Abstract
Using the method of moments, we prove that any polynomial moment of the solution of the homogeneous Boltzmann equation with hard potentials or hard spheres is bounded provided that a moment of order strictly higher than 2 exists initially. We also give partial results of convergence towards the Maxwellian equilibrium in the case of soft potentials. Finally, exponential as well as Maxwellian estimates are introduced for the Kac equation.
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References
L. Arkeryd, On the Boltzmann equation. I and II, Arch. Rational Mech. Anal. 45 (1972), 1–34.
L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation, Arch. Rational Mech. Anal. 77 (1981), 11–21.
L. Arkeryd, Stability in L 1 for the spatially homogeneous Boltzmann equation, Arch. Rational Mech. Anal. 103 (1988), 151–167.
A. V. Bobylev, Exact solutions of the nonlinear Boltzmann equation and the theory of relaxation of a Maxwellian gas, Teor. Math. Phys. 60 (1984), 280.
C. Cercignani, The Boltzmann equation and its applications, Berlin (1988).
S. Chapman & T. G. Cowling, The mathematical theory of non-uniform gases, London (1952).
L. Desvillettes, Convergence to equilibrium in large time for Boltzmann and B.G.K. equations, Arch. Rational Mech. Anal. 110 (1990), 73–91.
L. Desvillettes, Convergence to equilibrium in various situations for the solution of the Boltzmann equation. To appear in the Proceedings of the International Workshop on Nonlinear Kinetic Theory and Fluid Mechanics held at Rapallo in April 1992.
L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing, Tr. Th. and Stat. Phys. 21 (1992), 259–276.
R. DiPerna & P.-L. Lions, On the Cauchy problem for Boltzmann equation, Global existence and weak stability, Ann. Math 130 (1989), 321–366.
R. DiPerna & P.-L. Lions, Global solutions of Boltzmann equation and the entropy inequality, Arch. Rational Mech. Anal. 114 (1991), 47–55.
T. Elmroth, Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range, Arch. Rational Mech. Anal. 82 (1983), 1–12.
T. Elmroth, On the H-function and convergence towards equilibrium for a space homogeneous molecular density, SIAM J. Appl. Math. 44 (1984), 150–159.
H. Grad, Principles of the kinetic theory of gases, Flügge's Handbuch der Physik 12 (1958), 205–294.
M. Kac, Probality and related topics in the physical sciences, New York (1959).
H. P. McKean, Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas, Arch. Rational Mech. Anal. 21 (1966), 347–367.
R. Pettersson, Existence theorems for the linear, space inhomogeneous transport equation, IMA J. Appl. Math. 30 (1983), 81–105.
R. Pettersson, On solutions and higher moments for the linear Boltzmann equation with infinite range forces, IMA J. Appl. Math. 38 (1987), 151–166.
C. Truesdell, On the pressures and the flux of energy in a gas according to Maxwell's kinetic theory II, J. Rational Mech. Anal. 5 (1956), 55–128.
C. Truesdell & R. Muncaster, Fundamentals of Maxwell kinetic theory of a simple monoatomic gas, New York (1980).
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Desvillettes, L. Some applications of the method of moments for the homogeneous Boltzmann and Kac equations. Arch. Rational Mech. Anal. 123, 387–404 (1993). https://doi.org/10.1007/BF00375586
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DOI: https://doi.org/10.1007/BF00375586