zbMATH — the first resource for mathematics

On the spatially homogeneous Boltzmann equation. (English) Zbl 0946.35075
This interesting paper deals with the spatially homogeneous Boltzmann equation \[ \partial f/\partial t(t,v)= Q(f,f)(t,v),\;(t,v)\in (0,+ \infty) \times \mathbb{R}^3,\;f(0,v)= f_0(v),\;v\in\mathbb{R}^3, \] where \(f(t,v)\) is a nonnegative function, which describes the time evolution of the distribution of particles, which move with velocity \(v\), and \(Q(f,f)(t,v)\) is the collision operator. The main result is that for any initial data in \(L^1_2(\mathbb{R}^3)\) with finite mass and energy, there exists a unique solution \(f\in C([0, +\infty)\); \(L^1_2 (\mathbb{R}^3))\) for which the same two quantities are conserved. Here \(L^1_2 (\mathbb{R}^3)\) denotes the space of all functions \(f\) such that \(\|f\|_{1,s}= \int_{\mathbb{R}^3} f(v)(1+|v|^2)dv\) is bounded. Another interesting statement is that any solution satisfying certain bounds on moments of order \(s<2\) must necessarily have bounded energy. A time discretization of the Boltzmann equation with \(f_0\in L^1_s (\mathbb{R}^3)\) \((s\geq 2)\) is considered as well. Some estimates of the rate of convergence for the explicit and implicit Euler schemes are given. Two auxiliary results are of independent interest: a sharpened form of the Povzner inequality, and a regularity result for an iterated gain term.

35Q35 PDEs in connection with fluid mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
Full Text: DOI Numdam EuDML
[1] Arkeryd, L., On the Boltzmann equation, Arch. rational mech. anal., Vol. 34, 1-34, (1972)
[2] Carleman, T., Problèmes mathématiques dans la théorie cinétique des gaz, (1957), Almqvist and Wiksell Uppsala · Zbl 0077.23401
[3] Cercignani, C., The theory and application of the Boltzmann equation, (1988), Springer New York
[4] Cercignani, C.; Illner, R.; Pulvirenti, M., The mathematical theory of dilute gases, (1994), Springer New York · Zbl 0813.76001
[5] Desvillettes, L., Some applications of the method of moments for the homogeneous Boltzmann and Kac equation, Arch. rational mech. anal., Vol. 123, n^{o} 4, 387-395, (1993) · Zbl 0784.76081
[6] Diblasio, G., Differentiability of spatially homogeneous solutions of the Boltzmann equation, Commun. math. phys., Vol. 38, 331-340, (1974)
[7] Elmroth, T., Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range, Arch. rational mech. anal., Vol. 82, 1-12, (1983) · Zbl 0503.76091
[8] Gabetta, E.; Pareschi, L.; Toscani, G., Relaxation schemes for nonlinear kinetic equations, SIAM J. num. anal., Vol. 34, 2168-2194, (1997) · Zbl 0897.76071
[9] Gustafsson, T., Global Lp-properties for the spatially homogeneous Boltzmann equation, Arch. rational mech. anal., Vol. 103, 1-38, (1988) · Zbl 0656.76067
[10] Lions, P.-L.; Lions, P.-L.; Lions, P.-L., Compactness in Boltzmann’s equation via Fourier integral operators and applications I-III, J. math. Kyoto univ., J. math. Kyoto univ., J. math. Kyoto univ., Vol. 34, n^{o}. 3, 539-584, (1994) · Zbl 0884.35124
[11] Povzner, A.J., About the Boltzmann equation in kinetic gas theory, Mat. sborn, Vol. 58, 65-86, (1962) · Zbl 0128.22502
[12] Ringeisen, E., Contributions à l’étude mathématique des equations cinétiques, ()
[13] Wennberg, B., On moments and uniqueness for solutions to the space homogeneous Boltzmann equation, Transport theory stat. phys., Vol. 24, 4, 533-539, (1994) · Zbl 0812.76080
[14] {\scB. Wennberg}, Entropy dissipation and moment production for the Boltzmann equation, to appear in J. Statist. Phys. · Zbl 0935.82035
[15] Wennberg, B., The povzner inequality and moments in the Boltzmann equation, Rendiconti del circolo matematico di Palermo, ser II, suppl. 45, 673-681, (1996) · Zbl 0909.76089
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.