Convergence to self-similar solutions for the homogeneous Boltzmann equation. (English) Zbl 1379.82019
Authors’ abstract: The Boltzmann H-theorem implies that the solution to the Boltzmann equation tends to an equilibrium, that is, a Maxwellian when time tends to infinity. This has been proved in varies settings when the initial energy is finite. However, when the initial energy is infinite, the time asymptotic state is no longer described by a Maxwellian, but a self-similar solution obtained by Bobylev-Cercignani. The purpose of this paper is to rigorously justify this for the spatially homogeneous problem with a Maxwellian molecule type cross section without angular cutoff.
MSC:
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
| 35Q20 | Boltzmann equations |
| 35C06 | Self-similar solutions to PDEs |
References:
| [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 MR 1765272 · Zbl 0968.76076 |
| [2] | Arkeryd, L.: Asymptotic behaviour of the Boltzmann equation with infinite range forces. Comm. Math. Phys. 86, 475-484 (1982)Zbl 0514.35075 MR 0679196 · Zbl 0514.35075 |
| [3] | Bobylev, A. V.: The method of the Fourier transform in the theory of the Boltzmann equation for Maxwell molecules. Dokl. Akad. Nauk SSSR 225, 1041-1044 (1975) (in Russian) Zbl 0361.76077 MR 0398379 |
| [4] | Bobylev, A. V.: The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules. In: Mathematical Physics Reviews, Vol. 7, Soviet Sci. Rev. Sect. C Math. Phys. Rev. 7, Harwood, Chur, 111-233 (1988)Zbl 0850.76619 MR 1128328 · Zbl 0850.76619 |
| [5] | Bobylev, A. V., Cercignani, C.: Exact eternal solutions of the Boltzmann equation. J. Statist. Phys. 106, 1019-1038 (2002)Zbl 1001.82090 MR 1889600 · Zbl 1001.82090 |
| [6] | Bobylev, A. V., Cercignani, C.: Self-similar solutions of the Boltzmann equation and their applications. J. Statist. Phys. 106, 1039-1071 (2002)Zbl 1001.82091 MR 1889601 · Zbl 1001.82091 |
| [7] | Cannone, M., Karch, G.: Infinite energy solutions to the homogeneous Boltzmann equation. Comm. Pure Appl. Math. 63, 747-778 (2010)Zbl 1205.35180 MR 2649362 · Zbl 1205.35180 |
| [8] | Cannone, M., Karch, G.: On self-similar solutions to the homogeneous Boltzmann equation. Kinetic Related Models 6, 801-808 (2013)Zbl 1302.82084 MR 3177629 · Zbl 1302.82084 |
| [9] | Gabetta, E., Toscani, G., Wennberg, B.: Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation. J. Statist. Phys. 81, 901-934 (1995) Zbl 1081.82616 MR 1361302 · Zbl 1081.82616 |
| [10] | Ikenberry, E., Truesdell, C.: On the pressure and the flux of energy according to Maxwell’s kinetic energy I. J. Ration. Mech. Anal. 5, 1-54 (1956)Zbl 0070.23504 MR 0075725 · Zbl 0070.23504 |
| [11] | Morimoto, Y.: A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules. Kinetic Related Models 5, 551-561 (2012)Zbl 1255.35177 MR 2972452 · Zbl 1255.35177 |
| [12] | Morimoto, Y., Wang, S.-K., Yang, T.: A new characterization and global regularity of infinite energy solutions to the homogeneous Boltzmann equation. J. Math. Pures Appl. (9) 103, 809- 829 (2015)Zbl 1308.35155 MR 3310275 · Zbl 1308.35155 |
| [13] | Morimoto, Y., Yang, T.: Smoothing effect of the homogeneous Boltzmann equation with measure initial datum. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 32, 429-442 (2015) Zbl 1321.35125 MR 3325244 · Zbl 1321.35125 |
| [14] | Pulvirenti, A., Toscani, G.: The theory of the nonlinear Boltzmann equation for Maxwell molecules in Fourier representation. Ann. Mat. Pura Appl. 171, 181-204 (1996) Zbl 0874.45005 MR 1441870 · Zbl 0874.45005 |
| [15] | Tanaka, H.: Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Gebiete 46, 67-105 (1978)Zbl 0389.60079 MR 0512334 · Zbl 0389.60079 |
| [16] | Toscani, G., Villani, C.: Probability metrics and uniqueness of the solution to the Boltzmann equations for Maxwell gas. J. Statist. Phys. 94, 619-637 (1999)Zbl 0958.82044 MR 1675367 · Zbl 0958.82044 |
| [17] | Villani, C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal. 143, 273-307 (1998)Zbl 0912.45011 MR 1650006 · Zbl 0912.45011 |
| [18] | Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Handbook of Mathematical Fluid Dynamics, Vol. I, North-Holland, Amsterdam, 71-305 (2002) Zbl 1170.82369 MR 1942465 · Zbl 1170.82369 |
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