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Global existence in \(L^ 1\) for the Enskog equation and convergence of the solutions to solutions of the Boltzmann equation. (English) Zbl 0780.76066

Since R. J. DiPerna and P.-L. Lions [Ann. of Math., II. Ser. 130, No. 2, 321-366 (1989; Zbl 0698.45010)] proved the first global existence theorem in \(L^ 1\) with large initial data for the Boltzmann equation, their method has been used by several people. In this paper, making use of the same ideas, the authors give an existence theorem for the Enskog equation in a box, in \(L^ 1\) for initial data \(f_ 0\) with finite mass, energy and entropy. They also indicate that the result can be extended to the case of \(\mathbb{R}^ 3\), provided one additional assumption, that \(x^ 2 f_ 0\in L^ 1\), is satisfied. In the second part of the paper, they prove the weak convergence of solutions of the Enskog equation to solutions of the Boltzmann equation, by letting the diameter \(\sigma\) of the particles tend to zero.

MSC:

76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
45K05 Integro-partial differential equations

Citations:

Zbl 0698.45010
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References:

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