation

scholarly journals Incompressible hydrodynamic approximation with viscous heating to the Boltzmann equation

2017 ◽  
Vol 27 (12) ◽  
pp. 2261-2296 ◽  
Author(s):  
Yan Guo ◽  
Shuangqian Liu
Keyword(s):  
Boltzmann Equation ◽  
Initial Data ◽  
Kinetic Equations ◽  
Navier Stokes ◽  
Viscous Heating ◽  
Text Setting ◽  
Hydrodynamic Approximation ◽  
Energy Conversation ◽  
The Boltzmann Equation

The incompressible Navier–Stokes–Fourier (INSF) system with viscous heating was first derived from the Boltzmann equation in the form of the diffusive scaling by Bardos–Levermore–Ukai–Yang [Kinetic equations: Fluid dynamical limits and viscous heating, Bull. Inst. Math. Acad. Sin.[Formula: see text] 3 (2008) 1–49]. The purpose of this paper is to justify such an incompressible hydrodynamic approximation to the Boltzmann equation in [Formula: see text] setting in a periodic box. Based on an odd–even expansion of the solution with respect to the microscopic velocity, the diffusive coefficients are determined by the INSF system with viscous heating and the super-Burnett functions. More importantly, the remainder of the expansion is proven to decay exponentially in time via an [Formula: see text] approach on the condition that the initial data satisfies the mass, momentum and energy conversation laws.

scholarly journals Local well-posedness of the Boltzmann equation with polynomially decaying initial data

2020 ◽  
Vol 13 (4) ◽  
pp. 837-867 ◽  
Author(s):  
Christopher Henderson ◽  
◽  
Stanley Snelson ◽  
Andrei Tarfulea ◽  
◽  
...  
Keyword(s):  
Boltzmann Equation ◽  
Initial Data ◽  
Well Posedness ◽  
The Boltzmann Equation

Generalized Invariant Sets for the Boltzmann Equation

1997 ◽  
Vol 07 (04) ◽  
pp. 457-476 ◽  
Author(s):  
T. Goudon
Keyword(s):  
Boltzmann Equation ◽  
Initial Data ◽  
Global Existence ◽  
Existence Of Solutions ◽  
Invariant Set ◽  
Invariant Sets ◽  
Soft Or ◽  
Initial Value ◽  
Global Existence Of Solutions ◽  
The Boltzmann Equation

We are interested in the initial value problem for the Boltzmann equation, when the initial data u0 belongs to a set B0 = {δ0m1 (0,x,v) ≤ u0(x,v) ≤ C0m2 (0,x,v)} where m1, m2 are traveling Maxwellians. We consider soft or Maxwell's interactions with cutoff (7/3 < s ≤ 5) and C0 smaller than a bound depending on the coefficients of m2. We obtain global existence of solutions remaining in a "generalized invariant set" Bt ⊂ B∞, characterized by these particular states.

The Boltzmann Equation and the H Theorem (1872–1875)

2018 ◽  
Author(s):  
Olivier Darrigol
Keyword(s):  
Heat Conduction ◽  
Boltzmann Equation ◽  
Transport Phenomena ◽  
Dilute Gas ◽  
The Boltzmann Equation ◽  
H Theorem ◽  
Irreversible Evolution

This chapter covers Boltzmann’s writings about the Boltzmann equation and the H theorem in the period 1872–1875, through which he succeeded in deriving the irreversible evolution of the distribution of molecular velocities in a dilute gas toward Maxwell’s distribution. Boltzmann also used his equation to improve on Maxwell’s theory of transport phenomena (viscosity, diffusion, and heat conduction). The bulky memoir of 1872 and the eponymous equation probably are Boltzmann’s most famous achievements. Despite the now often obsolete ways of demonstration, despite the lengthiness of the arguments, and despite hidden difficulties in the foundations, Boltzmann there displayed his constructive skills at their best.

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