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Global solutions of Boltzmann’s equation and the entropy inequality. (English) Zbl 0724.45011
This paper is to some extent an addendum to the earlier remarkable paper of the authors [Ann. Math., II. Ser. 130, No.2, 321-366 (1989; Zbl 0698.45010)]. In that paper they use the natural (formal) conservation laws of mass, momentum, and entropy associated with the Boltzmann equation to establish global (in space-time) existence of solutions to a modified version of the Boltzmann equation for data \(f_ 0\) satisfying \[ f_ 0\geq 0,\quad \int_{{\mathbb{R}}^ N\times {\mathbb{R}}^ N}f_ 0(1+| \xi |^ 2+| x|^ 2+| \log f_ 0|)dx d\xi <\infty. \] (The authors termed this form of the Boltzmann equation as “renormalized”.)
The verification of one important stability question was left open in the original paper. If \(\{f_ n\}\) is a sequence of solutions of the renormalized Boltzmann equation corresponding to initial data \(f_{0n}\) at \(t=0\) and \[ \sup_{n}\int_{{\mathbb{R}}^ N\times {\mathbb{R}}^ N}\int dx d\xi f^ n_ 0\{1+| x|^ 2+| \xi |^ 2+\log | f_{0n}| \}<\infty,\quad f_{0n}\geq 0, \] what can be said regarding preservation of equality regarding the rate of dissipation of total entropy?
In this paper the authors prove that the sequence \(\{f_ n\}\) possesses a convergent subsequence, converging to a solution f of the renormalized Boltzmann equation which satisfies the original entropy rate dissipation equality as an inequality. The proof is based on the weak stability theory of the authors’ earlier paper.

45K05 Integro-partial differential equations
82C40 Kinetic theory of gases in time-dependent statistical mechanics
Full Text: DOI
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[3] R. J. DiPerna & P. L. Lions. On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. 130 (1989), pp. 321-366. · Zbl 0698.45010
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