Cooling process for inelastic Boltzmann equations for hard spheres. II: Self-similar solutions and tail behavior. (English) Zbl 1135.82030
Summary: This is the second part of our work with M. Rodriguez Ricard [cf. Part I, J. Stat. Phys. 124, No. 2-4, 655–702 (2006; Zbl 1135.82325)].
We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres, in the framework of so-called constant normal restitution coefficients. We prove the existence of self-similar solutions, and we give pointwise estimates on their tail. We also give general estimates on the tail and the regularity of generic solutions. In particular we prove Haff’s law on the rate of decay of temperature, as well as the algebraic decay of singularities. The proofs are based on the regularity study of a rescaled problem, with the help of the regularity properties of the gain part of the Boltzmann collision integral, well-known in the elastic case, and which are extended here in the context of granular gases.
We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres, in the framework of so-called constant normal restitution coefficients. We prove the existence of self-similar solutions, and we give pointwise estimates on their tail. We also give general estimates on the tail and the regularity of generic solutions. In particular we prove Haff’s law on the rate of decay of temperature, as well as the algebraic decay of singularities. The proofs are based on the regularity study of a rescaled problem, with the help of the regularity properties of the gain part of the Boltzmann collision integral, well-known in the elastic case, and which are extended here in the context of granular gases.
MSC:
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
| 82D05 | Statistical mechanics of gases |
| 82B40 | Kinetic theory of gases in equilibrium statistical mechanics |
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
Citations:
Zbl 1135.82325References:
| [1] | F. Abrahamsson, Strong L 1 convergence to equilibrium without entropy conditions for the Boltzmann equation, Comm. Partial Differential Equations 24:1501–1535 (1999). · Zbl 1059.35014 · doi:10.1080/03605309908821472 |
| [2] | R. Alexandre, L. Desvillettes, C. Villani, and B. Wennberg, Entropy dissipation and long range interactions, Arch. Ration. Mech. Anal. 152:327–355 (2000). · Zbl 0968.76076 · doi:10.1007/s002050000083 |
| [3] | M. Balabane, Équations différentielles, cours de l’École des Ponts et Chaussées (1985). · Zbl 0601.35072 |
| [4] | D. Benedetto, E. Caglioti, and M. Pulvirenti, A kinetic equation for granular media, Math. Mod. Numér. Anal. 31:615–641 (1997). · Zbl 0888.73006 |
| [5] | A. V. Bobylev, Moment inequalities for the Boltzmann equation and applications to the spatially homogeneous problems, J. Statist. Phys. 88:1183–1214 (1997). · Zbl 0979.82049 · doi:10.1007/BF02732431 |
| [6] | A. V. Bobylev, J. A. Carillo, and I. Gamba, On some properties of kinetic and hydrodynamics equations for inelastic interactions, J. Statist. Phys. 98:743–773 (2000). · Zbl 1056.76071 · doi:10.1023/A:1018627625800 |
| [7] | A. V. Bobylev and C. Cercignani, Self-Similar asymptotics for the Boltzmann equation with Inelastic and elastic interactions, J. Statist. Phys. 110:333–375 (2003). · Zbl 1134.82324 |
| [8] | A. V. Bobylev, C. Cercignani, and G. Toscani, Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials, J. Statist. Phys. 111:403–417 (2003). · Zbl 1119.82318 |
| [9] | A. V. Bobylev, I. Gamba, and V. Panferov, Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions, J. Statist. Phys. 116:1651–1682 (2004). · Zbl 1097.82021 · doi:10.1023/B:JOSS.0000041751.11664.ea |
| [10] | F. Bouchut and L. Desvillettes, A proof of smoothing properties of the positive part of Boltzmann’s kernel, Rev. Mat. Iberoamericana 14:47–61 (1998). · Zbl 0912.45014 |
| [11] | N. V. Brilliantov and T. Pöeschel, Kinetic Theory of Granular Gases (Oxford Graduate Texts. Oxford University Press, Oxford, 2004). |
| [12] | E. Caglioti and C. Villani, Homogeneous Cooling States are not always good approximations to granular flows, Arch. Ration. Mech. Anal. 163:329–343 (2002). · Zbl 1053.74012 · doi:10.1007/s002050200204 |
| [13] | T. Carleman, Sur la théorie de l’équation intégrodifférentielle de Boltzmann, Acta Math. 60 91–146 (1932). |
| [14] | C. Cercignani, Recent developments in the mechanics of granular materials, in Fisica matematica e ingegneria delle strutture (Pitagora Editrice, Bologna, 1995), pp. 119–132. |
| [15] | L. Desvillettes and C. Mouhot, About L p estimates for the spatially homogeneous Boltzmann equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 22:127–142 (2005). · Zbl 1077.76060 · doi:10.1016/j.anihpc.2004.03.002 |
| [16] | M. H. Ernst and R. Brito, Driven inelastic Maxwell molecules with high energy tails, Phys. Rev. E 65:1–4 (2002). · Zbl 1015.82030 · doi:10.1103/PhysRevE.65.040301 |
| [17] | M. H. Ernst and R. Brito, Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails, J. Statist. Phys. 109:407–432 (2002). · Zbl 1015.82030 · doi:10.1023/A:1020437925931 |
| [18] | M. Escobedo, S. Mischler, and M. Rodriguez Ricard, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire 22:99–125 (2005). · Zbl 1130.35025 · doi:10.1016/j.anihpc.2004.06.001 |
| [19] | I. Gamba, V. Panferov, and C. Villani, On the Boltzmann equation for diffusively excited granular media, Comm. Math. Phys. 246:503–541 (2004). · Zbl 1106.82031 · doi:10.1007/s00220-004-1051-5 |
| [20] | I. Gamba, V. Panferov, and C. Villani Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation. Work in progress. · Zbl 1273.76373 |
| [21] | T. Gustafsson, L p -estimates for the nonlinear spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal. 92:23–57 (1986). · Zbl 0619.76100 |
| [22] | T. Gustafsson, Global L p -properties for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal. 103:1–38 (1988). · Zbl 0656.76067 · doi:10.1007/BF00292919 |
| [23] | H. Grad, Asymptotic theory of the Boltzmann equation. II Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l’UNESCO, Paris, 1962), Vol. I, pp 26–59, New York, 1963. |
| [24] | P. K. Haff, Grain flow as a fluid-mechanical phenomenon, J. Fluid Mech. 134 (1983). · Zbl 0537.76005 |
| [25] | H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Arch. Ration. Mech. Anal. 172:407–428 (2004). · Zbl 1116.82025 |
| [26] | P.-L. Lions, Compactness in Boltzmann’s equation via Fourier integral operators and applications I, II, III, J. Math. Kyoto Univ. 34:391–427, 429–461, 539–584 (1994). · Zbl 0831.35139 |
| [27] | X. Lu, A direct method for the regularity of the gain term in the Boltzmann equation, J. Math. Anal. Appl. 228 (1998), 409–435. · Zbl 0913.76081 · doi:10.1006/jmaa.1998.6141 |
| [28] | S. Mischler, C. Mouhot, and M. Rodriguez Ricard, Cooling process for inelastic Boltzmann equations for hard spheres, Part I: The Cauchy problem, to appear in J. Statist. Phys. · Zbl 1135.82325 |
| [29] | S. Mischler and C. Mouhot, Convergence to self-similarity for a Boltzmann equation of dissipative hard spheres with small inelasticity work in progress · Zbl 1178.82056 |
| [30] | S. Mischler and B. Wennberg, 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 |
| [31] | C. Mouhot, Quantitative lower bound for the full Boltzmann equation, Part I: Periodic boundary conditions, Comm. Partial Differential Equations 30:881–917 (2005). · Zbl 1112.76061 · doi:10.1081/PDE-200059299 |
| [32] | C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Ration. Mech. Anal. 173 (2004), 169–212. · Zbl 1063.76086 · doi:10.1007/s00205-004-0316-7 |
| [33] | A. Pulvirenti and B. Wennberg, A Maxwellian lower bound for solutions to the Boltzmann equation, Comm. Math. Phys. 183:145–160 (1997). · Zbl 0866.76077 · doi:10.1007/BF02509799 |
| [34] | C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, Vol. I, 71–305 (North-Holland, Amsterdam, 2002). · Zbl 1170.82369 |
| [35] | B. Wennberg, Regularity in the Boltzmann equation and the Radon transform, Comm. Partial Differential Equations 19:2057–2074 (1994). · Zbl 0818.35128 · doi:10.1080/03605309408821082 |
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