We prove propagation of regularity, uniformly in time, for the scaled solutions of the inelastic Maxwell model for any value of the coefficient of restitution. The result follows from the uniform in time control of the tails of the Fourier transform of the solution, normalized in order to have constant energy. By standard arguments this implies the convergence of the scaled solution towards the stationary state in Sobolev and norms in the case of regular initial data as well as the convergence of the original solution to the corresponding self-similar cooling state. In the case of weak inelasticity, similar results have been established by Carlen, Carrillo and Carvalho (2009) in [11] via a precise control of the growth of the Fisher information.
@article{AIHPC_2010__27_2_719_0, author = {Furioli, G. and Pulvirenti, A. and Terraneo, E. and Toscani, G.}, title = {Convergence to self-similarity for the Boltzmann equation for strongly inelastic Maxwell molecules}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, volume = {27}, year = {2010}, pages = {719-737}, doi = {10.1016/j.anihpc.2009.11.005}, mrnumber = {2595198}, zbl = {05690778}, language = {en}, url = {http://dml.mathdoc.fr/item/AIHPC_2010__27_2_719_0} }
Furioli, G.; Pulvirenti, A.; Terraneo, E.; Toscani, G. Convergence to self-similarity for the Boltzmann equation for strongly inelastic Maxwell molecules. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) pp. 719-737. doi : 10.1016/j.anihpc.2009.11.005. http://gdmltest.u-ga.fr/item/AIHPC_2010__27_2_719_0/
[1] Self-similarity in random collision processes, Phys. Rev. E 68 (2003)
, , , ,[2] Decay rates in probability metrics towards homogeneous cooling states for the inelastic Maxwell model, J. Statist. Phys. 124 no. 2–4 (2006), 625-653 | MR 2264621 | Zbl 1135.82028
, , ,[3] Asymptotics of the fast diffusion equation via entropy estimates, Arch. Rational Mech. Anal. 191 (2009), 347-385 | MR 2481073 | Zbl 1178.35214
, , , , ,[4] The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, Mathematical Physics Reviews, vol. 7, Soviet Sci. Rev. Sect. C Math. Phys. Rev. vol. 7, Harwood Academic Publ., Chur (1988), 111-233 | MR 1128328 | Zbl 0850.76619
,[5] Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions, J. Statist. Phys. 110 no. 1–2 (2003), 333-375 | MR 1966332 | Zbl 1134.82324
, ,[6] On some properties of kinetic and hydrodynamic equations for inelastic interactions, J. Statist. Phys. 98 no. 3–4 (2000), 743-773 | MR 1749231 | Zbl 1056.76071
, , ,[7] Generalized kinetic Maxwell type models of granular gases, Mathematical Models of Granular Matter, Lecture Notes in Math. vol. 1937, Springer, Berlin (2008), 23-57 | MR 2436467 | Zbl 1298.76209
, , ,[8] Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials, J. Statist. Phys. 111 no. 1–2 (2003), 403-417 | MR 1964277 | Zbl 1119.82318
, , ,[9] Tanaka theorem for inelastic Maxwell models, Comm. Math. Phys. 276 no. 2 (2007), 287-314 | MR 2346391 | Zbl 1136.82033
, ,[10] Kinetic approach to long time behavior of linearized fast diffusion equations, J. Statist. Phys. 128 no. 4 (2007), 883-925 | MR 2344717 | Zbl 1131.82030
, ,[11] Strong convergence towards homogeneous cooling states for dissipative Maxwell models, Annales IHP Non Linear Analysis 26 no. 5 (2009), 1675-1700 | Numdam | MR 2566705 | Zbl 1175.82046
, , ,[12] Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas, Comm. Math. Phys. 199 no. 3 (1999), 521-546 | MR 1669689 | Zbl 0927.76088
, , ,[13] Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana 19 (2003), 1-48 | MR 2053570 | Zbl 1073.35127
, , ,[14] Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Rational Mech. Anal. 179 (2006), 217-263 | MR 2209130 | Zbl 1082.76105
, , ,[15] Asymptotic -decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J. 49 no. 1 (2000), 113-142 | MR 1777035 | Zbl 0963.35098
, ,[16] Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma (7) 6 (2007), 75-198 | MR 2355628 | Zbl 1142.82018
, ,[17] Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules, Trans. Amer. Math. Soc. 361 (2009), 1731-1747 | MR 2465814 | Zbl 1159.76044
, , ,[18] Strong convergence towards self-similarity for one-dimensional dissipative Maxwell models, J. Funct. Anal. 257 no. 7 (2009), 2291-2324 | MR 2548036 | Zbl 1180.82150
, , , ,[19] On the Boltzmann equation for diffusively excited granular media, Comm. Math. Phys. 246 no. 3 (2004), 503-541 | MR 2053942 | Zbl 1106.82031
, , ,[20] A strengthened central limit theorem for smooth densities, J. Funct. Anal. 129 no. 1 (1995), 148-167 | MR 1322646 | Zbl 0822.60018
, ,[21] Topics in Mass Transportation, Grad. Stud. Math. vol. 58 (2003)
,[22] Mathematics of granular materials, J. Statist. Phys. 124 no. 2–4 (2006), 781-822 | MR 2264625 | Zbl 1134.82040
,