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Boltzmann asymptotics with diffuse reflection boundary conditions. (English) Zbl 0877.76063
This paper deals with the nonlinear Boltzmann equation \((\partial_t+ \xi \cdot \nabla_x) f=Q(f,f)\), where \(Q\) is the collision operator. The collisionless case is first considered, and then strong \(L^1\)-convergence to a stationary solution when \(t\to+ \infty\) is proved in a bounded domain, with constant temperature on the boundary.

MSC:
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82B40 Kinetic theory of gases in equilibrium statistical mechanics
45K05 Integro-partial differential equations
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[1] Arkeryd, L.: Some examples of NSA methods in kinetic theory. Lecture Notes Math. 1551. Berlin-Heidelburg-New York: Springer 1993. · Zbl 0810.76076
[2] Arkeryd, L., Cercignani, C.: A global existence theorem for the initial-boundary value problem for the Boltzmann equation when the boundaries are not isothermal. Arch. Rat. Mechs. Anal.125, 271-287 (1993). · Zbl 0789.76075
[3] Arkeryd, L., Cercignani, C., Illner, R.: Measure solutions of the steady Boltzmann equation in a slab. Comm. Math. Phys.142, 285-296 (1991). · Zbl 0733.76063
[4] Arkeryd, L., Maslova, N.: On diffuse reflection at the boundary for the Boltzmann equation and related equations. J. Stat. Phys.77 (1994). · Zbl 0839.76073
[5] Arkeryd, L., Nouri, A.: A compactness result related to the stationary Boltzmann equation in a slab, with applications to the existence theory. Ind. Univ. Math. Journ. 44, Nb3, 815-839 (1995). · Zbl 0853.45015
[6] Bose, C., Grzegorczyk, P., Illner, R.: Asymptotic behavior of one-dimensional discrete velocity models in a slab. Preprint. 1993. · Zbl 0829.76076
[7] Carleman, T.: Th?orie cin?tique des gaz. Uppsala: Almqvist & Wiksell. 1957. · Zbl 0077.23401
[8] Cercignani, C.: The Boltzmann Equation and its Applications. Berlin: Springer. 1988. · Zbl 0646.76001
[9] Cercignani, C.: Equilibrium states and trend to equilibrium in a gas, according to the Boltzmann equation. Rend. di Matematica10, 77-95 (1990). · Zbl 0723.76079
[10] Desvillettes, L.: Convergence to equilibrium in large time for Boltzmann and B.G.K. equations. Arch. Rat. Mech. Anal.110, 73-91 (1990). · Zbl 0705.76070
[11] Diperna, R. J. Lions, P. L., Meyer, Y.:L p regularity of velocity averages. Ann. I.H.P. Anal. Non Lin8, 271-287 (1991). · Zbl 0763.35014
[12] Golse, F., Lions, P. L., Perthame, B., Sentis, R.: Regularity of the moments of the solution of a transport equation. J. Funct. Anal.76, 110-125 (1988). · Zbl 0652.47031
[13] Hamdache, K.: Weak solutions of the Boltzmann equation. Arch. Rat. Mechs. Anal.119, 309-353 (1992). · Zbl 0777.76084
[14] Lions, P. L.: Compactness in Boltzmann’s equations via Fourier integral operators and applications. J. Math. Kyoto Univ.34, 391-427 (1994). · Zbl 0831.35139
[15] Pettersson, R.: On weak and strong convergence to equilibrium for solutions to the linear Boltzmann equation. J. Stat. Phys.72, 355-380 (1993). · Zbl 1099.82516
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