scholarly journals Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules

2018 ◽  
Vol 11 (3) ◽  
pp. 597-613 ◽  
Author(s):  
Milana Pavić-Čolić ◽  
◽  
Maja Tasković ◽  
Keyword(s):  
Boltzmann Equation ◽  
Stretched Exponential ◽  
Maxwell Molecules ◽  
Exponential Moments ◽  
Kac Equation

Grad's Second Problem and Its Solution Within the Framework of Burnett Hydrodynamics

2020 ◽  
Vol 142 (10) ◽  
Author(s):  
Ravi Sudam Jadhav ◽  
Amit Agrawal
Keyword(s):  
Boltzmann Equation ◽  
Hard Sphere ◽  
Pressure Field ◽  
Higher Order ◽  
Transport Equations ◽  
Normal Solution ◽  
Uniform Pressure ◽  
One Dimensional ◽  
Maxwell Molecules ◽  
Flow Domain

Abstract In his seminal work, Grad not only derived 13 moment equations but also suggested two problems to check his derived equations. These problems are highly instructive as they bring out the character of the equations by examining their solutions to these problems. In this work, we propose Grad's second problem as the potential benchmark problem for checking the accuracy of different sets of higher-order transport equations. The problem definition can be stated as: examination of steady-state solution for a gas at rest in infinite domain upon application of a one-dimensional heat flux. With gas at rest (no bulk velocity), the interest lies in obtaining the solution for pressure and temperature. The problem is particularly interesting with respect to the solution for pressure when Maxwell and hard-sphere molecules are considered. For Maxwell molecules, it is well known that the exact normal solution of Boltzmann equation gives uniform pressure with no stresses in the flow domain. In the case of hard-sphere molecules, direct simulation Monte Carlo (DSMC) results predict nonuniform pressure field giving rise to stresses in the flow domain. The simplistic nature of the problem and interesting results for pressure for different interaction potentials makes it an ideal test problem for examining the accuracy of higher-order transport equations. The proposed problem is solved within the framework of Burnett hydrodynamics for hard-sphere and Maxwell molecules. For hard-sphere molecules, it is observed that the Burnett order stresses do not become zero; they rather give rise to a pressure gradient in a direction opposite to that of temperature gradient, consistent with the DSMC results. For Maxwell molecules, the numerical solution of Burnett equations predicts uniform pressure field and one-dimensional temperature field, consistent with the exact normal solution of the Boltzmann equation.

scholarly journals Uniform moment propagation for the Becker--Döring equations

2018 ◽  
Vol 149 (04) ◽  
pp. 995-1015
Author(s):  
José A. Cãnizo ◽  
Amit Einav ◽  
Bertrand Lods
Keyword(s):  
Maximum Principle ◽  
Rates Of Convergence ◽  
Convergence To Equilibrium ◽  
Stretched Exponential ◽  
The Maximum Principle ◽  
Exponential Moments ◽  
Time Propagation

AbstractWe show uniform-in-time propagation of algebraic and stretched exponential moments for the Becker--Döring equations. Our proof is based upon a suitable use of the maximum principle together with known rates of convergence to equilibrium.

scholarly journals Upper Maxwellian bounds for the Boltzmann equation with pseudo-Maxwell molecules

2017 ◽  
Vol 10 (3) ◽  
pp. 573-585 ◽  
Author(s):  
Alexander V. Bobylev ◽  
◽  
Irene M. Gamba ◽  
Keyword(s):  
Boltzmann Equation ◽  
Maxwell Molecules ◽  
The Boltzmann Equation
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