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Rates of convergence to equilibrium for collisionless kinetic equations in slab geometry. (English) Zbl 1401.82030

The paper addresses the existence of an invariant density and rates of convergence to equilibrium for one-dimensional collisionless (named free) transport equations, with mass preserving partly diffuse boundary operators. In [Acta Appl. Math. 147, No. 1, 19–38 (2017; Zbl 1369.82030)], M. Mokhtar-Kharroubi and R. Rudnicki have developed a convergence theory to equilibrium for a general class of monoenergetic free transport equations in slab geometry with azimuthal symmetry and with abstract boundary operators. In this model, the existence of an invariant density is established for a general class of partly diffuse boundary operators.
It is the departure point for the more refined analysis of the present paper, whose major purpose is to derive a quantified version (with algebraic rates) of the convergence theory, by employing a quantified version of Ingham’s Tauberian theorem [R. Chill and D. Seifert, Bull. Lond. Math. Soc. 48, No. 3, 519–532 (2016; Zbl 1359.40005)]. A general theory is provided, based on some natural structural conditions on the boundary operators in the vicinity of the tangential velocities to the slab. The analysis is restricted to monoenergetic models, albeit there is a mention (Remark \(33\)) that non-monoenergetic free models in slab geometry are amenable to similar methods.
The most important result of the paper is Theorem 29, which provides rigorous estimates for the rates of convergence with \(L^1\) initial data. The authors point out that the developed formalism in principle can be extended to multidimensional geometries with partly diffuse boundary operators. As a byproduct of the construction, a number of preliminary results of independent interest are given and some related open problems are pointed out. For comparative reasons it is also pointed out that there exists a substantial literature on rates of convergence to equilibrium for collisional (linear or nonlinear) kinetic equations, relying mostly on entropy methods. In particular, collisional kinetic equations with soft potentials are known to exhibit algebraic rates of convergence.

MSC:

82C40 Kinetic theory of gases in time-dependent statistical mechanics
82C70 Transport processes in time-dependent statistical mechanics
35R60 PDEs with randomness, stochastic partial differential equations
40E05 Tauberian theorems
47D06 One-parameter semigroups and linear evolution equations
35Q20 Boltzmann equations
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References:

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