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On asymptotics of the Boltzmann equation when the collisions become grazing. (English) Zbl 0769.76059

(Introduction.) The dynamics of a rarefied monoatomic gas is usually described by the Boltzmann equation: \[ {\partial f\over\partial t}+v\nabla_ xf=Q(f,f) \tag{1} \] where \(f(t,x,v)\) is the density of particles which at time \(t\) and point \(x\) move with velocity \(v\), and \(Q\) is a quadratic collision kernel taking into account any collisions preserving momentum and kinetic energy.
When almost each collision is grazing (i.e. the difference between velocities before and after the collision is very small), phenomenological arguments introduced by Landau or Chapman and Cowling ensure that the solution of equation (1) tends to the solution of the Fokker-Planck-Landau equation: \[ {\partial f\over\partial t}+v\Delta_ xf=P(f,f) \] with \[ P(f,f)=\nabla_ v\cdot\int_{v_ 1\in\mathbb{R}^ 3}\Gamma(| v-v_ 1|)\;\left\{I-{(v-v_ 1)\otimes(v-v_ 1)\over| v-v_ 1|^ 2}\right\} \{f(v_ 1)\nabla_ vf(v)- f(v)\nabla_{v_ 1}f(v_ 1)\}dv_ 1, \] where \(\Gamma\) is a nonnegative function depending only on the form of \(Q\).
The autor introduces an asymptotics of equation (1) leading to equation (2). He also computes the function \(\Gamma\) in some simple cases. In the section 3 he gives a mathematical proof of the above asymptotics, within the context of linearized equations. Finally, in section 4 he extends the previous results to the case of the Kač equation.

MSC:

76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
45K05 Integro-partial differential equations
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[1] DOI: 10.1007/978-1-4612-1039-9
[2] Chapman S., The mathematical theory of non-uniform gases (1939) · Zbl 0063.00782
[3] Degond P., Math. Mod. and Meth. in Appl. Sc.
[4] DOI: 10.2307/1971423 · Zbl 0698.45010
[5] Grad H., Flugge’s Handbuch der, Physik 12 pp 205– (1958)
[6] DOI: 10.1007/BF01211098 · Zbl 0609.76083
[7] DOI: 10.1007/BF01468142 · Zbl 0599.76088
[8] Kač M., Proc. 3rd Berkeley Symposium on Math. Statis. and Probab. 3 pp 171– (1956)
[9] Lifshitz E. M., Physical kinetics (1981)
[10] DOI: 10.1007/BF00264463 · Zbl 1302.60049
[11] Truesdell C., Fundamentals of Maxwell kinetic theory of a simple monoatomic gas (1980)
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