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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.

##### MSC:
 45K05 Integro-partial differential equations 82C40 Kinetic theory of gases in time-dependent statistical mechanics
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##### References:
 [1] L. Arkeryd. On the long time behaviour of the Boltzmann equation in a periodic box. Preprint. [2] C. Bardos, F. Golse & D. Levermore. In preparation, personal communication. [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 [4] R. J. DiPerna & P. L. Lions, Solutions globales de l’équation de Boltzmann. C. R. Acad. Sci. Paris 306 (1988), pp. 343-346. · Zbl 0662.35016 [5] R. J. DiPerna & P. L. Lions. Solutions globales de l’équation de Boltzmann. In Séminaire Equations aux Dérivées Partielles, École Polytechnique, Palaiseau, 1987-88. [6] P. Gérard. In Séminaire Bourbaki, Astérisque, SMF, Paris, 1988. [7] K. Hamdache. In preparation. [8] J. Polewczak. Global existence in L 1 for the modified nonlinear Enskog equation in ?3. Preprint. · Zbl 0719.35071
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