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On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. (English) Zbl 1162.82316
Summary: As part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations, started in [Commun. Pure Appl. Math. 54, No. 1, 1–42 (2001; Zbl 1029.82032)], we derive estimates on the rate of convergence to equilibrium for solutions of the Boltzmann equation, like \(O(t^{-\infty})\). Our results hold conditionally to some strong but natural estimates of smoothness, decay at large velocities and strict positivity, which at the moment have only been established in certain particular cases.
Among the most important steps in our proof are:
1) quantitative variants of Boltzmann’s \(H\)-theorem, based on symmetry features, hypercontractivity and information-theoretical tools;
2) a new, quantitative version of the instability of the hydrodynamic description for non-small Knudsen number;
3) some functional inequalities with geometrical content, in particular the Korn-type inequality; and
4) the study of a system of coupled differential inequalities of second order.
We also briefly point out the particular role of conformal velocity fields, when they are allowed by the geometry of the problem.

MSC:
82C40 Kinetic theory of gases in time-dependent statistical mechanics
35B40 Asymptotic behavior of solutions to PDEs
35F20 Nonlinear first-order PDEs
Software:
Boltzmann
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References:
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