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Well-posedness and large time behaviour for the non-cutoff Kac equation with a Gaussian thermostat. (English) Zbl 1187.82101
Summary: We consider here a Kac equation with a Gaussian thermostat in the case of a non-cutoff cross section. Under the sole assumptions of finite mass and finite energy for the initial data, we prove the existence of a global in time solution for which mass and energy are preserved. Then, via Fourier transform techniques, we show that this solution is smooth, unique and converges to the corresponding stationary state.

82C40Kinetic theory of gases (time-dependent statistical mechanics)
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[1] Alexandre, R., Desvillettes, L., Villani, C., Wennberg, B.: Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152, 327--355 (2000) · Zbl 0968.76076 · doi:10.1007/s002050000083
[2] Bagland, V., Wennberg, B., Wondmagegne, Y.: Stationary states for the noncutoff Kac equation with a Gaussian thermostat. Nonlinearity 20, 583--604 (2007) · Zbl 1171.82311 · doi:10.1088/0951-7715/20/3/003
[3] Bonetto, F., Daems, D., Lebowitz, J.L.: Properties of stationary nonequilibrium states in the thermostatted periodic Lorentz gas: the one particle system. J. Stat. Phys. 101, 35--60 (2000) · Zbl 0973.82024 · doi:10.1023/A:1026414222092
[4] Bonetto, F., Daems, D., Lebowitz, J.L., Ricci, V.: Properties of stationary nonequilibrium states in the thermostatted periodic Lorentz gas: the multiparticle system. Phys. Rev. E 65, 051204 (2002) 9 pages · doi:10.1103/PhysRevE.65.051204
[5] Carlen, E.A., Gabetta, E., Toscani, G.: Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Commun. Math. Phys. 199, 521--546 (1999) · Zbl 0927.76088 · doi:10.1007/s002200050511
[6] Carrillo, J.A., Toscani, G.: Contractive probability metrics and asymptotic behavior of dissipative kinetic equations. Riv. Mat. Univ. Parma 6(7), 75--198 (2007) · Zbl 1142.82018
[7] Chernov, N.I., Eyink, G.L., Lebowitz, J.L., Sinai, Ya.G.: Steady-state electrical conduction in the periodic Lorentz gas. Commun. Math. Phys. 154, 569--601 (1993) · Zbl 0780.58050 · doi:10.1007/BF02102109
[8] Chernov, N.I., Eyink, G.L., Lebowitz, J.L., Sinai, Ya.G.: Derivation of Ohm’s law in a deterministic mechanical model. Phys. Rev. Lett. 70, 2209--2212 (1993) · doi:10.1103/PhysRevLett.70.2209
[9] Dellacherie, C., Meyer, P.A.: Probabilités et Potentiels. Hermann, Paris (1975). Chapitres I à IV · Zbl 0323.60039
[10] Desvillettes, L.: About the regularizing properties of the non-cut-off Kac equation. Commun. Math. Phys. 168, 417--440 (1995) · Zbl 0827.76081 · doi:10.1007/BF02101556
[11] Desvillettes, L., Golse, F.: On the smoothing properties of a model Boltzmann equation without Grad’s cutoff assumption. In: Proceedings of the 21st International Symposium on Rarefied Gas Dynamics, vol. 1, pp. 47--54 (1999)
[12] Desvillettes, L., Wennberg, B.: Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff. Commun. Partial Diff. Equs. 29, 133--155 (2004) · Zbl 1103.82020 · doi:10.1081/PDE-120028847
[13] DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511--547 (1989) · Zbl 0696.34049 · doi:10.1007/BF01393835
[14] Dudley, R.M.: Real Analysis and Probability. Cambridge Studies in Advanced Mathematics, vol. 74. CUP, Cambridge (2002) · Zbl 1023.60001
[15] Dunford, N., Schwartz, J.T.: Linear Operators. Part I: General Theory. Interscience Publishers, New York (1958) · Zbl 0084.10402
[16] Edwards, R.E.: Functional Analysis. Theory and Applications. Holt, Rinehart and Winston, New York/Toronto/London (1965) · Zbl 0182.16101
[17] Evans, D.J., Morriss, G.P.: Statistical Mechanics of Nonequilibrium Liquids. Academic Press, London (1990) · Zbl 1145.82301
[18] Evans, D.J., Hoover, Wm.G., Failor, B.H., Moran, B., Ladd, A.J.C.: Nonequilibrium molecular dynamics via Gauss’s principle of least constraint. Phys. Rev. A 28, 1016--1021 (1983) · doi:10.1103/PhysRevA.28.1016
[19] Fournier, N.: Strict positivity of a solution to a one-dimensional Kac equation without cutoff. J. Stat. Phys. 99, 725--749 (2000) · Zbl 0959.82020 · doi:10.1023/A:1018683226672
[20] Gabetta, E., Pareschi, L.: About the non-cutoff Kac equation: uniqueness and asymptotic behaviour. Commun. Appl. Nonlinear Anal. 4, 1--20 (1997) · Zbl 0873.45006
[21] Gabetta, E., Toscani, G., Wennberg, B.: Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation. J. Stat. Phys. 81, 901--934 (1995) · Zbl 1081.82616 · doi:10.1007/BF02179298
[22] Hoover, Wm.G.: Molecular Dynamics. Lecture Notes in Physics, vol. 258. Springer, Berlin (1986)
[23] Kac, M.: Foundations of kinetic theory. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954--1955, Berkeley and Los Angeles, 1956, vol. III, pp. 171--197. University of California Press, Berkeley (1956)
[24] Laurençot, Ph.: The Lifshitz-Slyozov equation with encounters. Math. Models Methods Appl. Sci. 11, 731--748 (2001) · Zbl 1013.35054 · doi:10.1142/S0218202501001070
[25] Lê Châu-Hoàn: Etude de la classe des opérateurs m-accrétifs de L 1({$\Omega$}) et accrétifs dans L {$\Omega$}). PhD thesis, Université de Paris VI (1977)
[26] Mischler, S., Wennberg, B.: On the spatially homogeneous Boltzmann equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 16, 467--501 (1999) · Zbl 0946.35075 · doi:10.1016/S0294-1449(99)80025-0
[27] Moran, B., Hoover, Wm.G., Bestiale, S.: Diffusion in a periodic Lorentz gas. J. Stat. Phys. 48, 709--726 (1987) · Zbl 1084.82563 · doi:10.1007/BF01019693
[28] Morris, G.P., Dettmann, C.P.: Thermostats: analysis and application. Chaos 8, 321--336 (1998) · Zbl 0977.80002 · doi:10.1063/1.166314
[29] Ruelle, D.: Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J. Stat. Phys. 95, 393--468 (1999) · Zbl 0934.37010 · doi:10.1023/A:1004593915069
[30] Sundén, M., Wennberg, B.: Brownian approximation and Monte Carlo simulation of the non-cutoff Kac equation. J. Stat. Phys. 130, 295--312 (2008) · Zbl 1139.82035 · doi:10.1007/s10955-007-9424-8
[31] Toscani, G., Villani, C.: Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Stat. Phys. 94, 619--637 (1999) · Zbl 0958.82044 · doi:10.1023/A:1004589506756
[32] van Beijeren, H., Dorfman, J.R., Cohen, E.G.D., Posch, H.A., Dellago, Ch.: Lyapunov exponents from kinetic theory for a dilute field driven Lorentz gas. Phys. Rev. Lett. 77, 1974--1977 (1996)
[33] Vrabie, I.I.: Compactness Methods for Nonlinear Evolutions, 2nd edn. Longman Scientific and Technical, Harlow (1995) · Zbl 0842.47040
[34] Wennberg, B., Wondmagegne, Y.: Stationary states for the Kac equation with a Gaussian thermostat. Nonlinearity 17, 633--648 (2004) · Zbl 1049.76058 · doi:10.1088/0951-7715/17/2/016
[35] Wennberg, B., Wondmagegne, Y.: The Kac equation with a thermostatted force field. J. Stat. Phys. 124, 859--880 (2006) · Zbl 1134.82041 · doi:10.1007/s10955-005-9020-8